January 01, 1970
This paper develops a multipole expansion method for the quasi-periodic elastic single layer potential \(\mathcal{S}_D^{\alpha,0}\) associated with the Kelvin tensor in one-dimensional periodic arrays. A key step in this approach is the derivation of translation addition theorems for the real vector spherical harmonics \(V_{lm}\), \(W_{lm}\), and \(X_{lm}\). These addition theorems enable the exact calculation of all matrix entries of \(\mathcal{S}_D^{\alpha,0}\) in closed form. By working entirely within the spherical harmonic basis, the proposed analytical method overcomes the poor convergence and mesh-dependent issues commonly caused by the direct surface discretization of weakly singular kernels. Additionally, the involved infinite sums are evaluated exactly using polylogarithm functions, which eliminates the need for series truncation. As an application, the integral equation \(\mathcal{S}_D^{\alpha,0}[f]=\varphi\) is reduced to a linear system. This framework is further extended to dimer geometries consisting of two disjoint balls in each cell, where the off-diagonal matrices are explicitly formulated via the Lerch transcendent.
Real vector spherical harmonics,Addition theorem,Quasi-periodic single layer potential,Multipole expansion,Polylogarithm
In the study of wave propagation in periodic resonator structures, layer potential techniques combined with Floquet-Bloch theory provide a powerful analytical framework. In the acoustic setting, a multipole expansion method based on the scalar spherical harmonic basis has been developed in [1] to analyze subwavelength band gaps and topological edge modes in one-dimensional resonator chains. A key ingredient of that approach is the addition theorem for scalar spherical waves (see [1]), which allows the quasi-periodic single layer potential to be represented explicitly as a matrix in the spherical harmonic basis, analytically computable lattice sums.
The present paper develops the elastic analogue of this framework. The governing operator is now the Lamé system, whose fundamental solution is the Kelvin tensor, and the natural basis on the boundary of a spherical resonator consists of the three families of real vector spherical harmonics \(V_{lm}\), \(W_{lm}\), \(X_{lm}\) rather than scalar spherical harmonics [2]. The central difficulty is that addition theorems for these vector harmonics under translation are not available in the literature, yet they are indispensable for computing the matrix representation of the quasi-periodic elastic single layer potential \(\mathcal{S}_D^{\alpha,0}\).
A further motivation for this approach is numerical. The kernel of the elastic single layer potential \(\mathcal{S}_D^{\alpha,0}\) is weakly singular, and direct discretization of the operator on the sphere via surface meshes leads to bad convergence and numerical results that are sensitive to the choice of mesh. By working entirely in the vector spherical harmonic basis and deriving analytical expressions for all matrix entries, our method is completely mesh-free and free of numerical discretization error. Moreover, the infinite lattice sums that appear in the entries of \(\mathbf{M}(\alpha)\) are evaluated in closed form via polylogarithm functions[3] and, in the dimer case, the Lerch transcendent[4], thereby avoiding the need for any truncation of infinite series.
This gap is filled by deriving addition theorems for all three families of real vector spherical harmonics, and applying them to compute in closed form all entries of the matrix \(\mathbf{M}(\alpha)\) of \(\mathcal{S}_D^{\alpha,0}\) in the vector spherical harmonic basis. The framework is further extended to the dimer geometry, with coupling matrices \(\mathbf{M}^{12}\) and \(\mathbf{M}^{21}\) expressed through the Lerch transcendent. These results provide the analytic foundation for a multipole expansion method for elastic subwavelength resonator chains [5], [6], directly paralleling and generalizing the scalar approach of [1].
In this section, the single layer potential for the elastic Kelvin Green tensor, real vector spherical harmonics and the addition theorems for solid harmonics are introduced. The underlying space is \(\mathbb{R}^3\) throughout this discussion. Denote by \(B_\rho(x_0)\) the ball centered at \(x_0\) with radius \(\rho\) and \(\partial B_\rho(x_0)\) its boundary. In particular, let \(\mathbb{S}^2\) denote the unit sphere in \(\mathbb{R}^3\).
Define the Lamé operator corresponding to the Lamé constants \(\lambda, \mu\) by \[\mathcal{L}^{\lambda, \mu}\boldsymbol{u} := -\mu\Delta \boldsymbol{u} -(\lambda+ \mu)\nabla\nabla\cdot
\boldsymbol{u}.\] This definition is consistent with [2], [7] and [8]–[10] up to a sign. It is worth mentioning that one should reverse the sign when using the results in this paper if the operator being considered is \(-\mathcal{L}^{\lambda, \mu}\). Meanwhile, the fundamental solution \(\boldsymbol{G}(x)\) is given by [7]–[10]
\[\label{eq-Kelvin-tensor}\boldsymbol{G}_{ij}(x):=\dfrac{1}{8\pi\vert x\vert }\cdot
\left(\dfrac{\lambda+3\mu}{\lambda+2\mu}\delta_{ij}+\dfrac{\lambda+\mu}{\lambda+2\mu}\dfrac{x_ix_j}{\vert x\vert^2}\right),\tag{1}\] called the Kelvin solution tensor, where \(\delta_{ij}\) is the Kronecker
symbol.
Using the Kelvin tensor 1 , the single layer potential on \(L^2(\partial B_\rho(x_0))^3\) is defined by \[\mathcal{S}[\psi](x):=\int_{\partial B_\rho (x_0)}
\boldsymbol{G}(x-y)\psi(y)d\sigma(y),\quad x\in\mathbb{R}^3.\] This operator is well-defined because of the weak singularity of \(\boldsymbol{G}\)[8], [10].
Define the quasi-periodic single layer potential by \[\label{eq-def-S95D}
\mathcal{S}_D^{\alpha,0}[\psi](x) := \int_{\partial D} \boldsymbol{G}^{\alpha,0}(x-y) \psi(y) \mathrm{d}\sigma(y), \quad x \in \partial D,\tag{2}\] where \(D=B_\rho(0)\) with \(\rho<1/2\) and \(\boldsymbol{G}^{\alpha,0}(x-y)=\sum\limits_{n\in \mathbb{Z}}\boldsymbol{G}(x-y-\mathbf{a})e^{in\alpha}\) with \(\mathbf{a}=(n,0,0)\).
The operator \(\mathcal{S}_D^{\alpha,0}[\psi](x)\) can be decomposed as \[\label{eq-operator-S}\mathcal{S}_D^{\alpha,0}[\psi](x)=\mathcal{S}_D[\psi](x)+\sum\limits_{n\neq 0}\mathcal{S}_{D+n}[\psi](x)e^{in\alpha},\tag{3}\] where the operator \((L^2(\partial
D))^3\xlongrightarrow{\mathcal{S}_{D+n}}(L^2(\partial D))^3\) is defined by \[\mathcal{S}_{D+n}[\psi](x)=\int_{\partial D}\boldsymbol{G}(x-y-\mathbf{a})\psi(y)d\sigma(y).\] By making the change of variable \(y'=y+\mathbf{a}\), one obtains: \[\mathcal{S}_{D+n}[\psi](x)=\int_{\partial D+\mathbf{a}}\boldsymbol{G}(x-y)\psi(y-\mathbf{a})d\sigma(y).\] It can be regarded as an operator from \((L^2(\partial D+\mathbf{a}))^3\) to \((L^2(\partial D))^3\).
The real spherical harmonics on \(\mathbb{S}^2\) are well known. To distinguish the real and complex spherical harmonics, their indices are placed in different positions. For instance, \(Y_l^m\) is a complex spherical harmonic and \(Y_{lm}\) is the real one. Furthermore, the relation between them is given by [11]: \[\label{eq-relation-complex-real-Y}Y_{lm}=\left\{ \begin{align} \dfrac{1}{\sqrt{2}}(Y_l^{-m}+(-1)^m Y_l^m)&, \quad m>0,\\ Y_l^0\qquad\qquad &,\quad m=0,\\ \dfrac{i}{\sqrt{2}}(Y_l^m-(-1)^mY_l^{-m})&,\quad m<0. \end{align}\right.\tag{4}\] The real vector spherical harmonics are defined by \[\begin{align} V_{lm}(\hat{\mathbf{r}})&:=\nabla_SY_{lm}(\hat{\mathbf{r}})-(l+1)Y_{lm}(\hat{\mathbf{r}})\hat{\mathbf{r}},\\ W_{lm}(\hat{\mathbf{r}})&:=\nabla_S Y_{lm}(\hat{\mathbf{r}})+lY_{lm}(\hat{\mathbf{r}})\hat{\mathbf{r}},\\ X_{lm}(\hat{\mathbf{r}})&:=\hat{\mathbf{r}}\times\nabla_SY_{lm}(\hat{\mathbf{r}}) \end{align}\] where \(\nabla_S=\hat{\theta}\dfrac{\partial}{\partial \theta}+\hat{\phi}\dfrac{1}{\sin \theta}\dfrac{\partial}{\partial \phi}\) is the surface gradient operator and \(r\), \(\theta\), and \(\phi\) are the spherical polar coordinates of the 3-dimensional vector \(\mathbf{r}\)[2]. The notations \(Y_{lm}^1,Y_{lm}^2,Y_{lm}^3\) are used to represent \(V_{lm}, W_{lm},X_{lm}\) respectively in this paper. Meanwhile, the complex vector spherical harmonics can be derived from the relation 4 : \[\begin{align} V_l^m(\hat{\mathbf{r}})&=\nabla_SY_l^m(\hat{\mathbf{r}})-(l+1)Y_l^m(\hat{\mathbf{r}})\hat{\mathbf{r}},\\ W_l^m(\hat{\mathbf{r}})&=\nabla_S Y_l^m(\hat{\mathbf{r}})+lY_l^m(\hat{\mathbf{r}})\hat{\mathbf{r}},\\ X_l^m(\hat{\mathbf{r}})&=\hat{\mathbf{r}}\times\nabla_SY_l^m(\hat{\mathbf{r}}). \end{align}\] Precisely, \[V_\lambda^\mu=\left\{ \begin{align} \dfrac{(-1)^\mu}{\sqrt{2}}(V_{\lambda\mu}+i V_{\lambda,-\mu})&, \quad \mu>0,\\ V_{\lambda 0}\qquad\qquad &,\quad \mu=0,\\ \dfrac{1}{\sqrt{2}}(V_{\lambda,-\mu}-iV_{\lambda\mu})&,\quad \mu<0, \end{align}\right.\] \[W_\lambda^\mu=\left\{ \begin{align} \dfrac{(-1)^\mu}{\sqrt{2}}(W_{\lambda\mu}+i W_{\lambda,-\mu})&, \quad \mu>0,\\ W_{\lambda 0}\qquad\qquad &,\quad \mu=0,\\ \dfrac{1}{\sqrt{2}}(W_{\lambda,-\mu}-iW_{\lambda\mu})&,\quad \mu<0, \end{align}\right.\] and \[X_\lambda^\mu=\left\{ \begin{align} \dfrac{(-1)^\mu}{\sqrt{2}}(X_{\lambda\mu}+i X_{\lambda,-\mu})&, \quad \mu>0,\\ X_{\lambda 0}\qquad\qquad &,\quad \mu=0,\\ \dfrac{1}{\sqrt{2}}(X_{\lambda,-\mu}-iX_{\lambda\mu})&,\quad \mu<0. \end{align}\right.\]
Define two complex harmonics: \[\label{solid-harmonics}R_l^m(\mathbf{r})=\sqrt{\dfrac{4\pi}{2l+1}}\vert \mathbf{r}\vert^lY_l^m(\hat{\mathbf{r}}),\quad I_l^m(\mathbf{r})=\sqrt{\dfrac{4\pi}{2l+1}}\dfrac{Y_l^m(\hat{\mathbf{r}})}{\vert \mathbf{r}\vert^{l+1}}.\tag{5}\] The translation addition theorem of the regular solid harmonic gives a finite expansion[12], [13]: \[R_{l}^{m}(\mathbf{r} + \mathbf{a}) = \sum_{\lambda=0}^{l} {2l \choose 2\lambda}^{1/2} \sum_{\mu=-\lambda}^{\lambda} R_{\lambda}^{\mu}(\mathbf{r}) R_{l-\lambda}^{m-\mu}(\mathbf{a}) \langle \lambda, \mu; l - \lambda, m - \mu | l m \rangle,\] where the Clebsch-Gordan coefficient is given by \[\langle \lambda, \mu; l - \lambda, m - \mu | l m \rangle = {l + m\choose \lambda + \mu}^{1/2} {l - m\choose\lambda - \mu}^{1/2} {2l \choose 2\lambda}^{-1/2}.\]
The similar expansion for irregular solid harmonics gives an infinite series[12], [13]: \[I_{l}^{m}(\mathbf{r} + \mathbf{a}) = \sum_{\lambda=0}^{\infty} {2l + 2\lambda + 1 \choose 2\lambda}^{1/2} \sum_{\mu=-\lambda}^{\lambda} R_{\lambda}^{\mu}(\mathbf{r}) I_{l+\lambda}^{m-\mu}(\mathbf{a}) \langle \lambda, \mu; l + \lambda, m - \mu | l m \rangle\] with \(|\mathbf{r}| \leq |\mathbf{a}|\). The quantity between pointed brackets is again a Clebsch-Gordan coefficient, \[\langle \lambda, \mu; l + \lambda, m - \mu | l m \rangle = (-1)^{\lambda+\mu} {l + \lambda - m + \mu\choose \lambda + \mu}^{1/2}{l + \lambda + m - \mu\choose\lambda - \mu} ^{1/2} {2l + 2\lambda + 1 \choose 2\lambda}^{-1/2}.\]
For the complex valued vector spherical harmonics \(V_l^m\), \(W_l^m\) and \(X_l^m\), their addition theorems are derived with the aid of the following lemmas:
Lemma 1. The solid harmonics \(R_l^m(\mathbf{r})\) and \(I_l^m(\mathbf{r})\) are defined by 5 as before. Then
\(\nabla I_l^m(\mathbf{r})=\sqrt{\dfrac{4\pi}{2l+1}}r^{-l-2}V_l^m(\hat{\mathbf{r}})\).
\(\nabla R_\lambda^\mu(\mathbf{r})=\sqrt{\dfrac{4\pi}{2\lambda+1}}r^{\lambda-1}W_\lambda^\mu(\hat{\mathbf{r}})\).
\(\mathbf{r}\times \nabla I_l^m(\mathbf{r})=\sqrt{\dfrac{4\pi}{2l+1}} r^{-l-1}X_l^m(\hat{\mathbf{r}})\)
\(\mathbf{r}\times \nabla R_\lambda^\mu(\mathbf{r})=\sqrt{\dfrac{4\pi}{2\lambda+1}}r^\lambda X_\lambda^\mu(\hat{\mathbf{r}})\).
\(\mathbf{a}\times \nabla R_\lambda^\mu(\mathbf{r})=\sqrt{\dfrac{4\pi}{2\lambda+1}}r^{\lambda-1}(\mathbf{a}\times W_\lambda^\mu(\hat{\mathbf{r}}))\)
Theorem 1. Let \(\mathbf{r}'=\mathbf{r}+\mathbf{a}\) with \(\vert \mathbf{a}\vert>r\). It can be derived smoothly that \[\label{eq-addcomplex-V}{r'}^{-l-2}V_l^m(\hat{\mathbf{r}'}) =\sum\limits_{\lambda=1}^\infty\sum\limits_{\mu=-\lambda}^\lambda (-1)^{\lambda+\mu} \sqrt{\dfrac{2l+1}{2\lambda+1}} r^{\lambda-1} I_{l+\lambda}^{m-\mu}(\mathbf{a})\sqrt{{l+\lambda+\mu-m\choose \lambda+\mu}{l+\lambda+m-\mu\choose \lambda-\mu}}W_\lambda^\mu(\hat{\mathbf{r}}),\qquad{(1)}\] \[{r'}^{l-1}W_l^m(\hat{\mathbf{r}'})=\sum\limits_{\lambda=0}^l\sum\limits_{\mu=-\lambda}^\lambda \sqrt{\dfrac{2l+1}{2\lambda+1}}r^{\lambda-1}R_{l-\lambda}^{m-\mu}(\mathbf{a})\sqrt{{l+m\choose \lambda+\mu}{l-m\choose \lambda-\mu}}W_\lambda^\mu(\hat{\mathbf{r}}),\] \[\label{eq-addcomplex-X}\begin{align}{r'}^{-l-1}X_l^m(\hat{\mathbf{r}'})=\sum\limits_{\lambda=1}^\infty\sum\limits_{\mu=-\lambda}^\lambda(-1)^{\lambda+\mu}\sqrt{\dfrac{2l+1}{2\lambda+1}} r^{\lambda-1}I_{l+\lambda}^{m-\mu}(\mathbf{a})& \sqrt{{l+\lambda+\mu-m\choose \lambda+\mu}{l+\lambda+m-\mu\choose \lambda-\mu}}\\ &\left[r X_\lambda^\mu(\hat{\mathbf{r}})+\mathbf{a}\times W_\lambda^\mu(\hat{\mathbf{r}})\right], \end{align}\qquad{(2)}\] where \(r=\vert \mathbf{r}\vert\) and \(r'=\vert \mathbf{r}+\mathbf{a}\vert\).
These results can be moved to real valued spherical harmonics case.
To obtain the addition theorem for \({r'}^{l-1}W_{lm}\), for instance, one can apply the addition theorem for the complex-valued harmonics \(W_l^m\) and, conversely, transform the terms \(W_\lambda^\mu\) back into the real-valued \(W_{\lambda\mu}\). Then the addition theorem is obtained, while keeping complex-valued terms \(R_{l-\lambda}^{m-\mu}\) and discarding \(W_0^0\) which vanishes.
Theorem 2. Denote the coefficient \(\sqrt{\dfrac{2l+1}{2\lambda+1}}\sqrt{{l+m\choose \lambda+\mu}{l-m\choose \lambda-\mu}}R_{l-\lambda}^{m-\mu}(\mathbf{a})\) by \(A_{l,\lambda}^{m,\mu}(\mathbf{a})\). Then, when \(m>0\), \[\begin{align} {r'}^{l-1}W_{lm}(\mathbf{r}') =&\sum\limits_{\lambda=1}^l\sum\limits_{\mu=-\lambda}^\lambda r^{\lambda-1}\dfrac{1}{\sqrt{2}}\left(A_{l,\lambda}^{-m,\mu}(\mathbf{a})+(-1)^mA_{l,\lambda}^{m,\mu}(\mathbf{a})\right)W_\lambda^\mu(\hat{\mathbf{r}})\\ =&\sum\limits_{\lambda=1}^l\sum\limits_{\mu=-\lambda}^{-1} r^{\lambda-1}\dfrac{1}{2}\left(A_{l,\lambda}^{-m,\mu}(\mathbf{a})+(-1)^mA_{l,\lambda}^{m,\mu}(\mathbf{a})\right)[W_{\lambda,-\mu}(\hat{\mathbf{r}})-iW_{\lambda,\mu}(\hat{\mathbf{r}})]\\ &+\sum\limits_{\lambda=1}^l\sum\limits_{\mu=1}^\lambda r^{\lambda-1}\dfrac{(-1)^{\mu}}{2}\left(A_{l,\lambda}^{-m,\mu}(\mathbf{a})+(-1)^mA_{l,\lambda}^{m,\mu}(\mathbf{a})\right)[W_{\lambda,\mu}(\hat{\mathbf{r}})+iW_{\lambda,-\mu}(\hat{\mathbf{r}})]\\ &+\sum\limits_{\lambda=1}^l r^{\lambda-1}\dfrac{1}{\sqrt{2}}\left(A_{l,\lambda}^{-m,0}(\mathbf{a})+(-1)^mA_{l,\lambda}^{m,0}(\mathbf{a})\right)W_{\lambda,0}(\hat{\mathbf{r}}). \end{align}\] Similarly, when \(m<0\), \[\begin{align} {r'}^{l-1}W_{lm}(\mathbf{r}') =&\sum\limits_{\lambda=1}^l\sum\limits_{\mu=-\lambda}^\lambda r^{\lambda-1}\dfrac{i}{\sqrt{2}}\left(A_{l,\lambda}^{m,\mu}(\mathbf{a})-(-1)^mA_{l,\lambda}^{-m,\mu}(\mathbf{a})\right)W_\lambda^\mu(\hat{\mathbf{r}})\\ =&\sum\limits_{\lambda=1}^l\sum\limits_{\mu=-\lambda}^{-1} r^{\lambda-1}\dfrac{i}{2}\left(A_{l,\lambda}^{m,\mu}(\mathbf{a})-(-1)^mA_{l,\lambda}^{-m,\mu}(\mathbf{a})\right)[W_{\lambda,-\mu}(\hat{\mathbf{r}})-iW_{\lambda,\mu}(\hat{\mathbf{r}})]\\ &+\sum\limits_{\lambda=1}^l\sum\limits_{\mu=1}^\lambda r^{\lambda-1}\dfrac{(-1)^{\mu}}{2}\cdot i\left(A_{l,\lambda}^{m,\mu}(\mathbf{a})-(-1)^mA_{l,\lambda}^{-m,\mu}(\mathbf{a})\right)[W_{\lambda,\mu}(\hat{\mathbf{r}})+iW_{\lambda,-\mu}(\hat{\mathbf{r}})]\\ &+\sum\limits_{\lambda=1}^l r^{\lambda-1}\dfrac{i}{\sqrt{2}}\left(A_{l,\lambda}^{m,0}(\mathbf{a})-(-1)^mA_{l,\lambda}^{-m,0}(\mathbf{a})\right)W_{\lambda,0}(\hat{\mathbf{r}}). \end{align}\] When \(m=0\), \[\begin{align} {r'}^{l-1}W_{l0}(\mathbf{r}')=&{r'}^{l-1}W_l^0(\mathbf{r}')=\sum\limits_{\lambda=1}^l\sum\limits_{\mu=-\lambda}^\lambda r^{\lambda-1}A_{l,\lambda}^{0,\mu}(\mathbf{a})W_\lambda^\mu(\hat{\mathbf{r}})\\ =&\sum\limits_{\lambda=1}^l\sum\limits_{\mu=-\lambda}^{-1} r^{\lambda-1}\dfrac{1}{\sqrt{2}}A_{l,\lambda}^{0,\mu}(\mathbf{a})[W_{\lambda,-\mu}(\hat{\mathbf{r}})-iW_{\lambda,\mu}(\hat{\mathbf{r}})]\\ &+\sum\limits_{\lambda=1}^l\sum\limits_{\mu=1}^{\lambda} r^{\lambda-1}\dfrac{(-1)^\mu}{\sqrt{2}}A_{l,\lambda}^{0,\mu}(\mathbf{a})[W_{\lambda,\mu}(\hat{\mathbf{r}})+iW_{\lambda,-\mu}(\hat{\mathbf{r}})]\\ &+\sum\limits_{\lambda=1}^lA_{l,\lambda}^{0,0}(\mathbf{a})W_{\lambda,0}(\hat{\mathbf{r}}). \end{align}\]
In this way, the addition theorems for real-valued vector spherical harmonics are obtained as follows.
Theorem 3. Denote the coefficient in the addition theorem for \({r'}^{-l-2}V_{lm}\) by \(H_{l,\lambda}^{m,\mu}\), i.e., \[\boxed{H_{l,\lambda}^{m,\mu}:=(-1)^{\lambda+\mu} \sqrt{\dfrac{2l+1}{2\lambda+1}} I_{l+\lambda}^{m-\mu}(\mathbf{a}) \sqrt{{l+\lambda+\mu-m\choose \lambda+\mu}{l+\lambda+m-\mu\choose \lambda-\mu}}.}\] When \(m>0\), \[\begin{align}{r'}^{-l-2}V_{lm}(\mathbf{r}') =&\sum\limits_{\lambda=1}^\infty\sum\limits_{\mu=-\lambda}^\lambda r^{\lambda-1} \dfrac{1}{\sqrt{2}}\left(H_{l,\lambda}^{-m,\mu}(\mathbf{a})+(-1)^mH_{l,\lambda}^{m,\mu}(\mathbf{a})\right)W_\lambda^\mu(\hat{\mathbf{r}})\\ =&\sum\limits_{\lambda=1}^\infty\sum\limits_{\mu=-\lambda}^{-1}r^{\lambda-1} \dfrac{1}{2}\left(H_{l,\lambda}^{-m,\mu}(\mathbf{a})+(-1)^mH_{l,\lambda}^{m,\mu}(\mathbf{a})\right)[W_{\lambda,-\mu}(\hat{\mathbf{r}})-iW_{\lambda,\mu}(\hat{\mathbf{r}})]\\ &+\sum\limits_{\lambda=1}^\infty\sum\limits_{\mu=1}^{\lambda}r^{\lambda-1} \dfrac{(-1)^\mu}{2}\left(H_{l,\lambda}^{-m,\mu}(\mathbf{a})+(-1)^mH_{l,\lambda}^{m,\mu}(\mathbf{a})\right)[W_{\lambda,\mu}(\hat{\mathbf{r}})+iW_{\lambda,-\mu}(\hat{\mathbf{r}})]\\ &+\sum\limits_{\lambda=1}^\infty r^{\lambda-1} \dfrac{1}{\sqrt{2}}\left(H_{l,\lambda}^{-m,0}(\mathbf{a})+(-1)^mH_{l,\lambda}^{m,0}(\mathbf{a})\right)W_{\lambda,0}(\hat{\mathbf{r}}) \end{align}\]
When \(m<0\), \[\begin{align}{r'}^{-l-2}V_{lm}(\mathbf{r}')& =\sum\limits_{\lambda=1}^\infty\sum\limits_{\mu=-\lambda}^\lambda r^{\lambda-1} \dfrac{i}{\sqrt{2}}\left(H_{l,\lambda}^{m,\mu}(\mathbf{a})-(-1)^mH_{l,\lambda}^{-m,\mu}(\mathbf{a})\right)W_\lambda^\mu(\hat{\mathbf{r}})\\ &=\sum\limits_{\lambda=1}^\infty\sum\limits_{\mu=-\lambda}^{-1} r^{\lambda-1}\dfrac{i}{2}\bigl(H_{l,\lambda}^{m,\mu}(\mathbf{a})-(-1)^mH_{l,\lambda}^{-m,\mu}(\mathbf{a})\bigr)[W_{\lambda,-\mu}(\hat{\mathbf{r}})-iW_{\lambda,\mu}(\hat{\mathbf{r}})]\\ &\quad+\sum\limits_{\lambda=1}^\infty\sum\limits_{\mu=1}^\lambda r^{\lambda-1}\dfrac{(-1)^{\mu}}{2}\cdot i\bigl(H_{l,\lambda}^{m,\mu}(\mathbf{a})-(-1)^mH_{l,\lambda}^{-m,\mu}(\mathbf{a})\bigr)[W_{\lambda,\mu}(\hat{\mathbf{r}})+iW_{\lambda,-\mu}(\hat{\mathbf{r}})]\\ &\quad+\sum\limits_{\lambda=1}^\infty r^{\lambda-1}\dfrac{i}{\sqrt{2}}\bigl(H_{l,\lambda}^{m,0}(\mathbf{a})-(-1)^mH_{l,\lambda}^{-m,0}(\mathbf{a})\bigr)W_{\lambda,0}(\hat{\mathbf{r}}). \end{align}\] When \(m=0\), \[\begin{align}{r'}^{-l-2}V_{lm}(\mathbf{r}')&=\sum\limits_{\lambda=1}^\infty\sum\limits_{\mu=-\lambda}^\lambda r^{\lambda-1} H_{l,\lambda}^{0,\mu}(\mathbf{a})W_\lambda^\mu(\hat{\mathbf{r}})\\ &=\sum\limits_{\lambda=1}^\infty\sum\limits_{\mu=-\lambda}^{-1} r^{\lambda-1}\dfrac{1}{\sqrt{2}}H_{l,\lambda}^{0,\mu}(\mathbf{a})[W_{\lambda,-\mu}(\hat{\mathbf{r}})-iW_{\lambda,\mu}(\hat{\mathbf{r}})]\\ &\quad+\sum\limits_{\lambda=1}^\infty\sum\limits_{\mu=1}^{\lambda} r^{\lambda-1}\dfrac{(-1)^\mu}{\sqrt{2}}H_{l,\lambda}^{0,\mu}(\mathbf{a})[W_{\lambda,\mu}(\hat{\mathbf{r}})+iW_{\lambda,-\mu}(\hat{\mathbf{r}})]\\ &\quad+\sum\limits_{\lambda=1}^\infty r^{\lambda-1}H_{l,\lambda}^{0,0}(\mathbf{a})\,W_{\lambda,0}(\hat{\mathbf{r}}). \end{align}\]
By Eq.@eq:eq-addcomplex-V , it follows that \[\begin{align} {r'}^{-l}V_l^m(\hat{\mathbf{r}'}) &={r'}^{2}\cdot{r'}^{-l-2}V_l^m(\hat{\mathbf{r}'})={r'}^{2}\cdot \sum\limits_{\lambda=1}^\infty\sum\limits_{\mu=-\lambda}^{\lambda}H_{l,\lambda}^{m,\mu}(\mathbf{a}) r^{\lambda-1}W_\lambda^\mu(\hat{\mathbf{r}})\\ &=\sum\limits_{\lambda=1}^\infty\sum\limits_{\mu=-\lambda}^{\lambda}H_{l,\lambda}^{m,\mu}(\mathbf{a})((r^{\lambda+1}+a^2r^{\lambda-1})W_\lambda^\mu(\hat{\mathbf{r}})+2r^{\lambda}\hat{\mathbf{r}}\cdot \mathbf{a}W_\lambda^\mu(\hat{\mathbf{r}})).\end{align}\] From [14], it follows that
Lemma 2. \(\hat{\mathbf{r}}\cdot \mathbf{a}=\sqrt{\dfrac{4\pi}{3}}\sum\limits_{q=-1}^{1}(-1)^q a_{-q}Y_1^q(\hat{\mathbf{r}}),\) where \(a_{-q}:=\mathbf{a}\cdot \hat{\mathbf{e}}_{-q}\), more precisely, \[\label{eq-a-q}\mathbf{a}\cdot \hat{\mathbf{e}}_{-q}=\left\{ \begin{align} \dfrac{1}{\sqrt{2}}(a_x-ia_y)&,\quad q=1,\\ a_z\quad&,\quad q=0\\ -\dfrac{1}{\sqrt{2}}(a_x+ia_y)&,\quad q=-1 \end{align}\right.\qquad{(3)}\]
Remark 1. When \(\mathbf{b} = (-n, 0, 0)\), \[b_{-q} = \begin{cases} \dfrac{n}{\sqrt{2}} & q = -1 \\ 0 & q = 0 \\ -\dfrac{n}{\sqrt{2}} & q = 1 \end{cases}\]
From [14], they imply that \[\label{eq-W-vecY}W_\lambda^\mu(\hat{\mathbf{r}})=\sqrt{\lambda(2\lambda+1)}\vec{Y}_{\lambda,\lambda-1,\mu}(\hat{\mathbf{r}}),\tag{6}\] \[\label{eq-vecY-expand}\vec{Y}_{\lambda,\lambda-1,\mu}(\hat{\mathbf{r}})=\sum\limits_{m_1=-1}^{1}\langle \lambda-1,\mu-m_1; 1, m_1\mid \lambda\mu\rangle Y_{\lambda-1}^{\mu-m_1}(\hat{\mathbf{r}})\chi_{1,m_1},\tag{7}\] \[Y_1^q(\hat{\mathbf{r}})Y_{\lambda-1}^{\mu-m_1}(\hat{\mathbf{r}})=\sum\limits_{\vert\lambda-2\vert \leqslant l'\leqslant\lambda,\atop l'\neq\lambda-1} \sqrt{\dfrac{3(2\lambda-1)}{4\pi(2l'+1)}}\langle 1,q;\lambda-1,\mu-m_1\mid l',q+\mu-m_1\rangle\langle 1,0;\lambda-1,0\mid l',0\rangle Y_{l'}^{q+\mu-m_1}(\hat{\mathbf{r}}).\] Therefore, it follows that \[\label{eq-raWlambdamu}\begin{align} (\hat{\mathbf{r}}\cdot\mathbf{a})W_\lambda^\mu(\hat{\mathbf{r}}) &=\sum\limits_{q=-1}^{1}\sum\limits_{m_1=-1}^{1}\sum\limits_{\vert\lambda-2\vert \leqslant l'\leqslant\lambda,\atop l'\neq\lambda-1}\sqrt{\dfrac{\lambda(2\lambda+1)(2\lambda-1)}{2l'+1}}(-1)^qa_{-q}\\ \langle\lambda-1,\mu-m_1;1,m_1\mid \lambda,\mu\rangle&\langle 1,q;\lambda-1,\mu-m_1\mid l',q+\mu-m_1\rangle\langle 1,0;\lambda-1,0\mid l',0\rangle Y_{l'}^{q+\mu-m_1}(\hat{\mathbf{r}})\chi_{1,m_1}. \end{align}\tag{8}\]
Remark 2. If \(\lambda=1\), then \(l'\) only takes value on \(1\).
Now the last term \(Y_{l'}^{q+\mu-m_1}(\hat{\mathbf{r}})\chi_{1,m_1}\) should be recoupled. From [14] and the orthonormal relation: \[\sum\limits_{j,m}\langle l,k;1,m_s\mid j,m\rangle\langle l,k';1,m_s'\mid j,m\rangle=\delta_{kk'}\delta_{m_sm_s'},\] it is obtained that \[\label{eq-Ykl39}Y_{l'}^k(\hat{\mathbf{r}})\chi_{1,m_1}= \sum\limits_{\vert l'-1\vert\leqslant j\leqslant \vert l'+1\vert}\langle l',k;1,m_1\mid j,k+m_1\rangle \vec{Y}_{j,l',k+m_1}(\hat{\mathbf{r}}).\tag{9}\]
Remark 3. If \(l'=0\), then \(j\) only takes value on \(1\). And if this is the case, \(\vec{Y}_{1,0,k+m_1}(\hat{\mathbf{r}})=\dfrac{1}{\sqrt{3}}W_1^{k+m_1}(\hat{\mathbf{r}})\) by [14].
By [14], \((\hat{\mathbf{r}}\cdot\mathbf{a})W_\lambda^\mu(\hat{\mathbf{r}})\) is the linear combination of vector harmonics since \[\vec{Y}_{l'-1,l',k+m_1}=\dfrac{1}{\sqrt{l'(2l'-1)}}V_{l'-1}^{k+m_1},\] \[\vec{Y}_{l',l',k+m_1}=\dfrac{-i}{\sqrt{l'(l'+1)}}X_{l'}^{k+m_1},\] \[\vec{Y}_{l'+1,l',k+m_1}=\dfrac{1}{\sqrt{(l'+1)(2l'+3)}}W_{l'+1}^{k+m_1}.\] Combining Eq 8 and Eq 9 where \(k=q+\mu-m_1\), it follows that
Theorem 4. \[\begin{align}{r'}^{-l}V_l^m(\hat{\mathbf{r}'}) &=\sum\limits_{\lambda=1}^\infty \sum\limits_{\mu=-\lambda}^{\lambda}H_{l,\lambda}^{m,\mu}(\mathbf{a})(r^{\lambda+1}+a^2r^{\lambda-1})W_\lambda^\mu(\hat{\mathbf{r}})\\ &+\sum\limits_{\lambda=1}^\infty \sum\limits_{\mu=-\lambda}^{\lambda}\sum\limits_{q=-1}^{1}\sum\limits_{m_1=-1}^{1}\sum\limits_{\vert\lambda-2\vert \leqslant l'\leqslant\lambda,\atop l'\neq\lambda-1}\sum\limits_{\vert l'-1\vert\leqslant j\leqslant \vert l'+1\vert} H_{l,\lambda}^{m,\mu}(\mathbf{a})2r^{\lambda}K_{l',j,\lambda}^{m_1,\mu,q}a_{-q}\vec{Y}_{j,l',q+\mu}(\hat{\mathbf{r}}) \end{align}\] where \(\boxed{K_{l',j,\lambda}^{m_1,\mu,q}}=\sqrt{\dfrac{\lambda(2\lambda+1)(2\lambda-1)}{2l'+1}}(-1)^q\langle\lambda-1,\mu-m_1;1,m_1\mid \lambda,\mu\rangle\langle 1,q;\lambda-1,\mu-m_1\mid l',q+\mu-m_1\rangle\langle 1,0;\lambda-1,0\mid l',0\rangle\langle l',q+\mu-m_1;1,m_1\mid j,q+\mu\rangle\) and \[\vec{Y}_{l'-1,l',q+\mu}=\dfrac{1}{\sqrt{l'(2l'-1)}}V_{l'-1}^{q+\mu},\] \[\vec{Y}_{l',l',q+\mu}=\dfrac{-i}{\sqrt{l'(l'+1)}}X_{l'}^{q+\mu},\] \[\vec{Y}_{l'+1,l',q+\mu}=\dfrac{1}{\sqrt{(l'+1)(2l'+3)}}W_{l'+1}^{q+\mu}.\]
Given the addition theorem of \({r'}^{-l}V_l^m(\hat{\mathbf{r}'})\), it is not difficult to derive the addition theorem of \({r'}^{-l}W_l^m(\hat{\mathbf{r}'})\). Note that \[\label{eq-W-V61Y}W_l^m(\hat{\mathbf{r}})-V_l^m(\hat{\mathbf{r}})=(2l+1)Y_l^m(\hat{\mathbf{r}})\hat{\mathbf{r}}.\tag{10}\] It follows that \[{r'}^{-l}W_l^m(\hat{\mathbf{r}'})={r'}^{-l}V_l^m(\hat{\mathbf{r}'})+{r'}^{-l}(2l+1)Y_l^m(\hat{\mathbf{r}'})\hat{\mathbf{r}'}.\] The term \({r'}^{-l}Y_l^m(\hat{\mathbf{r}'})\hat{\mathbf{r}'}\) can be decomposed as \[\label{eq-tornapart} \begin{align}{r'}^{-l}Y_l^m(\hat{\mathbf{r}'})\hat{\mathbf{r}'}&=\sqrt{\dfrac{2l+1}{4\pi}}\mathbf{r}I_l^m(\hat{\mathbf{r}'})+\sqrt{\dfrac{2l+1}{4\pi}}\mathbf{a}I_l^m(\hat{\mathbf{r}'}). \end{align}\tag{11}\] Using the addition theorem of \(I_l^m\): \[I_l^m(\mathbf{r}+\mathbf{a})=\sum\limits_{\lambda=0}^\infty\sum\limits_{\mu=-\lambda}^\lambda\sqrt{\dfrac{2\lambda+1}{2l+1}}H_{l,\lambda}^{m,\mu}(\mathbf{a})R_\lambda^\mu(\mathbf{r}),\] the first term becomes \[\sqrt{\dfrac{2l+1}{4\pi}}\mathbf{r}I_l^m(\hat{\mathbf{r}'})=\sum\limits_{\lambda=0}^\infty\sum\limits_{\mu=-\lambda}^{\lambda}\dfrac{r^{\lambda+1}}{2\lambda+1}H_{l,\lambda}^{m,\mu}(\mathbf{a})(W_\lambda^\mu(\hat{\mathbf{r}})-V_\lambda^\mu(\hat{\mathbf{r}}))\] by the Eq 10 and the definition of \(R_\lambda^\mu\).
Lemma 3. Under the definition of \(a_{-q}\) in Eq ?? , the constant vector \(\mathbf{a}\) can be expanded as \[\label{eq-a-expand}\mathbf{a}=\sum\limits_{q=-1}^1(-1)^qa_{-q}\chi_{1,q}.\qquad{(4)}\]
By the addition theorem of \(I_l^m\) and Eq 9 , the second term in Eq 11 can be expressed as \[\begin{align} \sqrt{\dfrac{2l+1}{4\pi}}\mathbf{a}I_l^m(\hat{\mathbf{r}'})&=\sum\limits_{\lambda=0}^\infty\sum\limits_{\mu=-\lambda}^\lambda r^{\lambda} H_{l,\lambda}^{m,\mu}(\mathbf{a}) Y_\lambda^\mu(\hat{\mathbf{r}})\mathbf{a}\\ &=\sum\limits_{\lambda=0}^\infty\sum\limits_{\mu=-\lambda}^\lambda\sum\limits_{q=-1}^1\sum\limits_{\vert\lambda-1\vert\leqslant j\leqslant \vert\lambda+1\vert}(-1)^qa_{-q} r^{\lambda} H_{l,\lambda}^{m,\mu}(\mathbf{a}) \langle \lambda,\mu;1,q\mid j,\mu+q\rangle\vec{Y}_{j,\lambda,q+\mu}(\hat{\mathbf{r}}). \end{align}\]
Theorem 5. \[\begin{align} {r'}^{-l}W_l^m(\hat{\mathbf{r}'}) =&\sum\limits_{\lambda=1}^\infty \sum\limits_{\mu=-\lambda}^{\lambda}H_{l,\lambda}^{m,\mu}(\mathbf{a})(r^{\lambda+1}+a^2r^{\lambda-1})W_\lambda^\mu(\hat{\mathbf{r}})\\ &+\sum\limits_{\lambda=1}^\infty \sum\limits_{\mu=-\lambda}^{\lambda}\sum\limits_{q=-1}^{1}\sum\limits_{m_1=-1}^{1}\sum\limits_{\vert\lambda-2\vert \leqslant l'\leqslant\lambda,\atop l'\neq\lambda-1}\sum\limits_{\vert l'-1\vert\leqslant j\leqslant \vert l'+1\vert} H_{l,\lambda}^{m,\mu}(\mathbf{a})2r^{\lambda}K_{l',j,\lambda}^{m_1,\mu,q}a_{-q}\vec{Y}_{j,l',q+\mu}(\hat{\mathbf{r}})\\ &+(2l+1)\sum\limits_{\lambda=0}^\infty\sum\limits_{\mu=-\lambda}^{\lambda}\dfrac{r^{\lambda+1}}{2\lambda+1}H_{l,\lambda}^{m,\mu}(\mathbf{a})(W_\lambda^\mu(\hat{\mathbf{r}})-V_\lambda^\mu(\hat{\mathbf{r}}))\\ &+(2l+1)\sum\limits_{\lambda=0}^\infty\sum\limits_{\mu=-\lambda}^\lambda\sum\limits_{q=-1}^1\sum\limits_{\vert \lambda-1\vert\leqslant j\leqslant \vert\lambda+1\vert}(-1)^{q}a_{-q} r^\lambda H_{l,\lambda}^{m,\mu}(\mathbf{a})\langle \lambda,\mu;1,q\mid j,\mu+q\rangle\vec{Y}_{j,\lambda,q+\mu}(\hat{\mathbf{r}}). \end{align}\]
Thus, the real forms of addition theorems are given by
Theorem 6. When \(m>0\), \[\begin{align} &{r'}^{-l}V_{lm}(\hat{\mathbf{r}'})=\sum_{\lambda=1}^\infty\sum_{\mu=-\lambda}^\lambda\dfrac{1}{\sqrt{2}}\bigl(H_{l,\lambda}^{-m,\mu}(\mathbf{a})+(-1)^m H_{l,\lambda}^{m,\mu}(\mathbf{a})\bigr)(r^{\lambda+1}+a^2 r^{\lambda-1})W_\lambda^\mu(\hat{\mathbf{r}})\\ &+\sum_{\lambda=1}^\infty\sum_{\mu=-\lambda}^\lambda\sum_{q=-1}^1\sum_{m_1=-1}^1\sum_{\substack{|\lambda-2|\leqslant k\leqslant\lambda\\k\neq\lambda-1}}\sum_{|k-1|\leqslant j\leqslant|k+1|} 2r^\lambda a_{-q}\dfrac{1}{\sqrt{2}}\bigl(H_{l,\lambda}^{-m,\mu}(\mathbf{a})+(-1)^m H_{l,\lambda}^{m,\mu}(\mathbf{a})\bigr) K_{k,j,\lambda}^{m_1,\mu,q}\vec{Y}_{j,k,q+\mu}(\hat{\mathbf{r}}) \end{align}\] When \(m<0\), \[\begin{align} &{r'}^{-l}V_{lm}(\hat{\mathbf{r}'})=\sum_{\lambda=1}^\infty\sum_{\mu=-\lambda}^\lambda\dfrac{i}{\sqrt{2}}\bigl(H_{l,\lambda}^{m,\mu}(\mathbf{a})-(-1)^m H_{l,\lambda}^{-m,\mu}(\mathbf{a})\bigr)(r^{\lambda+1}+a^2 r^{\lambda-1})W_\lambda^\mu(\hat{\mathbf{r}})\\ &+\sum_{\lambda=1}^\infty\sum_{\mu=-\lambda}^\lambda\sum_{q=-1}^1\sum_{m_1=-1}^1\sum_{\substack{|\lambda-2|\leqslant k\leqslant\lambda\\k\neq\lambda-1}}\sum_{|k-1|\leqslant j\leqslant|k+1|} \dfrac{i}{\sqrt{2}}\bigl(H_{l,\lambda}^{m,\mu}(\mathbf{a})-(-1)^m H_{l,\lambda}^{-m,\mu}(\mathbf{a})\bigr)2r^\lambda K_{k,j,\lambda}^{m_1,\mu,q}a_{-q}\vec{Y}_{j,k,q+\mu}(\hat{\mathbf{r}}). \end{align}\] When \(m=0\), \[\begin{align} &{r'}^{-l}V_{l0}(\hat{\mathbf{r}'})=\sum_{\lambda=1}^\infty\sum_{\mu=-\lambda}^\lambda H_{l,\lambda}^{0,\mu}(\mathbf{a})(r^{\lambda+1}+a^2 r^{\lambda-1})W_\lambda^\mu(\hat{\mathbf{r}})\\ &+\sum_{\lambda=1}^\infty\sum_{\mu=-\lambda}^\lambda\sum_{q=-1}^1\sum_{m_1=-1}^1\sum_{\substack{|\lambda-2|\leqslant k\leqslant\lambda\\k\neq\lambda-1}}\sum_{|k-1|\leqslant j\leqslant|k+1|} H_{l,\lambda}^{0,\mu}(\mathbf{a})\,2r^\lambda K_{k,j,\lambda}^{m_1,\mu,q}a_{-q}\vec{Y}_{j,k,q+\mu}(\hat{\mathbf{r}}). \end{align}\] where \[W_\lambda^\mu=\left\{ \begin{align} \dfrac{(-1)^\mu}{\sqrt{2}}(W_{\lambda\mu}+i W_{\lambda,-\mu})&, \quad \mu>0,\\ W_{\lambda 0}\qquad\qquad &,\quad \mu=0,\\ \dfrac{1}{\sqrt{2}}(W_{\lambda,-\mu}-iW_{\lambda\mu})&,\quad \mu<0. \end{align}\right.\] and \[\vec{Y}_{k-1,k,q+\mu}= \left\{ \begin{align} \dfrac{(-1)^{q+\mu}}{ \sqrt{2k(2k-1)} }\bigl(V_{k-1,q+\mu}+i\,V_{k-1,-(q+\mu)}\bigr)&,q+\mu>0\\ \dfrac{1}{\sqrt{k(2k-1)}}\,V_{k-1,0}\quad &,q+\mu=0\\ \dfrac{1}{ \sqrt{2k(2k-1)} }\bigl(V_{k-1,-(q+\mu)}-i\,V_{k-1,q+\mu}\bigr)&,q+\mu<0 \end{align} \right.\] \[\vec{Y}_{k,k,q+\mu}= \left\{\begin{align}\dfrac{-i(-1)^{q+\mu}}{ \sqrt{2k(k+1)} }\bigl(X_{k,q+\mu}+i\,X_{k,-(q+\mu)}\bigr)&,q+\mu>0\\ \dfrac{-i}{\sqrt{k(k+1)}}\,X_{k0}\quad &,q+\mu=0\\ \dfrac{-i}{ \sqrt{2k(k+1)} }\bigl(X_{k,-(q+\mu)}-i\,X_{k,q+\mu}\bigr)&,q+\mu<0 \end{align}\right.\] \[\vec{Y}_{k+1,k,q+\mu}= \left\{\begin{align}\dfrac{(-1)^{q+\mu}}{ \sqrt{2(k+1)(2k+3)} }\bigl(W_{k+1,q+\mu}+i\,W_{k+1,-(q+\mu)}\bigr)&,q+\mu>0\\ \dfrac{1}{\sqrt{(k+1)(2k+3)}}\,W_{k+1,0}\quad &,q+\mu=0\\ \dfrac{1}{ \sqrt{2(k+1)(2k+3)} }\bigl(W_{k+1,-(q+\mu)}-i\,W_{k+1,q+\mu}\bigr)&,q+\mu<0 \end{align}\right.\]
Theorem 7. When \(m>0\), \[\begin{align}&{r'}^{-l}W_{lm}(\hat{\mathbf{r}'}) =\sum_{\lambda=1}^\infty\sum_{\mu=-\lambda}^\lambda\dfrac{1}{\sqrt{2}}\bigl(H_{l,\lambda}^{-m,\mu}(\mathbf{a})+(-1)^m H_{l,\lambda}^{m,\mu}(\mathbf{a})\bigr)(r^{\lambda+1}+a^2 r^{\lambda-1})W_\lambda^\mu(\hat{\mathbf{r}})\\ +&\sum_{\lambda=1}^\infty\sum_{\mu=-\lambda}^\lambda\sum_{q=-1}^1\sum_{m_1=-1}^1\sum_{\substack{|\lambda-2|\leqslant k\leqslant\lambda\\k\neq\lambda-1}}\sum_{|k-1|\leqslant j\leqslant|k+1|} \dfrac{1}{\sqrt{2}}2r^\lambda a_{-q}\bigl(H_{l,\lambda}^{-m,\mu}(\mathbf{a})+(-1)^m H_{l,\lambda}^{m,\mu}(\mathbf{a})\bigr) K_{k,j,\lambda}^{m_1,\mu,q}\vec{Y}_{j,k,q+\mu}(\hat{\mathbf{r}})\\ +&(2l+1)\sum_{\lambda=0}^\infty\sum_{\mu=-\lambda}^\lambda\dfrac{r^{\lambda+1}}{2\lambda+1}\dfrac{1}{\sqrt{2}}\bigl(H_{l,\lambda}^{-m,\mu}(\mathbf{a})+(-1)^m H_{l,\lambda}^{m,\mu}(\mathbf{a})\bigr)(W_\lambda^\mu(\hat{\mathbf{r}})-V_\lambda^\mu(\hat{\mathbf{r}}))\\ +&(2l+1)\sum_{\lambda=0}^\infty\sum_{\mu=-\lambda}^\lambda\sum_{q=-1}^1\sum_{|\lambda-1|\leqslant j\leqslant|\lambda+1|} \dfrac{(-1)^qa_{-q}r^\lambda}{\sqrt{2}}\bigl(H_{l,\lambda}^{-m,\mu}(\mathbf{a})+(-1)^m H_{l,\lambda}^{m,\mu}(\mathbf{a})\bigr) \\ &\langle\lambda,\mu;1,q\mid j,\mu+q\rangle\vec{Y}_{j,\lambda,q+\mu}(\hat{\mathbf{r}}). \end{align}\] When \(m<0\), \[\begin{align} &{r'}^{-l}W_{lm}(\hat{\mathbf{r}'}) =\sum_{\lambda=1}^\infty\sum_{\mu=-\lambda}^\lambda\dfrac{i}{\sqrt{2}}\bigl(H_{l,\lambda}^{m,\mu}(\mathbf{a})-(-1)^m H_{l,\lambda}^{-m,\mu}(\mathbf{a})\bigr)(r^{\lambda+1}+a^2 r^{\lambda-1})W_\lambda^\mu(\hat{\mathbf{r}})\\ +&\sum_{\lambda=1}^\infty\sum_{\mu=-\lambda}^\lambda\sum_{q=-1}^1\sum_{m_1=-1}^1\sum_{\substack{|\lambda-2|\leqslant k\leqslant\lambda\\k\neq\lambda-1}}\sum_{|k-1|\leqslant j\leqslant|k+1|} \dfrac{i}{\sqrt{2}}\bigl(H_{l,\lambda}^{m,\mu}(\mathbf{a})-(-1)^m H_{l,\lambda}^{-m,\mu}(\mathbf{a})\bigr)2r^\lambda K_{k,j,\lambda}^{m_1,\mu,q}a_{-q}\vec{Y}_{j,k,q+\mu}(\hat{\mathbf{r}})\\ +&(2l+1)\sum_{\lambda=0}^\infty\sum_{\mu=-\lambda}^\lambda\dfrac{r^{\lambda+1}}{2\lambda+1}\dfrac{i}{\sqrt{2}}\bigl(H_{l,\lambda}^{m,\mu}(\mathbf{a})-(-1)^m H_{l,\lambda}^{-m,\mu}(\mathbf{a})\bigr)(W_\lambda^\mu(\hat{\mathbf{r}})-V_\lambda^\mu(\hat{\mathbf{r}}))\\ +&(2l+1)\sum_{\lambda=0}^\infty\sum_{\mu=-\lambda}^\lambda\sum_{q=-1}^1\sum_{|\lambda-1|\leqslant j\leqslant|\lambda+1|} \dfrac{i(-1)^q a_{-q} r^\lambda}{\sqrt{2}}\bigl(H_{l,\lambda}^{m,\mu}(\mathbf{a})-(-1)^m H_{l,\lambda}^{-m,\mu}(\mathbf{a})\bigr)\\&\langle\lambda,\mu;1,q\mid j,\mu+q\rangle\vec{Y}_{j,\lambda,q+\mu}(\hat{\mathbf{r}}). \end{align}\] When \(m=0\), \[\begin{align} {r'}^{-l}W_{l0}(\hat{\mathbf{r}'}) =&\sum_{\lambda=1}^\infty\sum_{\mu=-\lambda}^\lambda H_{l,\lambda}^{0,\mu}(\mathbf{a})(r^{\lambda+1}+a^2 r^{\lambda-1})W_\lambda^\mu(\hat{\mathbf{r}})\\ &+\sum_{\lambda=1}^\infty\sum_{\mu=-\lambda}^\lambda\sum_{q=-1}^1\sum_{m_1=-1}^1\sum_{\substack{|\lambda-2|\leqslant k\leqslant\lambda\\k\neq\lambda-1}}\sum_{|k-1|\leqslant j\leqslant|k+1|} H_{l,\lambda}^{0,\mu}(\mathbf{a})\,2r^\lambda K_{k,j,\lambda}^{m_1,\mu,q}a_{-q}\vec{Y}_{j,k,q+\mu}(\hat{\mathbf{r}})\\ &+(2l+1)\sum_{\lambda=0}^\infty\sum_{\mu=-\lambda}^\lambda\dfrac{r^{\lambda+1}}{2\lambda+1}H_{l,\lambda}^{0,\mu}(\mathbf{a})(W_\lambda^\mu(\hat{\mathbf{r}})-V_\lambda^\mu(\hat{\mathbf{r}}))\\ &+(2l+1)\sum_{\lambda=0}^\infty\sum_{\mu=-\lambda}^\lambda\sum_{q=-1}^1\sum_{|\lambda-1|\leqslant j\leqslant|\lambda+1|} H_{l,\lambda}^{0,\mu}(\mathbf{a})(-1)^q a_{-q} r^\lambda\langle\lambda,\mu;1,q\mid j,\mu+q\rangle\vec{Y}_{j,\lambda,q+\mu}(\hat{\mathbf{r}}). \end{align}\] where \[\vec{Y}_{\lambda-1,\lambda,q+\mu}= \left\{ \begin{align} \dfrac{(-1)^{q+\mu}}{ \sqrt{2\lambda(2\lambda-1)} }\bigl(V_{\lambda-1,q+\mu}+i\,V_{\lambda-1,-(q+\mu)}\bigr)&,q+\mu>0\\ \dfrac{1}{\sqrt{\lambda(2\lambda-1)}}\,V_{\lambda-1,0}\quad &,q+\mu=0\\ \dfrac{1}{ \sqrt{2\lambda(2\lambda-1)} }\bigl(V_{\lambda-1,-(q+\mu)}-i\,V_{\lambda-1,q+\mu}\bigr)&,q+\mu<0 \end{align} \right.\] \[\vec{Y}_{\lambda,\lambda,q+\mu}= \left\{\begin{align}\dfrac{-i(-1)^{q+\mu}}{ \sqrt{2\lambda(\lambda+1)} }\bigl(X_{\lambda,q+\mu}+i\,X_{\lambda,-(q+\mu)}\bigr)&,q+\mu>0\\ \dfrac{-i}{\sqrt{\lambda(\lambda+1)}}\,X_{\lambda0}\quad &,q+\mu=0\\ \dfrac{-i}{ \sqrt{2\lambda(\lambda+1)} }\bigl(X_{\lambda,-(q+\mu)}-i\,X_{\lambda,q+\mu}\bigr)&,q+\mu<0 \end{align}\right.\] \[\vec{Y}_{\lambda+1,\lambda,q+\mu}= \left\{\begin{align}\dfrac{(-1)^{q+\mu}}{ \sqrt{2(\lambda+1)(2\lambda+3)} }\bigl(W_{\lambda+1,q+\mu}+i\,W_{\lambda+1,-(q+\mu)}\bigr)&,q+\mu>0\\ \dfrac{1}{\sqrt{(\lambda+1)(2\lambda+3)}}\,W_{\lambda+1,0}\quad &,q+\mu=0\\ \dfrac{1}{ \sqrt{2(\lambda+1)(2\lambda+3)} }\bigl(W_{\lambda+1,-(q+\mu)}-i\,W_{\lambda+1,q+\mu}\bigr)&,q+\mu<0 \end{align}\right.\]
Note that the coefficient of the addition theorem for \(X_l^m\) that multiplies the bracket \([r\,X_\lambda^\mu+\mathbf{a}\times W_\lambda^\mu]\) is identical to the coefficient that appeared in the \(V_{lm}\) case. Therefore, the following addition theorem is obtained.
Theorem 8. When \(m>0\), \[\begin{align} &{r'}^{-l-1}X_{lm}(\mathbf{r}')\\ =&\sum\limits_{\lambda=1}^\infty\sum\limits_{\mu=-\lambda}^{-1} r^{\lambda-1}\dfrac{1}{2}\bigl(H_{l,\lambda}^{-m,\mu}(\mathbf{a})+(-1)^mH_{l,\lambda}^{m,\mu}(\mathbf{a})\bigr)\Bigl\{r[X_{\lambda,-\mu}(\hat{\mathbf{r}})-iX_{\lambda,\mu}(\hat{\mathbf{r}})]+\mathbf{a}\times[W_{\lambda,-\mu}(\hat{\mathbf{r}})-iW_{\lambda,\mu}(\hat{\mathbf{r}})]\Bigr\}\\ +&\sum\limits_{\lambda=1}^\infty\sum\limits_{\mu=1}^\lambda r^{\lambda-1}\dfrac{(-1)^{\mu}}{2}\bigl(H_{l,\lambda}^{-m,\mu}(\mathbf{a})+(-1)^mH_{l,\lambda}^{m,\mu}(\mathbf{a})\bigr)\Bigl\{r[X_{\lambda,\mu}(\hat{\mathbf{r}})+iX_{\lambda,-\mu}(\hat{\mathbf{r}})]+\mathbf{a}\times[W_{\lambda,\mu}(\hat{\mathbf{r}})+iW_{\lambda,-\mu}(\hat{\mathbf{r}})]\Bigr\}\\ &\quad+\sum\limits_{\lambda=1}^\infty r^{\lambda-1}\dfrac{1}{\sqrt{2}}\bigl(H_{l,\lambda}^{-m,0}(\mathbf{a})+(-1)^mH_{l,\lambda}^{m,0}(\mathbf{a})\bigr)\Bigl\{r\,X_{\lambda,0}(\hat{\mathbf{r}})+\mathbf{a}\times W_{\lambda,0}(\hat{\mathbf{r}})\Bigr\}. \end{align}\] When \(m<0\), \[\begin{align} &{r'}^{-l-1}X_{lm}(\mathbf{r}')\\ =&\sum\limits_{\lambda=1}^\infty\sum\limits_{\mu=-\lambda}^{-1} r^{\lambda-1}\dfrac{i}{2}\bigl(H_{l,\lambda}^{m,\mu}(\mathbf{a})-(-1)^mH_{l,\lambda}^{-m,\mu}(\mathbf{a})\bigr)\Bigl\{r[X_{\lambda,-\mu}(\hat{\mathbf{r}})-iX_{\lambda,\mu}(\hat{\mathbf{r}})]+\mathbf{a}\times[W_{\lambda,-\mu}(\hat{\mathbf{r}})-iW_{\lambda,\mu}(\hat{\mathbf{r}})]\Bigr\}\\ +&\sum\limits_{\lambda=1}^\infty\sum\limits_{\mu=1}^\lambda r^{\lambda-1}\dfrac{(-1)^{\mu}}{2}\cdot i\bigl(H_{l,\lambda}^{m,\mu}(\mathbf{a})-(-1)^mH_{l,\lambda}^{-m,\mu}(\mathbf{a})\bigr)\Bigl\{r[X_{\lambda,\mu}(\hat{\mathbf{r}})+iX_{\lambda,-\mu}(\hat{\mathbf{r}})]\\&+\mathbf{a}\times[W_{\lambda,\mu}(\hat{\mathbf{r}})+ iW_{\lambda,-\mu}(\hat{\mathbf{r}})]\Bigr\}\\ &\quad+\sum\limits_{\lambda=1}^\infty r^{\lambda-1}\dfrac{i}{\sqrt{2}}\bigl(H_{l,\lambda}^{m,0}(\mathbf{a})-(-1)^mH_{l,\lambda}^{-m,0}(\mathbf{a})\bigr)\Bigl\{r\,X_{\lambda,0}(\hat{\mathbf{r}})+\mathbf{a}\times W_{\lambda,0}(\hat{\mathbf{r}})\Bigr\}. \end{align}\] When \(m=0\), \[\begin{align} {r'}^{-l-1}X_{l0}(\mathbf{r}') &=\sum\limits_{\lambda=1}^\infty\sum\limits_{\mu=-\lambda}^{-1} r^{\lambda-1}\dfrac{1}{\sqrt{2}}H_{l,\lambda}^{0,\mu}(\mathbf{a})\Bigl\{r[X_{\lambda,-\mu}(\hat{\mathbf{r}})-iX_{\lambda,\mu}(\hat{\mathbf{r}})]+\mathbf{a}\times[W_{\lambda,-\mu}(\hat{\mathbf{r}})-iW_{\lambda,\mu}(\hat{\mathbf{r}})]\Bigr\}\\ &\quad+\sum\limits_{\lambda=1}^\infty\sum\limits_{\mu=1}^{\lambda} r^{\lambda-1}\dfrac{(-1)^\mu}{\sqrt{2}}H_{l,\lambda}^{0,\mu}(\mathbf{a})\Bigl\{r[X_{\lambda,\mu}(\hat{\mathbf{r}})+iX_{\lambda,-\mu}(\hat{\mathbf{r}})]+\mathbf{a}\times[W_{\lambda,\mu}(\hat{\mathbf{r}})+iW_{\lambda,-\mu}(\hat{\mathbf{r}})]\Bigr\}\\ &\quad+\sum\limits_{\lambda=1}^\infty H_{l,\lambda}^{0,0}(\mathbf{a})\Bigl\{r\,X_{\lambda,0}(\hat{\mathbf{r}})+\mathbf{a}\times W_{\lambda,0}(\hat{\mathbf{r}})\Bigr\}. \end{align}\]
Note that it is difficult to compute the inner product involving the cross product \(\mathbf{a}\times W_{\lambda}^\mu\) in Eq. ?? . Therefore, the cross product must be converted to the linear combination of vector spherical harmonics.
Lemma 4. For the spherical basis \[\chi_{1,\pm 1}=\dfrac{1}{\sqrt{2}}\begin{pmatrix}\mp 1\\-i\\0\end{pmatrix},\quad \chi_{1,0}=\begin{pmatrix}\mp 1\\0\\0\end{pmatrix},\] the formula for their cross product is given by \[\chi_{1, m} \times \chi_{1, n} = i \cdot \text{sgn}(m - n) \chi_{1, m+n}.\]
Using Eq 6 , Eq 7 , Eq ?? , and Eq 9 , it yields that \[\begin{align}\mathbf{a}\times W_\lambda^{\mu}(\hat{\mathbf{r}})&=\sum\limits_{q=-1}^1\sum\limits_{m_1=-1}^1\sum\limits_{\vert\lambda-2\vert\leqslant j\leqslant\vert \lambda\vert}i (-1)^q\text{sgn}(q-m_1)a_{-q}\sqrt{\lambda(2\lambda+1)}\\ &\langle\lambda-1,\mu-m_1;1,m_1\mid \lambda,\mu\rangle \langle \lambda-1,\mu-m_1;1,q+m_1\mid j,\mu+q \rangle \vec{Y}_{j,\lambda-1,q+\mu}(\hat{\mathbf{r}}) \end{align}\] Therefore, the addition theorem for \(X_l^m\) ?? reads
\[\label{eq-addthm-X} \begin{align}&{r'}^{-l-1}X_l^m(\hat{\mathbf{r}'}) =\sum\limits_{\lambda=1}^\infty\sum\limits_{\mu=-\lambda}^\lambda(-1)^{\lambda+\mu}\sqrt{\dfrac{2l+1}{2\lambda+1}} r^{\lambda}I_{l+\lambda}^{m-\mu}(\mathbf{a}) \sqrt{{l+\lambda+\mu-m\choose \lambda+\mu}{l+\lambda+m-\mu\choose \lambda-\mu}}X_\lambda^\mu(\hat{\mathbf{r}})\\ &+\sum\limits_{\lambda=1}^\infty\sum\limits_{\mu=-\lambda}^\lambda\sum\limits_{q=-1}^1\sum\limits_{m_1=-1}^1\sum\limits_{\vert\lambda-2\vert\leqslant j\leqslant\vert \lambda\vert} i (-1)^{\lambda+\mu+q}\text{sgn}(q-m_1)a_{-q}\sqrt{\lambda(2l+1)} r^{\lambda-1}I_{l+\lambda}^{m-\mu}(\mathbf{a})\\ &\sqrt{{l+\lambda+\mu-m\choose \lambda+\mu}{l+\lambda+m-\mu\choose \lambda-\mu}} \langle\lambda-1,\mu-m_1;1,m_1\mid \lambda,\mu\rangle\\& \langle \lambda-1,\mu-m_1;1,q+m_1\mid j,\mu+q \rangle \vec{Y}_{j,\lambda-1,q+\mu}(\hat{\mathbf{r}}) \end{align}\tag{12}\]
Denote the coefficient \[\begin{align}\boxed{L_{l,j,\lambda}^{m,\mu,q,m_1}(\mathbf{a})}= &i (-1)^{\lambda+\mu+q}\text{sgn}(q-m_1)a_{-q}\sqrt{\lambda(2l+1)} I_{l+\lambda}^{m-\mu}(\mathbf{a}) \sqrt{{l+\lambda+\mu-m\choose \lambda+\mu}{l+\lambda+m-\mu\choose \lambda-\mu}}\\ &\langle\lambda-1,\mu-m_1;1,m_1\mid \lambda,\mu\rangle \langle \lambda-1,\mu-m_1;1,q+m_1\mid j,\mu+q \rangle \end{align}\]
Then Eq 12 becomes
\[\begin{align}{r'}^{-l-1}X_l^m(\hat{\mathbf{r}'}) =&\sum\limits_{\lambda=1}^\infty\sum\limits_{\mu=-\lambda}^\lambda r^{\lambda} H_{l,\lambda}^{m,\mu}(\mathbf{a})X_\lambda^\mu(\hat{\mathbf{r}})\\ &+\sum\limits_{\lambda=1}^\infty\sum\limits_{\mu=-\lambda}^\lambda\sum\limits_{q=-1}^1\sum\limits_{m_1=-1}^1\sum\limits_{\vert\lambda-2\vert\leqslant j\leqslant\vert \lambda\vert} r^{\lambda-1}L_{l,j,\lambda}^{m,\mu,q,m_1}(\mathbf{a})\vec{Y}_{j,\lambda-1,q+\mu}(\hat{\mathbf{r}}) \end{align}\] Hence, the real valued addition theorem for \({r'}^{-l-1}X_l^m(\hat{\mathbf{r}'})\) is given by
Theorem 9. When \(m>0\),
\[\begin{align}{r'}^{-l-1}X_{lm}(\hat{\mathbf{r}'}) =&\sum\limits_{\lambda=1}^\infty\sum\limits_{\mu=-\lambda}^\lambda r^{\lambda} \dfrac{1}{\sqrt{2}}(H_{l,\lambda}^{-m,\mu}(\mathbf{a})+(-1)^m H_{l,\lambda}^{m,\mu}(\mathbf{a}))X_\lambda^\mu(\hat{\mathbf{r}})\\ &+\sum\limits_{\lambda=1}^\infty\sum\limits_{\mu=-\lambda}^\lambda\sum\limits_{q=-1}^1\sum\limits_{m_1=-1}^1\sum\limits_{\vert\lambda-2\vert\leqslant j\leqslant\vert \lambda\vert} r^{\lambda-1}\dfrac{1}{\sqrt{2}}(L_{l,j,\lambda}^{-m,\mu,q,m_1}(\mathbf{a})\\&+(-1)^mL_{l,j,\lambda}^{m,\mu,q,m_1}(\mathbf{a}))\vec{Y}_{j,\lambda-1,q+\mu}(\hat{\mathbf{r}}) \end{align}\] When \(m<0\),
\[\begin{align}{r'}^{-l-1}X_{lm}(\hat{\mathbf{r}'}) =&\sum\limits_{\lambda=1}^\infty\sum\limits_{\mu=-\lambda}^\lambda r^{\lambda} \dfrac{i}{\sqrt{2}}(H_{l,\lambda}^{m,\mu}(\mathbf{a})-(-1)^m H_{l,\lambda}^{-m,\mu}(\mathbf{a}))X_\lambda^\mu(\hat{\mathbf{r}})\\ &+\sum\limits_{\lambda=1}^\infty\sum\limits_{\mu=-\lambda}^\lambda\sum\limits_{q=-1}^1\sum\limits_{m_1=-1}^1\sum\limits_{\vert\lambda-2\vert\leqslant j\leqslant\vert \lambda\vert} r^{\lambda-1}\dfrac{i}{\sqrt{2}}(L_{l,j,\lambda}^{m,\mu,q,m_1}(\mathbf{a})\\ &-(-1)^mL_{l,j,\lambda}^{-m,\mu,q,m_1}(\mathbf{a}))\vec{Y}_{j,\lambda-1,q+\mu}(\hat{\mathbf{r}}) \end{align}\]
When \(m=0\),
\[\begin{align}{r'}^{-l-1}X_{l0}(\hat{\mathbf{r}'}) =&\sum\limits_{\lambda=1}^\infty\sum\limits_{\mu=-\lambda}^\lambda r^{\lambda} H_{l,\lambda}^{0,\mu}(\mathbf{a})X_\lambda^\mu(\hat{\mathbf{r}})\\ &+\sum\limits_{\lambda=1}^\infty\sum\limits_{\mu=-\lambda}^\lambda\sum\limits_{q=-1}^1\sum\limits_{m_1=-1}^1\sum\limits_{\vert\lambda-2\vert\leqslant j\leqslant\vert \lambda\vert} r^{\lambda-1}L_{l,j,\lambda}^{0,\mu,q,m_1}(\mathbf{a})\vec{Y}_{j,\lambda-1,q+\mu}(\hat{\mathbf{r}}) \end{align}\]
where \[\vec{Y}_{\lambda-2,\lambda-1,q+\mu}= \left\{ \begin{align} \dfrac{(-1)^{q+\mu}}{ \sqrt{2(\lambda-1)(2\lambda-3)} }\bigl(V_{\lambda-2,q+\mu}+i\,V_{\lambda-2,-(q+\mu)}\bigr)&,q+\mu>0\\ \dfrac{1}{\sqrt{(\lambda-1)(2\lambda-3)}}\,V_{\lambda-2,0}\quad &,q+\mu=0\\ \dfrac{1}{ \sqrt{2(\lambda-1)(2\lambda-3)} }\bigl(V_{\lambda-2,-(q+\mu)}-i\,V_{\lambda-2,q+\mu}\bigr)&,q+\mu<0 \end{align} \right.\] \[\vec{Y}_{\lambda-1,\lambda-1,q+\mu}= \left\{\begin{align}\dfrac{-i(-1)^{q+\mu}}{ \sqrt{2(\lambda-1)(\lambda)} }\bigl(X_{\lambda-1,q+\mu}+i\,X_{\lambda-1,-(q+\mu)}\bigr)&,q+\mu>0\\ \dfrac{-i}{\sqrt{(\lambda-1)(\lambda)}}\,X_{\lambda-1,0}\quad &,q+\mu=0\\ \dfrac{-i}{ \sqrt{2(\lambda-1)(\lambda)} }\bigl(X_{\lambda-1,-(q+\mu)}-i\,X_{\lambda-1,q+\mu}\bigr)&,q+\mu<0 \end{align}\right.\] \[\vec{Y}_{\lambda,\lambda-1,q+\mu}= \left\{\begin{align}\dfrac{(-1)^{q+\mu}}{ \sqrt{2(\lambda)(2\lambda+1)} }\bigl(W_{\lambda,q+\mu}+i\,W_{\lambda,-(q+\mu)}\bigr)&,q+\mu>0\\ \dfrac{1}{\sqrt{(\lambda)(2\lambda+1)}}\,W_{\lambda,0}\quad &,q+\mu=0\\ \dfrac{1}{ \sqrt{2(\lambda)(2\lambda+1)} }\bigl(W_{\lambda,-(q+\mu)}-i\,W_{\lambda,q+\mu}\bigr)&,q+\mu<0 \end{align}\right.\]
Define the inner product over \(\mathbb{S}^2\) by \[(f,g)=\int_{\mathbb{S}^2}f\cdot g.\]
By [2], the orthogonal properties are given by \[\begin{align} &\int_{\mathbb{S}^2} V_{l m} \cdot W_{l' m'} = 0, \qquad \int_{\mathbb{S}^2} W_{l m} \cdot X_{l' m'} = 0,\qquad \int_{\mathbb{S}^2} X_{l m} \cdot V_{l' m'} = 0, \\ &\int_{\mathbb{S}^2} V_{l m} \cdot V_{l' m'} = \delta_{ll'} \delta_{m m'} (l+1)(2l+1), \quad \int_{\mathbb{S}^2} W_{l m} \cdot W_{l' m'} = \delta_{ll'} \delta_{m m'} l(2l+1), \\ &\int_{\mathbb{S}^2} X_{l m} \cdot X_{l' m'} = \delta_{ll'} \delta_{m m'} l(l+1). \end{align}\]
The following two lemmas are proved in [2]:
Lemma 5. The family of vector spherical harmonics gives a complete basis of \(L^2(\partial B_\rho(x_0))^3\) and any real function \(f\in L^2(\partial B_\rho(x_0))^3\) can be represented as \[f(x)=\sum\limits_{l=0}^\infty\sum\limits_{m=-l}^m \sum\limits_{k=1}^3F_{lm}^kY_{lm}^k\left(\dfrac{x-x_0}{\rho}\right)\]
Lemma 6. Let \(\underline{Y_{lm}}=(V_{lm},W_{lm},X_{lm}).\) Then
when \(|x| > 1\), it holds that \[(\mathcal{S}\underline{Y_{l m}})(x)
=
\underline{Y_{l m}}(\hat{x})
A^{out}_{\mathcal{S},l}(x),\] where the matrix \(A^{out}_{\mathcal{S},l}(x)\) is given by \[A^{out}_{\mathcal{S},l}(x)
=
\begin{bmatrix}
\dfrac{(3l+1)\mu+l\lambda}
{(2l+3)(2l+1)\mu(2\mu+\lambda)}
|x|^{-l-2}
&
\dfrac{l(\mu+\lambda)}
{2(2l+1)\mu(2\mu+\lambda)}
\left(|x|^{-l-2}-|x|^{-l}\right)
&
0
\\[12pt]
0
&
\dfrac{(3l+2)\mu+(l+1)\lambda}
{(2l-1)(2l+1)\mu(2\mu+\lambda)}
|x|^{-l}
&
0
\\[12pt]
0
&
0
&
\dfrac{1}{(2l+1)\mu}
|x|^{-l-1}
\end{bmatrix}.\]
Therefore, applying the Lemma for \(\partial D+(n,0,0)=\partial B_\rho(n,0,0)\) with \(\rho<1/2\), \(n\in\mathbb{Z}\) and \(n\neq 0\): \[\label{eq-S-D43nY}\mathcal{S}_{D+n}[Y^k_{lm}]=\rho
\sum\limits_{j=1}^3Y^j_{lm}(\widehat{x-\mathbf{a}})\left[A_{\mathcal{S},l}^{out}\left(\dfrac{x-\mathbf{a}}{\rho}\right)\right]_j^{k},\tag{13}\] where the superscript \(k\) of \(A_{\mathcal{S},l}^{out}\left(\dfrac{x-\mathbf{a}}{\rho}\right)\) stands for the \(k\)th column of the matrix and the subscript \(j\) the \(j\)th row. Let \(a_{ij}\) stand for the coefficient of the entries of \(A^{out}_{\mathcal{S},l}(x)\).
It is clear that \[\boxed{(V_{l'm'},\mathcal{S}_{D+n}[V_{lm}])=0},\quad \boxed{(X_{l'm'},\mathcal{S}_{D+n}[V_{lm}])=0},\quad \boxed{(W_{00},\mathcal{S}_{D+n}[V_{lm}])=0}.\] Let \(\mathbf{b}=-\mathbf{a}\). Now compute \((W_{l'm'},\mathcal{S}_{D+n}[V_{lm}])\) using Eq 13 .
Theorem 10. The expression of \(\boxed{(W_{l'm'},\mathcal{S}_{D+n}[V_{lm}])}\)(\(l'\geqslant 1\)) is given by:
If \(m>0\) and \(m'>0\), \[\begin{align}(W_{l'm'},\mathcal{S}_{D+n}[V_{lm}])=
\rho^{l+3}a_{11}[r^{l'-1}\dfrac{1}{2}(H_{l,l'}^{-m,-m'}(\mathbf{b})+(-1)^mH_{l,l'}^{m,-m'}(\mathbf{b}))l'(2l'+1)\\
+r^{l'-1}\dfrac{(-1)^{m'}}{2}(H_{l,l'}^{-m,m'}(\mathbf{b})+(-1)^mH_{l,l'}^{m,m'}(\mathbf{b}))l'(2l'+1)]
\end{align}\] If \(m>0\) and \(m'<0\), \[\begin{align}(W_{l'm'},\mathcal{S}_{D+n}[V_{lm}])=
\rho^{l+3}a_{11}[r^{l'-1}\dfrac{1}{2}(H_{l,l'}^{-m,m'}(\mathbf{b})+(-1)^mH_{l,l'}^{m,m'}(\mathbf{b}))(-i \cdot l'(2l'+1))\\
+r^{l'-1}\dfrac{(-1)^{m'}}{2}(H_{l,l'}^{-m,-m'}(\mathbf{b})+(-1)^mH_{l,l'}^{m,-m'}(\mathbf{b}))(i\cdot l'(2l'+1))]
\end{align}\] If \(m>0\) and \(m'=0\), \[\begin{align}(W_{l'm'},\mathcal{S}_{D+n}[V_{lm}])=
\rho^{l+3}a_{11}r^{l'-1}\dfrac{1}{\sqrt{2}}(H_{l,l'}^{-m,0}(\mathbf{b})+(-1)^mH_{l,l'}^{m,0}(\mathbf{b}))l'(2l'+1).
\end{align}\]
If \(m<0\) and \(m'>0\), \[\begin{align}(W_{l'm'},\mathcal{S}_{D+n}[V_{lm}])= \rho^{l+3}a_{11}[r^{l'-1}\dfrac{i}{2}(H_{l,l'}^{m,-m'}(\mathbf{b})-(-1)^m H_{l,l'}^{-m,-m'}(\mathbf{b}))l'(2l'+1)\\ +r^{l'-1}\dfrac{(-1)^{m'}}{2}\cdot i (H_{l,l'}^{m,m'}(\mathbf{b})-(-1)^mH_{l,l'}^{-m,m'}(\mathbf{b}))l'(2l'+1)] \end{align}\]
If \(m<0\) and \(m'<0\), \[\begin{align}(W_{l'm'},\mathcal{S}_{D+n}[V_{lm}])= \rho^{l+3}a_{11}[r^{l'-1}\dfrac{i}{2}(H_{l,l'}^{m,m'}(\mathbf{b})-(-1)^m H_{l,l'}^{-m,m'}(\mathbf{b}))(-i\cdot l'(2l'+1))\\ +r^{l'-1}\dfrac{(-1)^{m'}}{2}\cdot i (H_{l,l'}^{m,-m'}(\mathbf{b})-(-1)^mH_{l,l'}^{-m,-m'}(\mathbf{b}))(i\cdot l'(2l'+1))] \end{align}\]
If \(m<0\) and \(m'=0\), \[\begin{align}(W_{l'm'},\mathcal{S}_{D+n}[V_{lm}])= \rho^{l+3}a_{11}r^{l'-1}\dfrac{i}{\sqrt{2}}(H_{l,l'}^{m,0}(\mathbf{b})-(-1)^mH_{l,l'}^{-m,0}(\mathbf{b}))l'(2l'+1).\end{align}\]
If \(m=0\) and \(m'>0\), \[\begin{align}(W_{l'm'},\mathcal{S}_{D+n}[V_{lm}])= \rho^{l+3}a_{11}r^{l'-1} [\dfrac{1}{\sqrt{2}} H_{l,l'}^{0,-m'}(\mathbf{b})l'(2l'+1)+\dfrac{(-1)^{m'}}{\sqrt{2}}H_{l,l'}^{0,m'}(\mathbf{b})l'(2l'+1)] \end{align}\]
If \(m=0\) and \(m'<0\), \[\begin{align}(W_{l'm'},\mathcal{S}_{D+n}[V_{lm}])= \rho^{l+3}a_{11}r^{l'-1} [\dfrac{1}{\sqrt{2}} H_{l,l'}^{0,m'}(\mathbf{b})(-i\cdot l'(2l'+1))+\dfrac{(-1)^{m'}}{\sqrt{2}}H_{l,l'}^{0,-m'}(\mathbf{b})(i\cdot l'(2l'+1))] \end{align}\]
If \(m=0\) and \(m'=0\), \[\begin{align}(W_{l'm'},\mathcal{S}_{D+n}[V_{lm}])= \rho^{l+3}a_{11}r^{l'-1} [H_{l,l'}^{0,0}(\mathbf{b})l'(2l'+1)] \end{align}\] where \(a_{ij}\) stands for the coefficient of the entries of \(A^{out}_{\mathcal{S},l}(x)\).
It is clear that \((W_{0,0},\mathcal{S}_{D+n}[X_{lm}])=0, (X_{00},\mathcal{S}_{D+n}[X_{lm}])=0\).
Theorem 11. Entries for \(\boxed{(V_{l'm'},\mathcal{S}_{D+n}[X_{lm}])}\):
When \(m>0\), \(m'>0\), \[\begin{align}&(V_{l'm'},\mathcal{S}_{D+n}[X_{lm}])\\
=&\rho^{l+2}a_{33}r^{l'+1}\dfrac{\sqrt{(l'+1)(2l'+1)}}{2}\sum\limits_{q=-1}^{1}\sum\limits_{m_1=-1}^{1}[(-1)^{m'}(L_{l,l',l'+2}^{-m,m'-q,q,m_1}(\mathbf{b})+(-1)^mL_{l,l',l'+2}^{m,m'-q,q,m_1}(\mathbf{b}))\\
&+(L_{l,l',l'+2}^{-m,-m'-q,q,m_1}(\mathbf{b})+(-1)^mL_{l,l',l'+2}^{m,-m'-q,q,m_1}(\mathbf{b}))]
\end{align}\]
When \(m>0\), \(m'<0\), \[\begin{align}&(V_{l'm'},\mathcal{S}_{D+n}[X_{lm}])\\ =&\rho^{l+2}a_{33}r^{l'+1}\dfrac{\sqrt{(l'+1)(2l'+1)}}{2}\sum\limits_{q=-1}^{1}\sum\limits_{m_1=-1}^{1} [i\cdot (-1)^{m'}(L_{l,l',l'+2}^{-m,-m'-q,q,m_1}(\mathbf{b})+(-1)^mL_{l,l',l'+2}^{m,-m'-q,q,m_1}(\mathbf{b}))\\ &-i\cdot (L_{l,l',l'+2}^{-m,m'-q,q,m_1}(\mathbf{b})+(-1)^mL_{l,l',l'+2}^{m,m'-q,q,m_1}(\mathbf{b}))] \end{align}\] When \(m>0\), \(m'=0\), \[\begin{align}&(V_{l'm'},\mathcal{S}_{D+n}[X_{lm}])\\ =&\rho^{l+2}a_{33}r^{l'+1}\dfrac{\sqrt{(l'+1)(2l'+1)}}{\sqrt{2}}\sum\limits_{q=-1}^{1}\sum\limits_{m_1=-1}^{1} [ (L_{l,l',l'+2}^{-m,-q,q,m_1}(\mathbf{b})+(-1)^mL_{l,l',l'+2}^{m,-q,q,m_1}(\mathbf{b}))] \end{align}\]
When \(m<0\), \(m'>0\), \[\begin{align}&(V_{l'm'},\mathcal{S}_{D+n}[X_{lm}])\\ =&\rho^{l+2}a_{33}r^{l'+1}i\cdot \dfrac{\sqrt{(l'+1)(2l'+1)}}{2}\sum\limits_{q=-1}^{1}\sum\limits_{m_1=-1}^{1}[(-1)^{m'}(L_{l, l',l'+2}^{\textcolor{magenta}{m},m'-q,q,m_1}(\mathbf{b})-(-1)^mL_{l, l',l'+2}^{-m,m'-q,q,m_1}(\mathbf{b}))\\ &+(L_{l, l',l'+2}^{m,-m'-q,q,m_1}(\mathbf{b})-(-1)^mL_{l,l',l'+2}^{-m,-m'-q,q,m_1}(\mathbf{b}))] \end{align}\]
When \(m<0\), \(m'<0\), \[\begin{align}&(V_{l'm'},\mathcal{S}_{D+n}[X_{lm}])\\ =&\rho^{l+2}a_{33}r^{l'+1} \dfrac{\sqrt{(l'+1)(2l'+1)}}{2}\sum\limits_{q=-1}^{1}\sum\limits_{m_1=-1}^{1}[(L_{l,l',l'+2}^{m,m'-q,q,m_1}(\mathbf{b})-(-1)^mL_{l,l',l'+2}^{-m,m'-q,q,m_1}(\mathbf{b}))\\ &- (-1)^{m'}(L_{l,l',l'+2}^{m,-m'-q,q,m_1}(\mathbf{b})-(-1)^mL_{l,l',l'+2}^{-m,-m'-q,q,m_1}(\mathbf{b}))] \end{align}\] When \(m<0\), \(m'=0\), \[\begin{align}&(V_{l'm'},\mathcal{S}_{D+n}[X_{lm}])\\ =&\rho^{l+2}a_{33}r^{l'+1}i\cdot \dfrac{\sqrt{(l'+1)(2l'+1)}}{\sqrt{2}}\sum\limits_{q=-1}^{1}\sum\limits_{m_1=-1}^{1} [ L_{l,l',l'+2}^{m,-q,q,m_1}(\mathbf{b})-(-1)^mL_{l,l',l'+2}^{-m,-q,q,m_1}(\mathbf{b})] \end{align}\]
When \(m=0\), \(m'>0\), \[\begin{align}&(V_{l'm'},\mathcal{S}_{D+n}[X_{lm}])\\ =&\rho^{l+2}a_{33}r^{l'+1} \dfrac{\sqrt{(l'+1)(2l'+1)}}{\sqrt{2}}\sum\limits_{q=-1}^{1}\sum\limits_{m_1=-1}^{1} [(-1)^{m'} L_{l,l',l'+2}^{0,m'-q,q,m_1}(\mathbf{b})+L_{l,l',l'+2}^{0,-m'-q,q,m_1}(\mathbf{b})] \end{align}\]
When \(m=0\), \(m'<0\), \[\begin{align}&(V_{l'm'},\mathcal{S}_{D+n}[X_{lm}])\\ =&\rho^{l+2}a_{33}r^{l'+1} \dfrac{\sqrt{(l'+1)(2l'+1)}}{\sqrt{2}}\sum\limits_{q=-1}^{1}\sum\limits_{m_1=-1}^{1} [i\cdot (-1)^{m'} L_{l,l',l'+2}^{0,-m'-q,q,m_1}(\mathbf{b})-i\cdot L_{l,l',l'+2}^{0,m'-q,q,m_1}(\mathbf{b})] \end{align}\]
When \(m=0\), \(m'=0\), \[\begin{align}&(V_{l'm'},\mathcal{S}_{D+n}[X_{lm}])\\ =&\rho^{l+2}a_{33}r^{l'+1} \sqrt{(l'+1)(2l'+1)}\sum\limits_{q=-1}^{1}\sum\limits_{m_1=-1}^{1} [ L_{l,l',l'+2}^{0,-q,q,m_1}(\mathbf{b})] \end{align}\]
Remark 4. Numerical results show that all terms above vanish.
Theorem 12. Entries for \(\boxed{(W_{l'm'},\mathcal{S}_{D+n}[X_{lm}])}\):
When \(m>0, m'>0\), \[\begin{align}
&(W_{l'm'},\mathcal{S}_{D+n}[X_{lm}])\\
=&\;\rho^{l+2}a_{33}\,r^{l'-1}\,\frac{\sqrt{l'(2l'+1)}}{2}\sum\limits_{q=-1}^{1}\sum\limits_{m_1=-1}^{1}\bigl[(-1)^{m'}\bigl(L_{l,l',l'}^{-m,\,m'-q,\,q,\,m_1}(\mathbf{b})+(-1)^mL_{l,l',l'}^{m,\,m'-q,\,q,\,m_1}(\mathbf{b})\bigr)\\
&\qquad\qquad\qquad\qquad+\bigl(L_{l,l',l'}^{-m,\,-m'-q,\,q,\,m_1}(\mathbf{b})+(-1)^mL_{l,l',l'}^{m,\,-m'-q,\,q,\,m_1}(\mathbf{b})\bigr)\bigr]
\end{align}\] When \(m>0, m'<0\), \[\begin{align}
&(W_{l'm'},\mathcal{S}_{D+n}[X_{lm}])\\
=&\;\rho^{l+2}a_{33}\,r^{l'-1}\,\frac{\sqrt{l'(2l'+1)}}{2}\sum\limits_{q=-1}^{1}\sum\limits_{m_1=-1}^{1}\bigl[i(-1)^{m'}\bigl(L_{l,l',l'}^{-m,\,-m'-q,\,q,\,m_1}(\mathbf{b})+(-1)^mL_{l,l',l'}^{m,\,-m'-q,\,q,\,m_1}(\mathbf{b})\bigr)\\
&\qquad\qquad\qquad\qquad-i\bigl(L_{l,l',l'}^{-m,\,m'-q,\,q,\,m_1}(\mathbf{b})+(-1)^mL_{l,l',l'}^{m,\,m'-q,\,q,\,m_1}(\mathbf{b})\bigr)\bigr]
\end{align}\]
When \(m>0, m'=0\), \[\begin{align} &(W_{l'0},\mathcal{S}_{D+n}[X_{lm}])\\ =&\;\rho^{l+2}a_{33}\,r^{l'-1}\,\frac{\sqrt{l'(2l'+1)}}{\sqrt{2}}\sum\limits_{q=-1}^{1}\sum\limits_{m_1=-1}^{1}\bigl(L_{l,l',l'}^{-m,\,-q,\,q,\,m_1}(\mathbf{b})+(-1)^mL_{l,l',l'}^{m,\,-q,\,q,\,m_1}(\mathbf{b})\bigr) \end{align}\]
When \(m<0, m'>0\), \[\begin{align} &(W_{l'm'},\mathcal{S}_{D+n}[X_{lm}])\\ =&\;\rho^{l+2}a_{33}\,r^{l'-1}\,i\cdot\frac{\sqrt{l'(2l'+1)}}{2}\sum\limits_{q=-1}^{1}\sum\limits_{m_1=-1}^{1}\bigl[(-1)^{m'}\bigl(L_{l,l',l'}^{m,\,m'-q,\,q,\,m_1}(\mathbf{b})-(-1)^mL_{l,l',l'}^{-m,\,m'-q,\,q,\,m_1}(\mathbf{b})\bigr)\\ &\qquad\qquad\qquad\qquad+\bigl(L_{l,l',l'}^{m,\,-m'-q,\,q,\,m_1}(\mathbf{b})-(-1)^mL_{l,l',l'}^{-m,\,-m'-q,\,q,\,m_1}(\mathbf{b})\bigr)\bigr] \end{align}\] When \(m<0, m'<0\), \[\begin{align} &(W_{l'm'},\mathcal{S}_{D+n}[X_{lm}])\\ =&\;\rho^{l+2}a_{33}\,r^{l'-1}\,\frac{\sqrt{l'(2l'+1)}}{2}\sum\limits_{q=-1}^{1}\sum\limits_{m_1=-1}^{1}\bigl[\bigl(L_{l,l',l'}^{m,\,m'-q,\,q,\,m_1}(\mathbf{b})-(-1)^mL_{l,l',l'}^{-m,\,m'-q,\,q,\,m_1}(\mathbf{b})\bigr)\\ &\qquad\qquad\qquad\qquad-(-1)^{m'}\bigl(L_{l,l',l'}^{m,\,-m'-q,\,q,\,m_1}(\mathbf{b})-(-1)^mL_{l,l',l'}^{-m,\,-m'-q,\,q,\,m_1}(\mathbf{b})\bigr)\bigr] \end{align}\] When \(m<0, m'=0\), \[\begin{align} &(W_{l'0},\mathcal{S}_{D+n}[X_{lm}])\\ =&\;\rho^{l+2}a_{33}\,r^{l'-1}\,\frac{i\sqrt{l'(2l'+1)}}{\sqrt{2}}\sum\limits_{q=-1}^{1}\sum\limits_{m_1=-1}^{1}\bigl(L_{l,l',l'}^{m,\,-q,\,q,\,m_1}(\mathbf{b})-(-1)^mL_{l,l',l'}^{-m,\,-q,\,q,\,m_1}(\mathbf{b})\bigr) \end{align}\] When \(m=0, m'>0\), \[\begin{align} &(W_{l'm'},\mathcal{S}_{D+n}[X_{l0}])\\ =&\;\rho^{l+2}a_{33}\,r^{l'-1}\,\frac{\sqrt{l'(2l'+1)}}{\sqrt{2}}\sum\limits_{q=-1}^{1}\sum\limits_{m_1=-1}^{1}\bigl[(-1)^{m'}L_{l,l',l'}^{0,\,m'-q,\,q,\,m_1}(\mathbf{b})+L_{l,l',l'}^{0,\,-m'-q,\,q,\,m_1}(\mathbf{b})\bigr] \end{align}\]
When \(m=0, m'<0\), \[\begin{align} &(W_{l'm'},\mathcal{S}_{D+n}[X_{l0}])\\ =&\;\rho^{l+2}a_{33}\,r^{l'-1}\,\frac{\sqrt{l'(2l'+1)}}{\sqrt{2}}\sum\limits_{q=-1}^{1}\sum\limits_{m_1=-1}^{1}\bigl[i(-1)^{m'}L_{l,l',l'}^{0,\,-m'-q,\,q,\,m_1}(\mathbf{b})-i\,L_{l,l',l'}^{0,\,m'-q,\,q,\,m_1}(\mathbf{b})\bigr] \end{align}\]
When \(m=0, m'=0\), \[\begin{align} &(W_{l'0},\mathcal{S}_{D+n}[X_{l0}])\\ =&\;\rho^{l+2}a_{33}\,r^{l'-1}\,\sqrt{l'(2l'+1)}\sum\limits_{q=-1}^{1}\sum\limits_{m_1=-1}^{1}L_{l,l',l'}^{0,\,-q,\,q,\,m_1}(\mathbf{b}) \end{align}\]
Theorem 13. Entries for \(\boxed{(X_{l'm'},\mathcal{S}_{D+n}[X_{lm}])}\):
When \(m>0,\;m'>0\): \[\begin{align}
&(X_{l'm'},\mathcal{S}_{D+n}[X_{lm}])\\
=&\;\rho^{l+2}a_{33}r^{l'}\Biggl\{
\frac{l'(l'+1)}{2}\Bigl[(-1)^{m'}\bigl(H_{l,l'}^{-m,m'}(\mathbf{b})+(-1)^m H_{l,l'}^{m,m'}(\mathbf{b})\bigr)
+\bigl(H_{l,l'}^{-m,-m'}(\mathbf{b})+(-1)^m H_{l,l'}^{m,-m'}(\mathbf{b})\bigr)\Bigr]\\
&-\frac{i\sqrt{l'(l'+1)}}{2}\sum_{q=-1}^{1}\sum_{m_1=-1}^{1}\Bigl[(-1)^{m'}\bigl(L_{l,l',l'+1}^{-m,m'-q,q,m_1}(\mathbf{b})+(-1)^m L_{l,l',l'+1}^{m,m'-q,q,m_1}(\mathbf{b})\bigr)\\
&\qquad\qquad\qquad\qquad\quad+\bigl(L_{l,l',l'+1}^{-m,-m'-q,q,m_1}(\mathbf{b})+(-1)^m L_{l,l',l'+1}^{m,-m'-q,q,m_1}(\mathbf{b})\bigr)\Bigr]\Biggr\}
\end{align}\]
When \(m>0,\;m'<0\): \[\begin{align} &(X_{l'm'},\mathcal{S}_{D+n}[X_{lm}])\\ =&\;\rho^{l+2}a_{33}r^{l'}\Biggl\{ \frac{l'(l'+1)}{2}\Bigl[i(-1)^{m'}\bigl(H_{l,l'}^{-m,-m'}(\mathbf{b})+(-1)^m H_{l,l'}^{m,-m'}(\mathbf{b})\bigr) -i\bigl(H_{l,l'}^{-m,m'}(\mathbf{b})+(-1)^m H_{l,l'}^{m,m'}(\mathbf{b})\bigr)\Bigr]\\ &+\frac{\sqrt{l'(l'+1)}}{2}\sum_{q=-1}^{1}\sum_{m_1=-1}^{1}\Bigl[(-1)^{m'}\bigl(L_{l,l',l'+1}^{-m,-m'-q,q,m_1}(\mathbf{b})+(-1)^m L_{l,l',l'+1}^{m,-m'-q,q,m_1}(\mathbf{b})\bigr)\\ &\qquad\qquad\qquad\qquad\quad-\bigl(L_{l,l',l'+1}^{-m,m'-q,q,m_1}(\mathbf{b})+(-1)^m L_{l,l',l'+1}^{m,m'-q,q,m_1}(\mathbf{b})\bigr)\Bigr]\Biggr\} \end{align}\]
When \(m>0,\;m'=0\): \[\begin{align} &(X_{l'0},\mathcal{S}_{D+n}[X_{lm}])\\ =&\;\rho^{l+2}a_{33}r^{l'}\Biggl\{ \frac{l'(l'+1)}{\sqrt{2}}\bigl(H_{l,l'}^{-m,0}(\mathbf{b})+(-1)^m H_{l,l'}^{m,0}(\mathbf{b})\bigr)\\ &-\frac{i\sqrt{l'(l'+1)}}{\sqrt{2}}\sum_{q=-1}^{1}\sum_{m_1=-1}^{1}\bigl(L_{l,l',l'+1}^{-m,-q,q,m_1}(\mathbf{b})+(-1)^m L_{l,l',l'+1}^{m,-q,q,m_1}(\mathbf{b})\bigr)\Biggr\} \end{align}\]
When \(m<0,\;m'>0\): \[\begin{align} &(X_{l'm'},\mathcal{S}_{D+n}[X_{lm}])\\ =&\;\rho^{l+2}a_{33}r^{l'}\Biggl\{ \frac{l'(l'+1)}{2}\Bigl[i(-1)^{m'}\bigl(H_{l,l'}^{m,m'}(\mathbf{b})-(-1)^m H_{l,l'}^{-m,m'}(\mathbf{b})\bigr) +i\bigl(H_{l,l'}^{m,-m'}(\mathbf{b})-(-1)^m H_{l,l'}^{-m,-m'}(\mathbf{b})\bigr)\Bigr]\\ &+\frac{\sqrt{l'(l'+1)}}{2}\sum_{q=-1}^{1}\sum_{m_1=-1}^{1}\Bigl[(-1)^{m'}\bigl(L_{l,l',l'+1}^{m,m'-q,q,m_1}(\mathbf{b})-(-1)^m L_{l,l',l'+1}^{-m,m'-q,q,m_1}(\mathbf{b})\bigr)\\ &\qquad\qquad\qquad\qquad\quad+\bigl(L_{l,l',l'+1}^{m,-m'-q,q,m_1}(\mathbf{b})-(-1)^m L_{l,l',l'+1}^{-m,-m'-q,q,m_1}(\mathbf{b})\bigr)\Bigr]\Biggr\} \end{align}\]
When \(m<0,\;m'<0\): \[\begin{align} &(X_{l'm'},\mathcal{S}_{D+n}[X_{lm}])\\ =&\;\rho^{l+2}a_{33}r^{l'}\Biggl\{ \frac{l'(l'+1)}{2}\Bigl[\bigl(H_{l,l'}^{m,m'}(\mathbf{b})-(-1)^m H_{l,l'}^{-m,m'}(\mathbf{b})\bigr) -(-1)^{m'}\bigl(H_{l,l'}^{m,-m'}(\mathbf{b})-(-1)^m H_{l,l'}^{-m,-m'}(\mathbf{b})\bigr)\Bigr]\\ &+\frac{i\sqrt{l'(l'+1)}}{2}\sum_{q=-1}^{1}\sum_{m_1=-1}^{1}\Bigl[(-1)^{m'}\bigl(L_{l,l',l'+1}^{m,-m'-q,q,m_1}(\mathbf{b})-(-1)^m L_{l,l',l'+1}^{-m,-m'-q,q,m_1}(\mathbf{b})\bigr)\\ &\qquad\qquad\qquad\qquad\quad-\bigl(L_{l,l',l'+1}^{m,m'-q,q,m_1}(\mathbf{b})-(-1)^m L_{l,l',l'+1}^{-m,m'-q,q,m_1}(\mathbf{b})\bigr)\Bigr]\Biggr\} \end{align}\]
When \(m<0,\;m'=0\): \[\begin{align} &(X_{l'0},\mathcal{S}_{D+n}[X_{lm}])\\ =&\;\rho^{l+2}a_{33}r^{l'}\Biggl\{ \frac{il'(l'+1)}{\sqrt{2}}\bigl(H_{l,l'}^{m,0}(\mathbf{b})-(-1)^m H_{l,l'}^{-m,0}(\mathbf{b})\bigr)\\ &+\frac{\sqrt{l'(l'+1)}}{\sqrt{2}}\sum_{q=-1}^{1}\sum_{m_1=-1}^{1}\bigl(L_{l,l',l'+1}^{m,-q,q,m_1}(\mathbf{b})-(-1)^m L_{l,l',l'+1}^{-m,-q,q,m_1}(\mathbf{b})\bigr)\Biggr\} \end{align}\]
When \(m=0,\;m'>0\): \[\begin{align} &(X_{l'm'},\mathcal{S}_{D+n}[X_{l0}])\\ =&\;\rho^{l+2}a_{33}r^{l'}\Biggl\{ \frac{l'(l'+1)}{\sqrt{2}}\Bigl[(-1)^{m'}H_{l,l'}^{0,m'}(\mathbf{b})+H_{l,l'}^{0,-m'}(\mathbf{b})\Bigr]\\ &-\frac{i\sqrt{l'(l'+1)}}{\sqrt{2}}\sum_{q=-1}^{1}\sum_{m_1=-1}^{1}\Bigl[(-1)^{m'}L_{l,l',l'+1}^{0,m'-q,q,m_1}(\mathbf{b})+L_{l,l',l'+1}^{0,-m'-q,q,m_1}(\mathbf{b})\Bigr]\Biggr\} \end{align}\]
When \(m=0,\;m'<0\): \[\begin{align} &(X_{l'm'},\mathcal{S}_{D+n}[X_{l0}])\\ =&\;\rho^{l+2}a_{33}r^{l'}\Biggl\{ \frac{l'(l'+1)}{\sqrt{2}}\Bigl[i(-1)^{m'}H_{l,l'}^{0,-m'}(\mathbf{b})-iH_{l,l'}^{0,m'}(\mathbf{b})\Bigr]\\ &+\frac{\sqrt{l'(l'+1)}}{\sqrt{2}}\sum_{q=-1}^{1}\sum_{m_1=-1}^{1}\Bigl[(-1)^{m'}L_{l,l',l'+1}^{0,-m'-q,q,m_1}(\mathbf{b})-L_{l,l',l'+1}^{0,m'-q,q,m_1}(\mathbf{b})\Bigr]\Biggr\} \end{align}\]
When \(m=0,\;m'=0\): \[\begin{align} &(X_{l'0},\mathcal{S}_{D+n}[X_{l0}])\\ =&\;\rho^{l+2}a_{33}r^{l'}\Biggl\{ l'(l'+1)H_{l,l'}^{0,0}(\mathbf{b}) -i\sqrt{l'(l'+1)}\sum_{q=-1}^{1}\sum_{m_1=-1}^{1}L_{l,l',l'+1}^{0,-q,q,m_1}(\mathbf{b})\Biggr\} \end{align}\]
Theorem 14. Entries for \(\boxed{(V_{l'm'},\mathcal{S}_{D+n}[W_{lm}])}\):
When \(m>0\), \(m'>0\), \[\begin{align}
&(V_{l'm'},\mathcal{S}_{D+n}[W_{lm}])\\
=&\sum\limits_{\lambda\in\{l'+1,l'+3\}}\sum\limits_{q=-1}^{1}\sum\limits_{m_1=-1}^{1}\rho^{l+1} (a_{22}-a_{12}) r^\lambda b_{-q}\sqrt{(l'+1)(2l'+1)}
[(-1)^{m'}(H_{l,\lambda}^{-m,m'-q}(\mathbf{b})\\
&+(-1)^mH_{l,\lambda}^{m,m'-q}(\mathbf{b}))K_{l'+1,l',\lambda}^{m_1,m'-q,q}
+(H_{l,\lambda}^{-m,-m'-q}(\mathbf{b})+(-1)^mH_{l,\lambda}^{m,-m'-q}(\mathbf{b}))K_{l'+1,l',\lambda}^{m_1,-m'-q,q}]\\
&-\rho^{l+1} a_{22}r^{l'+1}\dfrac{(2l+1)(l'+1)}{2}[(-1)^{m'}(H_{l,l'}^{-m,m'}(\mathbf{b})+(-1)^mH_{l,l'}^{m,m'}(\mathbf{b}))+(H_{l,l'}^{-m,-m'}(\mathbf{b})+(-1)^mH_{l,l'}^{m,-m'}(\mathbf{b}))]\\
&+\rho^{l+1} a_{22}r^{l'+1}(2l+1)\dfrac{\sqrt{(l'+1)(2l'+1)}}{2} \sum\limits_{q=-1}^1(-1)^q b_{-q}[(-1)^{m'}(H_{l,l'+1}^{-m,m'-q}(\mathbf{b})+(-1)^mH_{l,l'+1}^{m,m'-q}(\mathbf{b}))\\
&\langle l'+1,m'-q;1,q\mid l',m'\rangle+(H_{l,l'+1}^{-m,-m'-q}(\mathbf{b})+(-1)^mH_{l,l'+1}^{m,-m'-q}(\mathbf{b}))\langle l'+1,-m'-q;1,q\mid l',-m'\rangle]
\end{align}\]
When \(m>0\), \(m'<0\),
\[\begin{align} &(V_{l'm'},\mathcal{S}_{D+n}[W_{lm}])\\ =&\sum\limits_{\lambda\in\{l'+1,l'+3\}}\sum\limits_{q=-1}^{1}\sum\limits_{m_1=-1}^{1}\rho^{l+1} (a_{22}-a_{12}) r^\lambda b_{-q}\sqrt{(l'+1)(2l'+1)} [i\cdot (-1)^{m'}(H_{l,\lambda}^{-m,-m'-q}(\mathbf{b})\\ &+(-1)^mH_{l,\lambda}^{m,-m'-q}(\mathbf{b}))K_{l'+1,l',\lambda}^{m_1,-m'-q,q} -i\cdot (H_{l,\lambda}^{-m,m'-q}(\mathbf{b})+(-1)^mH_{l,\lambda}^{m,m'-q}(\mathbf{b}))K_{l'+1,l',\lambda}^{m_1,m'-q,q}]\\ &-\rho^{l+1} a_{22}r^{l'+1}\dfrac{(2l+1)(l'+1)}{2}[i\cdot (-1)^{m'}(H_{l,l'}^{-m,-m'}(\mathbf{b})+(-1)^mH_{l,l'}^{m,-m'}(\mathbf{b}))\\ &-i\cdot (H_{l,l'}^{-m,m'}(\mathbf{b})+(-1)^mH_{l,l'}^{m,m'}(\mathbf{b}))]\\ &+\rho^{l+1} a_{22}r^{l'+1}(2l+1)\dfrac{\sqrt{(l'+1)(2l'+1)}}{2} \sum\limits_{q=-1}^1(-1)^q b_{-q}[i\cdot (-1)^{m'}(H_{l,l'+1}^{-m,-m'-q}(\mathbf{b})+(-1)^mH_{l,l'+1}^{m,-m'-q}(\mathbf{b}))\\ &\langle l'+1,-m'-q;1,q\mid l',-m'\rangle -i\cdot (H_{l,l'+1}^{-m,m'-q}(\mathbf{b})+(-1)^mH_{l,l'+1}^{m,m'-q}(\mathbf{b}))\langle l'+1,m'-q;1,q\mid l',m'\rangle] \end{align}\]
When \(m>0\), \(m'=0\), \[\begin{align} &(V_{l'm'},\mathcal{S}_{D+n}[W_{lm}])\\ =&\sum\limits_{\lambda\in\{l'+1,l'+3\}}\sum\limits_{q=-1}^{1}\sum\limits_{m_1=-1}^{1}\rho^{l+1} (a_{22}-a_{12}) r^\lambda b_{-q}\sqrt{2(l'+1)(2l'+1)} [(H_{l,\lambda}^{-m,-q}(\mathbf{b})+(-1)^mH_{l,\lambda}^{m,-q}(\mathbf{b}))K_{l'+1,l',\lambda}^{m_1,-q,q}]\\ &-\rho^{l+1} a_{22}r^{l'+1}\dfrac{(2l+1)(l'+1)}{\sqrt{2}}[H_{l,l'}^{-m,0}(\mathbf{b})+(-1)^mH_{l,l'}^{m,0}(\mathbf{b})]\\ &+\rho^{l+1} a_{22}r^{l'+1}(2l+1)\dfrac{\sqrt{(l'+1)(2l'+1)}}{\sqrt{2}} \sum\limits_{q=-1}^1(-1)^q b_{-q}(H_{l,l'+1}^{-m,-q}(\mathbf{b})+(-1)^mH_{l,l'+1}^{m,-q}(\mathbf{b}))\\ &\langle l'+1,-q;1,q\mid l',0\rangle \end{align}\]
When \(m<0\), \(m'>0\), \[\begin{align} &(V_{l'm'},\mathcal{S}_{D+n}[W_{lm}])\\ =&\sum\limits_{\lambda\in\{l'+1,l'+3\}}\sum\limits_{q=-1}^{1}\sum\limits_{m_1=-1}^{1}\rho^{l+1} (a_{22}-a_{12}) r^\lambda b_{-q}i\cdot\sqrt{(l'+1)(2l'+1)} [(-1)^{m'}(H_{l,\lambda}^{m,m'-q}(\mathbf{b})\\ &-(-1)^mH_{l,\lambda}^{-m,m'-q}(\mathbf{b}))K_{l'+1,l',\lambda}^{m_1,m'-q,q} +(H_{l,\lambda}^{m,-m'-q}(\mathbf{b})-(-1)^mH_{l,\lambda}^{-m,-m'-q}(\mathbf{b}))K_{l'+1,l',\lambda}^{m_1,-m'-q,q}]\\ &-\rho^{l+1} a_{22}r^{l'+1}i\cdot \dfrac{(2l+1)(l'+1)}{2}[(-1)^{m'}(H_{l,l'}^{m,m'}(\mathbf{b})-(-1)^mH_{l,l'}^{-m,m'}(\mathbf{b}))\\ &+(H_{l,l'}^{m,-m'}(\mathbf{b})-(-1)^mH_{l,l'}^{-m,-m'}(\mathbf{b}))]\\ &+\rho^{l+1} a_{22}r^{l'+1}i\cdot (2l+1)\dfrac{\sqrt{(l'+1)(2l'+1)}}{2} \sum\limits_{q=-1}^1(-1)^q b_{-q}[(-1)^{m'}(H_{l,l'+1}^{m,m'-q}(\mathbf{b})-(-1)^mH_{l,l'+1}^{-m,m'-q}(\mathbf{b}))\\ &\langle l'+1,m'-q;1,q\mid l',m'\rangle+(H_{l,l'+1}^{m,-m'-q}(\mathbf{b})-(-1)^mH_{l,l'+1}^{-m,-m'-q}(\mathbf{b}))\langle l'+1,-m'-q;1,q\mid l',-m'\rangle] \end{align}\]
When \(m<0\), \(m'<0\),
\[\begin{align} &(V_{l'm'},\mathcal{S}_{D+n}[W_{lm}])\\ =&\sum\limits_{\lambda\in\{l'+1,l'+3\}}\sum\limits_{q=-1}^{1}\sum\limits_{m_1=-1}^{1}\rho^{l+1} (a_{22}-a_{12}) r^\lambda b_{-q}\sqrt{(l'+1)(2l'+1)} [-(-1)^{m'}(H_{l,\lambda}^{m,-m'-q}(\mathbf{b})\\ &-(-1)^mH_{l,\lambda}^{-m,-m'-q}(\mathbf{b}))K_{l'+1,l',\lambda}^{m_1,-m'-q,q} + (H_{l,\lambda}^{m,m'-q}(\mathbf{b})-(-1)^mH_{l,\lambda}^{-m,m'-q}(\mathbf{b}))K_{l'+1,l',\lambda}^{m_1,m'-q,q}]\\ &-\rho^{l+1} a_{22}r^{l'+1}\dfrac{(2l+1)(l'+1)}{2} [- (-1)^{m'}(H_{l,l'}^{m,-m'}(\mathbf{b})-(-1)^mH_{l,l'}^{-m,-m'}(\mathbf{b}))\\ &+ (H_{l,l'}^{m,m'}(\mathbf{b})-(-1)^mH_{l,l'}^{-m,m'}(\mathbf{b}))]\\ &+\rho^{l+1} a_{22}r^{l'+1}(2l+1)\dfrac{\sqrt{(l'+1)(2l'+1)}}{2} \sum\limits_{q=-1}^1(-1)^q b_{-q}[-(-1)^{m'}(H_{l,l'+1}^{m,-m'-q}(\mathbf{b})-(-1)^mH_{l,l'+1}^{-m,-m'-q}(\mathbf{b}))\\ &\langle l'+1,-m'-q;1,q\mid l',-m'\rangle + (H_{l,l'+1}^{m,m'-q}(\mathbf{b})-(-1)^mH_{l,l'+1}^{-m,m'-q}(\mathbf{b}))\langle l'+1,m'-q;1,q\mid l',m'\rangle] \end{align}\]
When \(m<0\), \(m'=0\), \[\begin{align} &(V_{l'm'},\mathcal{S}_{D+n}[W_{lm}])\\ =&\sum\limits_{\lambda\in\{l'+1,l'+3\}}\sum\limits_{q=-1}^{1}\sum\limits_{m_1=-1}^{1}\rho^{l+1} (a_{22}-a_{12}) r^\lambda b_{-q}i\cdot \sqrt{2(l'+1)(2l'+1)} [(H_{l,\lambda}^{m,-q}(\mathbf{b})\\ &-(-1)^mH_{l,\lambda}^{-m,-q}(\mathbf{b}))K_{l'+1,l',\lambda}^{m_1,-q,q}]\\ &-\rho^{l+1} a_{22}r^{l'+1}i\cdot \dfrac{(2l+1)(l'+1)}{\sqrt{2}}[(H_{l,l'}^{m,0}(\mathbf{b})-(-1)^mH_{l,l'}^{-m,0}(\mathbf{b})]\\ &+\rho^{l+1} a_{22}r^{l'+1}i\cdot (2l+1)\dfrac{\sqrt{(l'+1)(2l'+1)}}{\sqrt{2}} \sum\limits_{q=-1}^1(-1)^q b_{-q}[ (H_{l,l'+1}^{m,-q}(\mathbf{b})-(-1)^mH_{l,l'+1}^{-m,-q}(\mathbf{b}))\\ &\langle l'+1,-q;1,q\mid l',0\rangle] \end{align}\]
When \(m=0\), \(m'>0\), \[\begin{align} &(V_{l'm'},\mathcal{S}_{D+n}[W_{lm}])\\ =&\sum\limits_{\lambda\in\{l'+1,l'+3\}}\sum\limits_{q=-1}^{1}\sum\limits_{m_1=-1}^{1}\rho^{l+1} (a_{22}-a_{12}) r^\lambda b_{-q}\sqrt{2(l'+1)(2l'+1)} [(-1)^{m'}H_{l,\lambda}^{0,m'-q}(\mathbf{b}) K_{l'+1,l',\lambda}^{m_1,m'-q,q}\\ &+H_{l,\lambda}^{0,-m'-q}(\mathbf{b})K_{l'+1,l',\lambda}^{m_1,-m'-q,q}]\\ &-\rho^{l+1} a_{22}r^{l'+1}\dfrac{(2l+1)(l'+1)}{\sqrt{2}}[(-1)^{m'}H_{l,l'}^{0,m'}(\mathbf{b})+H_{l,l'}^{0,-m'}(\mathbf{b})]\\ &+\rho^{l+1} a_{22}r^{l'+1}(2l+1)\dfrac{\sqrt{(l'+1)(2l'+1)}}{\sqrt{2}} \sum\limits_{q=-1}^1(-1)^q b_{-q}[(-1)^{m'}H_{l,l'+1}^{0,m'-q}(\mathbf{b})\\ &\langle l'+1,m'-q;1,q\mid l',m'\rangle+(H_{l,l'+1}^{0,-m'-q}(\mathbf{b})\langle l'+1,-m'-q;1,q\mid l',-m'\rangle] \end{align}\]
When \(m=0\), \(m'<0\),
\[\begin{align} &(V_{l'm'},\mathcal{S}_{D+n}[W_{lm}])\\ =&\sum\limits_{\lambda\in\{l'+1,l'+3\}}\sum\limits_{q=-1}^{1}\sum\limits_{m_1=-1}^{1}\rho^{l+1} (a_{22}-a_{12}) r^\lambda b_{-q}\sqrt{2(l'+1)(2l'+1)} [i\cdot (-1)^{m'}H_{l,\lambda}^{0,-m'-q}(\mathbf{b})K_{l'+1,l',\lambda}^{m_1,-m'-q,q}\\ & -i\cdot H_{l,\lambda}^{0,m'-q}(\mathbf{b})K_{l'+1,l',\lambda}^{m_1,m'-q,q}]\\ &-\rho^{l+1} a_{22}r^{l'+1}\dfrac{(2l+1)(l'+1)}{\sqrt{2}}[i\cdot (-1)^{m'}H_{l,l'}^{0,-m'}(\mathbf{b})-i\cdot H_{l,l'}^{0,m'}(\mathbf{b})]\\ &+\rho^{l+1} a_{22}r^{l'+1}(2l+1)\dfrac{\sqrt{(l'+1)(2l'+1)}}{\sqrt{2}} \sum\limits_{q=-1}^1(-1)^q b_{-q}[i\cdot (-1)^{m'} H_{l,l'+1}^{0,-m'-q}(\mathbf{b})\\ &\langle l'+1,-m'-q;1,q\mid l',-m'\rangle-i\cdot H_{l,l'+1}^{0,m'-q}(\mathbf{b})\langle l'+1,m'-q;1,q\mid l',m'\rangle] \end{align}\]
When \(m=0\), \(m'=0\), \[\begin{align} &(V_{l'm'},\mathcal{S}_{D+n}[W_{lm}])\\ =&\sum\limits_{\lambda\in\{l'+1,l'+3\}}\sum\limits_{q=-1}^{1}\sum\limits_{m_1=-1}^{1}\rho^{l+1} (a_{22}-a_{12}) r^\lambda b_{-q}\sqrt{4(l'+1)(2l'+1)} [H_{l,\lambda}^{0,-q}(\mathbf{b})K_{l'+1,l',\lambda}^{m_1,-q,q}]\\ &-\rho^{l+1} a_{22}r^{l'+1}(2l+1)(l'+1)[H_{l,l'}^{0,0}(\mathbf{b})]\\ &+\rho^{l+1} a_{22}r^{l'+1}(2l+1)\sqrt{(l'+1)(2l'+1)} \sum\limits_{q=-1}^1(-1)^q b_{-q}[ H_{l,l'+1}^{0,-q}(\mathbf{b})\langle l'+1,-q;1,q\mid l',0\rangle] \end{align}\]
Theorem 15. Entries for \(\boxed{(X_{l'm'},\mathcal{S}_{D+n}[W_{lm}])}\):
When \(m>0\), \(m'>0\), \[\begin{align}
&(X_{l'm'},\mathcal{S}_{D+n}[W_{lm}])\\
=&\sum\limits_{\lambda\in\{l',l'+2\mid l'\geqslant 1\}}\sum\limits_{q=-1}^{1}\sum\limits_{m_1=-1}^{1}-i\cdot \rho^{l+1} (a_{22}-a_{12}) r^\lambda b_{-q}\sqrt{l'(l'+1)}
[(-1)^{m'}(H_{l,\lambda}^{-m,m'-q}(\mathbf{b})\\
&+(-1)^mH_{l,\lambda}^{m,m'-q}(\mathbf{b}))K_{l',l',\lambda}^{m_1,m'-q,q}
+(H_{l,\lambda}^{-m,-m'-q}(\mathbf{b})+(-1)^mH_{l,\lambda}^{m,-m'-q}(\mathbf{b}))K_{l',l',\lambda}^{m_1,-m'-q,q}]\\
&- i\cdot \rho^{l+1} a_{22}r^{l'}(2l+1)\dfrac{\sqrt{l'(l'+1)}}{2} \sum\limits_{q=-1}^1(-1)^q b_{-q}[(-1)^{m'}(H_{l,l'}^{-m,m'-q}(\mathbf{b})+(-1)^mH_{l,l'}^{m,m'-q}(\mathbf{b}))\\
&\langle l',m'-q;1,q\mid l',m'\rangle+(H_{l,l'}^{-m,-m'-q}(\mathbf{b})+(-1)^mH_{l,l'}^{m,-m'-q}(\mathbf{b}))\langle l',-m'-q;1,q\mid l',-m'\rangle]
\end{align}\]
When \(m>0\), \(m'<0\), \[\begin{align} &(X_{l'm'},\mathcal{S}_{D+n}[W_{lm}])\\ =&\sum\limits_{\lambda\in\{l',l'+2\mid l'\geqslant 1\}}\sum\limits_{q=-1}^{1}\sum\limits_{m_1=-1}^{1}\rho^{l+1} (a_{22}-a_{12}) r^\lambda b_{-q}\sqrt{l'(l'+1)} [ (-1)^{m'}(H_{l,\lambda}^{-m,-m'-q}(\mathbf{b})\\ &+(-1)^mH_{l,\lambda}^{m,-m'-q}(\mathbf{b}))K_{l',l',\lambda}^{m_1,-m'-q,q} - (H_{l,\lambda}^{-m,m'-q}(\mathbf{b})+(-1)^mH_{l,\lambda}^{m,m'-q}(\mathbf{b}))K_{l',l',\lambda}^{m_1,m'-q,q}]\\ &+\rho^{l+1} a_{22}r^{l'}(2l+1)\dfrac{\sqrt{l'(l'+1)}}{2} \sum\limits_{q=-1}^1(-1)^q b_{-q}[ (-1)^{m'}(H_{l,l'}^{-m,-m'-q}(\mathbf{b})+(-1)^mH_{l,l'}^{m,-m'-q}(\mathbf{b}))\\ &\langle l',-m'-q;1,q\mid l',-m'\rangle - (H_{l,l'}^{-m,m'-q}(\mathbf{b})+(-1)^mH_{l,l'}^{m,m'-q}(\mathbf{b}))\langle l',m'-q;1,q\mid l',m'\rangle]\end{align}\]
When \(m>0\), \(m'=0\), \[\begin{align} &(X_{l'm'},\mathcal{S}_{D+n}[W_{lm}])\\ =&\sum\limits_{\lambda\in\{l',l'+2\mid l'\geqslant 1\}}\sum\limits_{q=-1}^{1}\sum\limits_{m_1=-1}^{1} -i\cdot \rho^{l+1} (a_{22}-a_{12}) r^\lambda b_{-q}\sqrt{2l'(l'+1)} [(H_{l,\lambda}^{-m,-q}(\mathbf{b})+(-1)^mH_{l,\lambda}^{m,-q}(\mathbf{b}))K_{l',l',\lambda}^{m_1,-q,q}]\\ &-i\cdot \rho^{l+1} a_{22}r^{l'}(2l+1)\dfrac{\sqrt{l'(l'+1)}}{\sqrt{2}} \sum\limits_{q=-1}^1(-1)^q b_{-q}[ (H_{l,l'}^{-m,-q}(\mathbf{b})+(-1)^mH_{l,l'}^{m,-q}(\mathbf{b}))\\ &\langle l',-q;1,q\mid l',0\rangle] \end{align}\]
When \(m<0\), \(m'>0\), \[\begin{align} &(X_{l'm'},\mathcal{S}_{D+n}[W_{lm}])\\ =&\sum\limits_{\lambda\in\{l',l'+2\mid l'\geqslant 1\}}\sum\limits_{q=-1}^{1}\sum\limits_{m_1=-1}^{1}\rho^{l+1} (a_{22}-a_{12}) r^\lambda b_{-q}\sqrt{l'(l'+1)} [(-1)^{m'}(H_{l,\lambda}^{m,m'-q}(\mathbf{b})\\ &-(-1)^mH_{l,\lambda}^{-m,m'-q}(\mathbf{b}))K_{l',l',\lambda}^{m_1,m'-q,q} +(H_{l,\lambda}^{m,-m'-q}(\mathbf{b})-(-1)^mH_{l,\lambda}^{-m,-m'-q}(\mathbf{b}))K_{l',l',\lambda}^{m_1,-m'-q,q}]\\ & +\rho^{l+1} a_{22}r^{l'} (2l+1)\dfrac{\sqrt{l'(l'+1)}}{2} \sum\limits_{q=-1}^1(-1)^q b_{-q}[(-1)^{m'}(H_{l,l'}^{m,m'-q}(\mathbf{b})-(-1)^mH_{l,l'}^{-m,m'-q}(\mathbf{b}))\\ &\langle l',m'-q;1,q\mid l',m'\rangle+(H_{l,l'}^{m,-m'-q}(\mathbf{b})-(-1)^mH_{l,l'}^{-m,-m'-q}(\mathbf{b}))\langle l',-m'-q;1,q\mid l',-m'\rangle] \end{align}\]
When \(m<0\), \(m'<0\), \[\begin{align} &(X_{l'm'},\mathcal{S}_{D+n}[W_{lm}])\\ =&\sum\limits_{\lambda\in\{l',l'+2\mid l'\geqslant 1\}}\sum\limits_{q=-1}^{1}\sum\limits_{m_1=-1}^{1}\rho^{l+1} (a_{22}-a_{12}) r^\lambda b_{-q}i\cdot \sqrt{l'(l'+1)} [(-1)^{m'}(H_{l,\lambda}^{m,-m'-q}(\mathbf{b})\\ &-(-1)^mH_{l,\lambda}^{-m,-m'-q}(\mathbf{b}))K_{l',l',\lambda}^{m_1,-m'-q,q} - (H_{l,\lambda}^{m,m'-q}(\mathbf{b})-(-1)^mH_{l,\lambda}^{-m,m'-q}(\mathbf{b}))K_{l',l',\lambda}^{m_1,m'-q,q}]\\ &-i\cdot \rho^{l+1} a_{22}r^{l'}(2l+1)\dfrac{\sqrt{l'(l'+1)}}{2} \sum\limits_{q=-1}^1(-1)^q b_{-q}[-(-1)^{m'}(H_{l,l'}^{m,-m'-q}(\mathbf{b})-(-1)^mH_{l,l'}^{-m,-m'-q}(\mathbf{b}))\\ &\langle l',-m'-q;1,q\mid l',-m'\rangle + (H_{l,l'}^{m,m'-q}(\mathbf{b})-(-1)^mH_{l,l'}^{-m,m'-q}(\mathbf{b}))\langle l',m'-q;1,q\mid l',m'\rangle] \end{align}\]
When \(m<0\), \(m'=0\), \[\begin{align} &(X_{l'm'},\mathcal{S}_{D+n}[W_{lm}])\\ =&\sum\limits_{\lambda\in\{l',l'+2\mid l'\geqslant 1\}}\sum\limits_{q=-1}^{1}\sum\limits_{m_1=-1}^{1} \rho^{l+1} (a_{22}-a_{12}) r^\lambda b_{-q} \sqrt{2l'(l'+1)} [(H_{l,\lambda}^{m,-q}(\mathbf{b})-(-1)^mH_{l,\lambda}^{-m,-q}(\mathbf{b}))K_{l',l',\lambda}^{m_1,-q,q}]\\ &+\rho^{l+1} a_{22}r^{l'} (2l+1)\dfrac{\sqrt{l'(l'+1)}}{\sqrt{2}} \sum\limits_{q=-1}^1(-1)^q b_{-q}[ (H_{l,l'}^{m,-q}(\mathbf{b})-(-1)^mH_{l,l'}^{-m,-q}(\mathbf{b}))\\ &\langle l',-q;1,q\mid l',0\rangle] \end{align}\]
When \(m=0\), \(m'>0\), \[\begin{align} &(X_{l'm'},\mathcal{S}_{D+n}[W_{lm}])\\ =&\sum\limits_{\lambda\in\{l',l'+2\mid l'\geqslant 1\}}\sum\limits_{q=-1}^{1}\sum\limits_{m_1=-1}^{1} -i\cdot \rho^{l+1} (a_{22}-a_{12}) r^\lambda b_{-q}\sqrt{2l'(l'+1)} [(-1)^{m'}H_{l,\lambda}^{0,m'-q}(\mathbf{b}) K_{l',l',\lambda}^{m_1,m'-q,q}\\ &+H_{l,\lambda}^{0,-m'-q}(\mathbf{b})K_{l',l',\lambda}^{m_1,-m'-q,q}]\\ &-i\cdot \rho^{l+1} a_{22}r^{l'}(2l+1)\dfrac{\sqrt{l'(l'+1)}}{\sqrt{2}} \sum\limits_{q=-1}^1(-1)^q b_{-q}[(-1)^{m'}H_{l,l'}^{0,m'-q}(\mathbf{b})\\ &\langle l',m'-q;1,q\mid l',m'\rangle+(H_{l,l'}^{0,-m'-q}(\mathbf{b})\langle l',-m'-q;1,q\mid l',-m'\rangle] \end{align}\]
When \(m=0\), \(m'<0\),
\[\begin{align} &(X_{l'm'},\mathcal{S}_{D+n}[W_{lm}])\\ =&\sum\limits_{\lambda\in\{l',l'+2\mid l'\geqslant 1\}}\sum\limits_{q=-1}^{1}\sum\limits_{m_1=-1}^{1}\rho^{l+1} (a_{22}-a_{12}) r^\lambda b_{-q}\sqrt{2l'(l'+1)} [(-1)^{m'}H_{l,\lambda}^{0,-m'-q}(\mathbf{b})K_{l',l',\lambda}^{m_1,-m'-q,q}\\ & - H_{l,\lambda}^{0,m'-q}(\mathbf{b})K_{l',l',\lambda}^{m_1,m'-q,q}]\\ &+\rho^{l+1} a_{22}r^{l'}(2l+1)\dfrac{\sqrt{l'(l'+1)}}{\sqrt{2}} \sum\limits_{q=-1}^1(-1)^q b_{-q}[ (-1)^{m'} H_{l,l'}^{0,-m'-q}(\mathbf{b})\\ &\langle l',-m'-q;1,q\mid l',-m'\rangle- H_{l,l'}^{0,m'-q}(\mathbf{b})\langle l',m'-q;1,q\mid l',m'\rangle] \end{align}\]
When \(m=0\), \(m'=0\), \[\begin{align} &(X_{l'm'},\mathcal{S}_{D+n}[W_{lm}])\\ =&\sum\limits_{\lambda\in\{l',l'+2\mid l'\geqslant 1\}}\sum\limits_{q=-1}^{1}\sum\limits_{m_1=-1}^{1} -i\cdot \rho^{l+1} (a_{22}-a_{12}) r^\lambda b_{-q}\sqrt{4l'(l'+1)} [H_{l,\lambda}^{0,-q}(\mathbf{b})K_{l',l',\lambda}^{m_1,-q,q}]\\ &-i\cdot \rho^{l+1} a_{22}r^{l'}(2l+1)\sqrt{l'(l'+1)} \sum\limits_{q=-1}^1(-1)^q b_{-q}[ H_{l,l'}^{0,-q}(\mathbf{b})\langle l',-q;1,q\mid l',0\rangle] \end{align}\]
Theorem 16. The entries \(\boxed{(W_{l'm'},\mathcal{S}_{D+n}[W_{lm}])}\)(\(l'\geqslant 1\)) are given by:
If \(m>0\) and \(m'>0\), \[\begin{align}&(W_{l'm'},\mathcal{S}_{D+n}[W_{lm}])\\
&=\dfrac{l'(2l'+1)}{2}[ \rho^{l+3}a_{12}r^{l'-1}+\rho^{l+1}(a_{22}-a_{12})(r^{l'+1}+a^2r^{l'-1})+\rho^{l+1}a_{22}r^{l'+1}\dfrac{2l+1}{2l'+1}]\\
&[(-1)^{m'}(H_{l,l'}^{-m,m'}(\mathbf{b})+(-1)^mH_{l,l'}^{m,m'}(\mathbf{b}))+(H_{l,l'}^{-m,-m'}(\mathbf{b})+(-1)^mH_{l,l'}^{m,-m'}(\mathbf{b}))]\\
&+\rho^{l+1}(a_{22}-a_{12})\sqrt{l'(2l'+1)}\sum\limits_{\lambda\in\{l'+1,l'-1\}}\sum\limits_{q=-1}^{q=1}\sum\limits_{m_1=-1}^{1}r^\lambda b_{-q}[(-1)^{m'}(H_{l,\lambda}^{-m,m'-q}(\mathbf{b})\\
&+(-1)^mH_{l,\lambda}^{m,m'-q}(\mathbf{b}))K_{l'-1,l',\lambda}^{m_1,m'-q,q}+(H_{l,\lambda}^{-m,-m'-q}(\mathbf{(}b))+(-1)^mH_{l,\lambda}^{m,-m'-q}(\mathbf{b}))K_{l'-1,l',\lambda}^{m_1,-m'-q,q}]\\
&+\rho^{l+1}a_{22}(2l+1)\dfrac{\sqrt{l'(2l'+1)}}{2}\sum\limits_{q=-1}^1(-1)^q b_{-q}r^{l'-1}[(-1)^{m'}(H_{l,l'-1}^{-m,m'-q}(\mathbf{b})+(-1)^mH_{l,l'-1}^{m,m'-q}(\mathbf{b}))\\
&\langle l'-1,m'-q;1,q\mid l',m' \rangle+(H_{l,l'-1}^{-m,-m'-q}(\mathbf{b})+(-1)^mH_{l,l'-1}^{m,-m'-q}(\mathbf{b}))\langle l'-1,-m'-q;1,q\mid l',-m' \rangle]
\end{align}\]
If \(m>0\) and \(m'<0\), \[\begin{align} &(W_{l'm'},\mathcal{S}_{D+n}[W_{lm}])\\ &=\dfrac{l'(2l'+1)}{2}\Bigl[ \rho^{l+3}a_{12}r^{l'-1}+\rho^{l+1}(a_{22}-a_{12})(r^{l'+1}+a^2r^{l'-1})+\rho^{l+1}a_{22}r^{l'+1}\dfrac{2l+1}{2l'+1}\Bigr]\\ &\quad\Bigl[i\cdot(-1)^{m'}(H_{l,l'}^{-m,-m'}(\mathbf{b})+(-1)^mH_{l,l'}^{m,-m'}(\mathbf{b}))-i\cdot(H_{l,l'}^{-m,m'}(\mathbf{b})+(-1)^mH_{l,l'}^{m,m'}(\mathbf{b}))\Bigr]\\ &+\rho^{l+1}(a_{22}-a_{12})\sqrt{l'(2l'+1)}\sum_{\lambda\in\{l'+1,l'-1\}}\sum_{q=-1}^{1}\sum_{m_1=-1}^{1}r^\lambda b_{-q}\Bigl[i\cdot(-1)^{m'}(H_{l,\lambda}^{-m,-m'-q}(\mathbf{b})\\ &\quad+(-1)^mH_{l,\lambda}^{m,-m'-q}(\mathbf{b}))K_{l'-1,l',\lambda}^{m_1,-m'-q,q}-i\cdot(H_{l,\lambda}^{-m,m'-q}(\mathbf{b})+(-1)^mH_{l,\lambda}^{m,m'-q}(\mathbf{b}))K_{l'-1,l',\lambda}^{m_1,m'-q,q}\Bigr]\\ &+\rho^{l+1}a_{22}(2l+1)\dfrac{\sqrt{l'(2l'+1)}}{2}\sum_{q=-1}^{1}(-1)^q b_{-q}r^{l'-1}\Bigl[i\cdot(-1)^{m'}(H_{l,l'-1}^{-m,-m'-q}(\mathbf{b})+(-1)^mH_{l,l'-1}^{m,-m'-q}(\mathbf{b}))\\ &\quad\langle l'-1,-m'-q;1,q\mid l',-m' \rangle-i\cdot(H_{l,l'-1}^{-m,m'-q}(\mathbf{b})+(-1)^mH_{l,l'-1}^{m,m'-q}(\mathbf{b}))\langle l'-1,m'-q;1,q\mid l',m' \rangle\Bigr] \end{align}\]
If \(m>0\) and \(m'=0\), \[\begin{align} &(W_{l'm'},\mathcal{S}_{D+n}[W_{lm}])\\ &=\dfrac{l'(2l'+1)}{\sqrt{2}}\Bigl[ \rho^{l+3}a_{12}r^{l'-1}+\rho^{l+1}(a_{22}-a_{12})(r^{l'+1}+a^2r^{l'-1})+\rho^{l+1}a_{22}r^{l'+1}\dfrac{2l+1}{2l'+1}\Bigr]\\ &\quad\Bigl[(H_{l,l'}^{-m,0}(\mathbf{b})+(-1)^mH_{l,l'}^{m,0}(\mathbf{b}))\Bigr]\\ &+\rho^{l+1}(a_{22}-a_{12})\sqrt{2l'(2l'+1)}\sum_{\lambda\in\{l'+1,l'-1\}}\sum_{q=-1}^{1}\sum_{m_1=-1}^{1}r^\lambda b_{-q}\Bigl[(H_{l,\lambda}^{-m,-q}(\mathbf{b})+(-1)^mH_{l,\lambda}^{m,-q}(\mathbf{b}))K_{l'-1,l',\lambda}^{m_1,-q,q}\Bigr]\\ &+\rho^{l+1}a_{22}(2l+1)\dfrac{\sqrt{l'(2l'+1)}}{\sqrt{2}}\sum_{q=-1}^{1}(-1)^q b_{-q}r^{l'-1}\Bigl[(H_{l,l'-1}^{-m,-q}(\mathbf{b})+(-1)^mH_{l,l'-1}^{m,-q}(\mathbf{b}))\langle l'-1,-q;1,q\mid l',0 \rangle\Bigr] \end{align}\]
If \(m<0\) and \(m'>0\), \[\begin{align} &(W_{l'm'},\mathcal{S}_{D+n}[W_{lm}])\\ &=\dfrac{l'(2l'+1)}{2}\Bigl[ \rho^{l+3}a_{12}r^{l'-1}+\rho^{l+1}(a_{22}-a_{12})(r^{l'+1}+a^2r^{l'-1})+\rho^{l+1}a_{22}r^{l'+1}\dfrac{2l+1}{2l'+1}\Bigr]\\ &\quad i\Bigl[(-1)^{m'}(H_{l,l'}^{m,m'}(\mathbf{b})-(-1)^mH_{l,l'}^{-m,m'}(\mathbf{b}))+(H_{l,l'}^{m,-m'}(\mathbf{b})-(-1)^mH_{l,l'}^{-m,-m'}(\mathbf{b}))\Bigr]\\ &+\rho^{l+1}(a_{22}-a_{12})\sqrt{l'(2l'+1)}\sum_{\lambda\in\{l'+1,l'-1\}}\sum_{q=-1}^{1}\sum_{m_1=-1}^{1}r^\lambda b_{-q}\;i\Bigl[(-1)^{m'}(H_{l,\lambda}^{m,m'-q}(\mathbf{b})\\ &\quad-(-1)^mH_{l,\lambda}^{-m,m'-q}(\mathbf{b}))K_{l'-1,l',\lambda}^{m_1,m'-q,q}+(H_{l,\lambda}^{m,-m'-q}(\mathbf{b})-(-1)^mH_{l,\lambda}^{-m,-m'-q}(\mathbf{b}))K_{l'-1,l',\lambda}^{m_1,-m'-q,q}\Bigr]\\ &+\rho^{l+1}a_{22}(2l+1)\dfrac{\sqrt{l'(2l'+1)}}{2}\sum_{q=-1}^{1}(-1)^q b_{-q}r^{l'-1}\;i\Bigl[(-1)^{m'}(H_{l,l'-1}^{m,m'-q}(\mathbf{b})-(-1)^mH_{l,l'-1}^{-m,m'-q}(\mathbf{b}))\\ &\quad\langle l'-1,m'-q;1,q\mid l',m' \rangle+(H_{l,l'-1}^{m,-m'-q}(\mathbf{b})-(-1)^mH_{l,l'-1}^{-m,-m'-q}(\mathbf{b}))\langle l'-1,-m'-q;1,q\mid l',-m' \rangle\Bigr] \end{align}\]
If \(m<0\) and \(m'<0\), \[\begin{align} &(W_{l'm'},\mathcal{S}_{D+n}[W_{lm}])\\ &=\dfrac{l'(2l'+1)}{2}\Bigl[ \rho^{l+3}a_{12}r^{l'-1}+\rho^{l+1}(a_{22}-a_{12})(r^{l'+1}+a^2r^{l'-1})+\rho^{l+1}a_{22}r^{l'+1}\dfrac{2l+1}{2l'+1}\Bigr]\\ &\quad\Bigl[-(-1)^{m'}(H_{l,l'}^{m,-m'}(\mathbf{b})-(-1)^mH_{l,l'}^{-m,-m'}(\mathbf{b}))+(H_{l,l'}^{m,m'}(\mathbf{b})-(-1)^mH_{l,l'}^{-m,m'}(\mathbf{b}))\Bigr]\\ &+\rho^{l+1}(a_{22}-a_{12})\sqrt{l'(2l'+1)}\sum_{\lambda\in\{l'+1,l'-1\}}\sum_{q=-1}^{1}\sum_{m_1=-1}^{1}r^\lambda b_{-q}\Bigl[-(-1)^{m'}(H_{l,\lambda}^{m,-m'-q}(\mathbf{b})\\ &\quad-(-1)^mH_{l,\lambda}^{-m,-m'-q}(\mathbf{b}))K_{l'-1,l',\lambda}^{m_1,-m'-q,q}+(H_{l,\lambda}^{m,m'-q}(\mathbf{b})-(-1)^mH_{l,\lambda}^{-m,m'-q}(\mathbf{b}))K_{l'-1,l',\lambda}^{m_1,m'-q,q}\Bigr]\\ &+\rho^{l+1}a_{22}(2l+1)\dfrac{\sqrt{l'(2l'+1)}}{2}\sum_{q=-1}^{1}(-1)^q b_{-q}r^{l'-1}\Bigl[-(-1)^{m'}(H_{l,l'-1}^{m,-m'-q}(\mathbf{b})-(-1)^mH_{l,l'-1}^{-m,-m'-q}(\mathbf{b}))\\ &\quad\langle l'-1,-m'-q;1,q\mid l',-m' \rangle+(H_{l,l'-1}^{m,m'-q}(\mathbf{b})-(-1)^mH_{l,l'-1}^{-m,m'-q}(\mathbf{b}))\langle l'-1,m'-q;1,q\mid l',m' \rangle\Bigr] \end{align}\]
If \(m<0\) and \(m'=0\), \[\begin{align} &(W_{l'm'},\mathcal{S}_{D+n}[W_{lm}])\\ &=\dfrac{l'(2l'+1)}{\sqrt{2}}\Bigl[ \rho^{l+3}a_{12}r^{l'-1}+\rho^{l+1}(a_{22}-a_{12})(r^{l'+1}+a^2r^{l'-1})+\rho^{l+1}a_{22}r^{l'+1}\dfrac{2l+1}{2l'+1}\Bigr]\\ &\quad i\Bigl[(H_{l,l'}^{m,0}(\mathbf{b})-(-1)^mH_{l,l'}^{-m,0}(\mathbf{b}))\Bigr]\\ &+\rho^{l+1}(a_{22}-a_{12})\sqrt{2l'(2l'+1)}\sum_{\lambda\in\{l'+1,l'-1\}}\sum_{q=-1}^{1}\sum_{m_1=-1}^{1}r^\lambda b_{-q}\;i\Bigl[(H_{l,\lambda}^{m,-q}(\mathbf{b})-(-1)^mH_{l,\lambda}^{-m,-q}(\mathbf{b}))K_{l'-1,l',\lambda}^{m_1,-q,q}\Bigr]\\ &+\rho^{l+1}a_{22}(2l+1)\dfrac{\sqrt{l'(2l'+1)}}{\sqrt{2}}\sum_{q=-1}^{1}(-1)^q b_{-q}r^{l'-1}\;i\Bigl[(H_{l,l'-1}^{m,-q}(\mathbf{b})-(-1)^mH_{l,l'-1}^{-m,-q}(\mathbf{b}))\langle l'-1,-q;1,q\mid l',0 \rangle\Bigr] \end{align}\]
If \(m=0\) and \(m'>0\), \[\begin{align} &(W_{l'm'},\mathcal{S}_{D+n}[W_{lm}])\\ &=\dfrac{l'(2l'+1)}{\sqrt{2}}\Bigl[ \rho^{l+3}a_{12}r^{l'-1}+\rho^{l+1}(a_{22}-a_{12})(r^{l'+1}+a^2r^{l'-1})+\rho^{l+1}a_{22}r^{l'+1}\dfrac{2l+1}{2l'+1}\Bigr]\\ &\quad\Bigl[(-1)^{m'}H_{l,l'}^{0,m'}(\mathbf{b})+H_{l,l'}^{0,-m'}(\mathbf{b})\Bigr]\\ &+\rho^{l+1}(a_{22}-a_{12})\sqrt{2l'(2l'+1)}\sum_{\lambda\in\{l'+1,l'-1\}}\sum_{q=-1}^{1}\sum_{m_1=-1}^{1}r^\lambda b_{-q}\Bigl[(-1)^{m'}H_{l,\lambda}^{0,m'-q}(\mathbf{b})K_{l'-1,l',\lambda}^{m_1,m'-q,q}\\ &+H_{l,\lambda}^{0,-m'-q}(\mathbf{b})K_{l'-1,l',\lambda}^{m_1,-m'-q,q}\Bigr]\\ &+\rho^{l+1}a_{22}(2l+1)\dfrac{\sqrt{l'(2l'+1)}}{\sqrt{2}}\sum_{q=-1}^{1}(-1)^q b_{-q}r^{l'-1}\Bigl[(-1)^{m'}H_{l,l'-1}^{0,m'-q}(\mathbf{b})\langle l'-1,m'-q;1,q\mid l',m' \rangle\\ &+H_{l,l'-1}^{0,-m'-q}(\mathbf{b})\langle l'-1,-m'-q;1,q\mid l',-m' \rangle\Bigr] \end{align}\]
If \(m=0\) and \(m'<0\), \[\begin{align} &(W_{l'm'},\mathcal{S}_{D+n}[W_{lm}])\\ &=\dfrac{l'(2l'+1)}{\sqrt{2}}\Bigl[ \rho^{l+3}a_{12}r^{l'-1}+\rho^{l+1}(a_{22}-a_{12})(r^{l'+1}+a^2r^{l'-1})+\rho^{l+1}a_{22}r^{l'+1}\dfrac{2l+1}{2l'+1}\Bigr]\\ &\quad i\Bigl[(-1)^{m'}H_{l,l'}^{0,-m'}(\mathbf{b})-H_{l,l'}^{0,m'}(\mathbf{b})\Bigr]\\ &+\rho^{l+1}(a_{22}-a_{12})\sqrt{2l'(2l'+1)}\sum_{\lambda\in\{l'+1,l'-1\}}\sum_{q=-1}^{1}\sum_{m_1=-1}^{1}r^\lambda b_{-q}\;i\Bigl[(-1)^{m'}H_{l,\lambda}^{0,-m'-q}(\mathbf{b})K_{l'-1,l',\lambda}^{m_1,-m'-q,q}\\ &-H_{l,\lambda}^{0,m'-q}(\mathbf{b})K_{l'-1,l',\lambda}^{m_1,m'-q,q}\Bigr]\\ &+\rho^{l+1}a_{22}(2l+1)\dfrac{\sqrt{l'(2l'+1)}}{\sqrt{2}}\sum_{q=-1}^{1}(-1)^q b_{-q}r^{l'-1}\;i\Bigl[(-1)^{m'}H_{l,l'-1}^{0,-m'-q}(\mathbf{b})\langle l'-1,-m'-q;1,q\mid l',-m' \rangle\\ &-H_{l,l'-1}^{0,m'-q}(\mathbf{b})\langle l'-1,m'-q;1,q\mid l',m' \rangle\Bigr] \end{align}\]
If \(m=0\) and \(m'=0\), \[\begin{align} &(W_{l'm'},\mathcal{S}_{D+n}[W_{lm}])\\ &=l'(2l'+1)\Bigl[ \rho^{l+3}a_{12}r^{l'-1}+\rho^{l+1}(a_{22}-a_{12})(r^{l'+1}+a^2r^{l'-1})+\rho^{l+1}a_{22}r^{l'+1}\dfrac{2l+1}{2l'+1}\Bigr]H_{l,l'}^{0,0}(\mathbf{b})\\ &+\rho^{l+1}(a_{22}-a_{12})\cdot 2\sqrt{l'(2l'+1)}\sum_{\lambda\in\{l'+1,l'-1\}}\sum_{q=-1}^{1}\sum_{m_1=-1}^{1}r^\lambda b_{-q}\;H_{l,\lambda}^{0,-q}(\mathbf{b})K_{l'-1,l',\lambda}^{m_1,-q,q}\\ &+\rho^{l+1}a_{22}(2l+1)\sqrt{l'(2l'+1)}\sum_{q=-1}^{1}(-1)^q b_{-q}r^{l'-1}\;H_{l,l'-1}^{0,-q}(\mathbf{b})\langle l'-1,-q;1,q\mid l',0 \rangle \end{align}\]
Lemma 7. Let \(\underline{Y_{lm}}=(Y^1_{lm},Y^2_{lm},Y^3_{lm})=(V_{lm},W_{lm},X_{lm}).\) Then, we have \[(\mathcal{S}_D\underline{Y_{l m}})(x) = \underline{Y_{l m}}(\hat{x}) A_{\mathcal{S}_D,l},\] where the matrix \(A_{\mathcal{S}_D,l}\) is given by \[A_{\mathcal{S}_D,l} = \begin{bmatrix} \dfrac{(3l+1)\mu+l\lambda} {(2l+3)(2l+1)\mu(2\mu+\lambda)} & 0 & 0 \\[12pt] 0 & \dfrac{(3l+2)\mu+(l+1)\lambda} {(2l-1)(2l+1)\mu(2\mu+\lambda)} & 0 \\[12pt] 0 & 0 & \dfrac{1}{(2l+1)\mu} \end{bmatrix}=\text{diag}(\tau_{\mathcal{S}_D,l}^1,\tau_{\mathcal{S}_D,l}^2,\tau_{\mathcal{S}_D,l}^3).\]
Therefore, \[( Y_{l'm'}^p,\mathcal{S}_D Y_{lm}^q)= \delta_{ll'}\delta_{mm'}\delta_{pq}\cdot \rho\cdot \tau_{\mathcal{S}_D}^p(l)\cdot \text{norm}_p(l)\] where \(\text{norm}_1(l)= (l+1)(2l+1)\), \(\text{norm}_2(l)= l(2l+1)\), \(\text{norm}_3(l)=l(l+1)\).
Define the entries \[\label{eq-M-entries}M_{(l'm'p),(lmq)}(\alpha) = (Y_{l'm'}^p,\mathcal{S}^{\alpha,0}_D Y_{lm}^q)= ( Y_{l'm'}^p,\mathcal{S}_D Y_{lm}^q) + \sum_{n\neq 0}e^{-in\alpha}(Y_{l'm'}^p,\mathcal{S}_{D+n} Y_{lm}^q).\tag{14}\]
Remark 5. Since the operator \(\mathcal{S}_D^{\alpha,0}\) is complex, there should be a conjugation on \(e^{in\alpha}\) when it is extracted from the second slot of inner product, hence \(e^{-in\alpha}\).
Definition 1. The polylogarithm function is defined by a power series in \(z\) generalizing the Mercator series, which is also a Dirichlet series in \(s\)[3]: \[\mathrm{Li}_s(z)=\sum\limits_{k=1}^\infty\dfrac{z^k}{k^s}.\]
Lemma 8. Denote the series \(\sum\limits_{n\neq 0}H_{l,\lambda}^{m,\mu}(\mathbf{b})e^{-in\alpha}\) by \(\mathscr{H}_{l,\lambda}^{m,\mu}(\alpha)\). Then \[\begin{align} \sum\limits_{n\neq 0}H_{l,\lambda}^{m,\mu}(\mathbf{b})e^{-in\alpha}=(-1)^{\lambda+\mu} \sqrt{\dfrac{2l+1}{2\lambda+1}}& \sqrt{{l+\lambda+\mu-m\choose \lambda+\mu}{l+\lambda+m-\mu\choose \lambda-\mu}} \sqrt{\dfrac{4\pi}{2(l+\lambda)+1}}\\ &\big(\mathrm{Li}_{l+\lambda+1}(e^{-i\alpha})Y_{l+\lambda}^{m-\mu}(\dfrac{\pi}{2},\pi)+\mathrm{Li}_{l+\lambda+1}(e^{i\alpha})Y_{l+\lambda}^{m-\mu}(\dfrac{\pi}{2},0)\big) \end{align}.\]
Theorem 17. The expression of \(\boxed{M_{(l'm'2),(lm1)}(\alpha)}\)(\(l'\geqslant 1\), otherwise this term vanishes) is given by:
If \(m>0\) and \(m'>0\), \[\begin{align}M_{(l'm'2),(lm1)}(\alpha)=
\rho^{l+3}a_{11}[r^{l'-1}\dfrac{1}{2}(\mathscr{H}_{l,l'}^{-m,-m'}(\alpha)+(-1)^m\mathscr{H}_{l,l'}^{m,-m'}(\alpha))l'(2l'+1)\\
+r^{l'-1}\dfrac{(-1)^{m'}}{2}(\mathscr{H}_{l,l'}^{-m,m'}(\alpha)+(-1)^m\mathscr{H}_{l,l'}^{m,m'}(\alpha))l'(2l'+1)]
\end{align}\] If \(m>0\) and \(m'<0\), \[\begin{align}M_{(l'm'2),(lm1)}(\alpha)=
\rho^{l+3}a_{11}[r^{l'-1}\dfrac{1}{2}(\mathscr{H}_{l,l'}^{-m,m'}(\alpha)+(-1)^m\mathscr{H}_{l,l'}^{m,m'}(\alpha))(-i \cdot l'(2l'+1))\\
+r^{l'-1}\dfrac{(-1)^{m'}}{2}(\mathscr{H}_{l,l'}^{-m,-m'}(\alpha)+(-1)^m\mathscr{H}_{l,l'}^{m,-m'}(\alpha))(i\cdot l'(2l'+1))]
\end{align}\] If \(m>0\) and \(m'=0\), \[\begin{align}M_{(l'm'2),(lm1)}(\alpha)=
\rho^{l+3}a_{11}r^{l'-1}\dfrac{1}{\sqrt{2}}(\mathscr{H}_{l,l'}^{-m,0}(\alpha)+(-1)^m\mathscr{H}_{l,l'}^{m,0}(\alpha))l'(2l'+1).
\end{align}\]
If \(m<0\) and \(m'>0\), \[\begin{align}M_{(l'm'2),(lm1)}(\alpha)= \rho^{l+3}a_{11}[r^{l'-1}\dfrac{i}{2}(\mathscr{H}_{l,l'}^{m,-m'}(\alpha)-(-1)^m \mathscr{H}_{l,l'}^{-m,-m'}(\alpha))l'(2l'+1)\\ +r^{l'-1}\dfrac{(-1)^{m'}}{2}\cdot i (\mathscr{H}_{l,l'}^{m,m'}(\alpha)-(-1)^m\mathscr{H}_{l,l'}^{-m,m'}(\alpha))l'(2l'+1)] \end{align}\]
If \(m<0\) and \(m'<0\), \[\begin{align}M_{(l'm'2),(lm1)}(\alpha)= \rho^{l+3}a_{11}[r^{l'-1}\dfrac{i}{2}(\mathscr{H}_{l,l'}^{m,m'}(\alpha)-(-1)^m \mathscr{H}_{l,l'}^{-m,m'}(\alpha))(-i\cdot l'(2l'+1))\\ +r^{l'-1}\dfrac{(-1)^{m'}}{2}\cdot i (\mathscr{H}_{l,l'}^{m,-m'}(\alpha)-(-1)^m\mathscr{H}_{l,l'}^{-m,-m'}(\alpha))(i\cdot l'(2l'+1))] \end{align}\]
If \(m<0\) and \(m'=0\), \[\begin{align}M_{(l'm'2),(lm1)}(\alpha)= \rho^{l+3}a_{11}r^{l'-1}\dfrac{i}{\sqrt{2}}(\mathscr{H}_{l,l'}^{m,0}(\alpha)-(-1)^m\mathscr{H}_{l,l'}^{-m,0}(\alpha))l'(2l'+1).\end{align}\]
If \(m=0\) and \(m'>0\), \[\begin{align}M_{(l'm'2),(lm1)}(\alpha)= \rho^{l+3}a_{11}r^{l'-1} [\dfrac{1}{\sqrt{2}} \mathscr{H}_{l,l'}^{0,-m'}(\alpha)l'(2l'+1)+\dfrac{(-1)^{m'}}{\sqrt{2}}\mathscr{H}_{l,l'}^{0,m'}(\alpha)l'(2l'+1)] \end{align}\]
If \(m=0\) and \(m'<0\), \[\begin{align}M_{(l'm'2),(lm1)}(\alpha)= \rho^{l+3}a_{11}r^{l'-1} [\dfrac{1}{\sqrt{2}} \mathscr{H}_{l,l'}^{0,m'}(\alpha)(-i\cdot l'(2l'+1))+\dfrac{(-1)^{m'}}{\sqrt{2}}\mathscr{H}_{l,l'}^{0,-m'}(\alpha)(i\cdot l'(2l'+1))] \end{align}\]
If \(m=0\) and \(m'=0\), \[\begin{align}M_{(l'm'2),(lm1)}(\alpha)= \rho^{l+3}a_{11}r^{l'-1} [\mathscr{H}_{l,l'}^{0,0}(\alpha)l'(2l'+1)] \end{align}\] where \(a_{ij}\) stands for the coefficient of the entries of \(A^{out}_{\mathcal{S},l}(x)\).
Lemma 9. Denote \(\sum\limits_{n\neq 0}L_{l,j,\lambda}^{m,\mu,q,m_1}(\mathbf{b})e^{-in\alpha}\) by \(\mathscr{L}_{l,j,\lambda}^{m,\mu,q,m_1}(\alpha)\). Then \[\begin{align} \mathscr{L}_{l,j,\lambda}^{m,\mu,q,m_1}(\alpha)&= i (-1)^{\lambda+\mu+q}\text{sgn}(q-m_1)\epsilon_{q}\sqrt{\lambda(2l+1)} \sqrt{{l+\lambda+\mu-m\choose \lambda+\mu}{l+\lambda+m-\mu\choose \lambda-\mu}}\\ &\langle\lambda-1,\mu-m_1;1,m_1\mid \lambda,\mu\rangle \langle \lambda-1,\mu-m_1;1,q+m_1\mid j,\mu+q \rangle \sqrt{\dfrac{4\pi}{2(l+\lambda)+1}}\\ &\big(\mathrm{Li}_{l+\lambda}(e^{-i\alpha})Y_{l+\lambda}^{m-\mu}(\dfrac{\pi}{2},\pi)-\mathrm{Li}_{l+\lambda}(e^{i\alpha})Y_{l+\lambda}^{m-\mu}(\dfrac{\pi}{2},0)\big) \end{align}\] where \[\epsilon_q=\left\{ \begin{align} \dfrac{1}{\sqrt{2}}&,\quad q=-1\\ 0&,\quad q=0\\ -\dfrac{1}{\sqrt{2}}&,\quad,q=1 \end{align}\right.\]
Theorem 18. The expression of \(\boxed{M_{(l'm',2),(lm,3)}(\alpha)}\)(\(l'\geqslant 1,l\geqslant 1\), otherwise this term vanishes) is given by When \(m>0, m'>0\), \[\begin{align} &M_{(l'm',2),(lm,3)}(\alpha)\\ =&\;\rho^{l+2}a_{33}\,r^{l'-1}\,\frac{\sqrt{l'(2l'+1)}}{2}\sum\limits_{q=-1}^{1}\sum\limits_{m_1=-1}^{1}\bigl[(-1)^{m'}\bigl(\mathscr{L}_{l,l',l'}^{-m,\,m'-q,\,q,\,m_1}(\alpha)+(-1)^m\mathscr{L}_{l,l',l'}^{m,\,m'-q,\,q,\,m_1}(\alpha)\bigr)\\ &\qquad\qquad\qquad\qquad+\bigl(\mathscr{L}_{l,l',l'}^{-m,\,-m'-q,\,q,\,m_1}(\alpha)+(-1)^m\mathscr{L}_{l,l',l'}^{m,\,-m'-q,\,q,\,m_1}(\alpha)\bigr)\bigr] \end{align}\] When \(m>0, m'<0\), \[\begin{align} &M_{(l'm',2),(lm,3)}(\alpha)\\ =&\;\rho^{l+2}a_{33}\,r^{l'-1}\,\frac{\sqrt{l'(2l'+1)}}{2}\sum\limits_{q=-1}^{1}\sum\limits_{m_1=-1}^{1}\bigl[i(-1)^{m'}\bigl(\mathscr{L}_{l,l',l'}^{-m,\,-m'-q,\,q,\,m_1}(\alpha)+(-1)^m\mathscr{L}_{l,l',l'}^{m,\,-m'-q,\,q,\,m_1}(\alpha)\bigr)\\ &\qquad\qquad\qquad\qquad-i\bigl(\mathscr{L}_{l,l',l'}^{-m,\,m'-q,\,q,\,m_1}(\alpha)+(-1)^m\mathscr{L}_{l,l',l'}^{m,\,m'-q,\,q,\,m_1}(\alpha)\bigr)\bigr] \end{align}\]
When \(m>0, m'=0\), \[\begin{align} &M_{(l',0,2),(lm,3)}(\alpha)\\ =&\;\rho^{l+2}a_{33}\,r^{l'-1}\,\frac{\sqrt{l'(2l'+1)}}{\sqrt{2}}\sum\limits_{q=-1}^{1}\sum\limits_{m_1=-1}^{1}\bigl(\mathscr{L}_{l,l',l'}^{-m,\,-q,\,q,\,m_1}(\alpha)+(-1)^m\mathscr{L}_{l,l',l'}^{m,\,-q,\,q,\,m_1}(\alpha)\bigr) \end{align}\]
When \(m<0, m'>0\), \[\begin{align} &M_{(l'm',2),(lm,3)}(\alpha)\\ =&\;\rho^{l+2}a_{33}\,r^{l'-1}\,i\cdot\frac{\sqrt{l'(2l'+1)}}{2}\sum\limits_{q=-1}^{1}\sum\limits_{m_1=-1}^{1}\bigl[(-1)^{m'}\bigl(\mathscr{L}_{l,l',l'}^{m,\,m'-q,\,q,\,m_1}(\alpha)-(-1)^m\mathscr{L}_{l,l',l'}^{-m,\,m'-q,\,q,\,m_1}(\alpha)\bigr)\\ &\qquad\qquad\qquad\qquad+\bigl(\mathscr{L}_{l,l',l'}^{m,\,-m'-q,\,q,\,m_1}(\alpha)-(-1)^m\mathscr{L}_{l,l',l'}^{-m,\,-m'-q,\,q,\,m_1}(\alpha)\bigr)\bigr] \end{align}\] When \(m<0, m'<0\), \[\begin{align} &M_{(l'm',2),(lm,3)}(\alpha)\\ =&\;\rho^{l+2}a_{33}\,r^{l'-1}\,\frac{\sqrt{l'(2l'+1)}}{2}\sum\limits_{q=-1}^{1}\sum\limits_{m_1=-1}^{1}\bigl[\bigl(\mathscr{L}_{l,l',l'}^{m,\,m'-q,\,q,\,m_1}(\alpha)-(-1)^m\mathscr{L}_{l,l',l'}^{-m,\,m'-q,\,q,\,m_1}(\alpha)\bigr)\\ &\qquad\qquad\qquad\qquad-(-1)^{m'}\bigl(\mathscr{L}_{l,l',l'}^{m,\,-m'-q,\,q,\,m_1}(\alpha)-(-1)^m\mathscr{L}_{l,l',l'}^{-m,\,-m'-q,\,q,\,m_1}(\alpha)\bigr)\bigr] \end{align}\] When \(m<0, m'=0\), \[\begin{align} &M_{(l',0,2),(lm,3)}(\alpha)\\ =&\;\rho^{l+2}a_{33}\,r^{l'-1}\,\frac{i\sqrt{l'(2l'+1)}}{\sqrt{2}}\sum\limits_{q=-1}^{1}\sum\limits_{m_1=-1}^{1}\bigl(\mathscr{L}_{l,l',l'}^{m,\,-q,\,q,\,m_1}(\alpha)-(-1)^m\mathscr{L}_{l,l',l'}^{-m,\,-q,\,q,\,m_1}(\alpha)\bigr) \end{align}\] When \(m=0, m'>0\), \[\begin{align} &M_{(l'm',2),(l,0,3)}(\alpha)\\ =&\;\rho^{l+2}a_{33}\,r^{l'-1}\,\frac{\sqrt{l'(2l'+1)}}{\sqrt{2}}\sum\limits_{q=-1}^{1}\sum\limits_{m_1=-1}^{1}\bigl[(-1)^{m'}\mathscr{L}_{l,l',l'}^{0,\,m'-q,\,q,\,m_1}(\alpha)+\mathscr{L}_{l,l',l'}^{0,\,-m'-q,\,q,\,m_1}(\alpha)\bigr] \end{align}\]
When \(m=0, m'<0\), \[\begin{align} &M_{(l'm',2),(l,0,3)}(\alpha)\\ =&\;\rho^{l+2}a_{33}\,r^{l'-1}\,\frac{\sqrt{l'(2l'+1)}}{\sqrt{2}}\sum\limits_{q=-1}^{1}\sum\limits_{m_1=-1}^{1}\bigl[i(-1)^{m'}\mathscr{L}_{l,l',l'}^{0,\,-m'-q,\,q,\,m_1}(\alpha)-i\,\mathscr{L}_{l,l',l'}^{0,\,m'-q,\,q,\,m_1}(\alpha)\bigr] \end{align}\]
When \(m=0, m'=0\), \[\begin{align} &M_{(l',0,2),(l,0,3)}(\alpha)\\ =&\;\rho^{l+2}a_{33}\,r^{l'-1}\,\sqrt{l'(2l'+1)}\sum\limits_{q=-1}^{1}\sum\limits_{m_1=-1}^{1}\mathscr{L}_{l,l',l'}^{0,\,-q,\,q,\,m_1}(\alpha) \end{align}\]
Lemma 10. Denote \(\sum\limits_{n\neq 0}b_{-q}H_{l,\lambda}^{m,\mu}(\mathbf{b})e^{-in\alpha}\) by \(\mathscr{A}_{l,\lambda}^{m,\mu}(\alpha,q)\), then \[\begin{align} \sum\limits_{n\neq 0}b_{-q}H_{l,\lambda}^{m,\mu}(\mathbf{b})e^{-in\alpha} =(-1)^{\lambda+\mu}\epsilon_q \sqrt{\dfrac{2l+1}{2\lambda+1}}& \sqrt{{l+\lambda+\mu-m\choose \lambda+\mu}{l+\lambda+m-\mu\choose \lambda-\mu}} \sqrt{\dfrac{4\pi}{2(l+\lambda)+1}}\\ & \big(\mathrm{Li}_{l+\lambda}(e^{-i\alpha})Y_{l+\lambda}^{m-\mu}(\dfrac{\pi}{2},\pi)-\mathrm{Li}_{l+\lambda}(e^{i\alpha})Y_{l+\lambda}^{m-\mu}(\dfrac{\pi}{2},0)\big) \end{align}.\]
Theorem 19. The expression of \(\boxed{M_{(l'm',1),(lm,2)}(\alpha)}\)(\(l\geqslant 1\), otherwise this term vanishes) is given by
When \(m>0\), \(m'>0\), \[\begin{align}
&M_{(l'm',1),(lm,2)}(\alpha)\\
=&\sum\limits_{\lambda\in\{l'+1,l'+3\}}\sum\limits_{q=-1}^{1}\sum\limits_{m_1=-1}^{1}\rho^{l+1} (a_{22}-a_{12}) r^\lambda \sqrt{(l'+1)(2l'+1)}
[(-1)^{m'}(\mathscr{A}_{l,\lambda}^{-m,m'-q}(\alpha,q)\\
&+(-1)^m\mathscr{A}_{l,\lambda}^{m,m'-q}(\alpha,q))K_{l'+1,l',\lambda}^{m_1,m'-q,q}
+(\mathscr{A}_{l,\lambda}^{-m,-m'-q}(\alpha,q)+(-1)^m\mathscr{A}_{l,\lambda}^{m,-m'-q}(\alpha,q))K_{l'+1,l',\lambda}^{m_1,-m'-q,q}]\\
&-\rho^{l+1} a_{22}r^{l'+1}\dfrac{(2l+1)(l'+1)}{2}[(-1)^{m'}(\mathscr{H}_{l,l'}^{-m,m'}(\alpha)+(-1)^m\mathscr{H}_{l,l'}^{m,m'}(\alpha))\\
&+(\mathscr{H}_{l,l'}^{-m,-m'}(\alpha)+(-1)^m\mathscr{H}_{l,l'}^{m,-m'}(\alpha))]\\
&+\rho^{l+1} a_{22}r^{l'+1}(2l+1)\dfrac{\sqrt{(l'+1)(2l'+1)}}{2} \sum\limits_{q=-1}^1(-1)^q [(-1)^{m'}(\mathscr{A}_{l,l'+1}^{-m,m'-q}(\alpha,q)+(-1)^m\mathscr{A}_{l,l'+1}^{m,m'-q}(\alpha,q))\\
&\langle l'+1,m'-q;1,q\mid l',m'\rangle+(\mathscr{A}_{l,l'+1}^{-m,-m'-q}(\alpha,q)+(-1)^m\mathscr{A}_{l,l'+1}^{m,-m'-q}(\alpha,q))\langle l'+1,-m'-q;1,q\mid l',-m'\rangle]
\end{align}\]
When \(m>0\), \(m'<0\),
\[\begin{align} &M_{(l'm',1),(lm,2)}(\alpha)\\ =&\sum\limits_{\lambda\in\{l'+1,l'+3\}}\sum\limits_{q=-1}^{1}\sum\limits_{m_1=-1}^{1}\rho^{l+1} (a_{22}-a_{12}) r^\lambda \sqrt{(l'+1)(2l'+1)} [i\cdot (-1)^{m'}(\mathscr{A}_{l,\lambda}^{-m,-m'-q}(\alpha,q)\\ &+(-1)^m\mathscr{A}_{l,\lambda}^{m,-m'-q}(\alpha,q))K_{l'+1,l',\lambda}^{m_1,-m'-q,q} -i\cdot (\mathscr{A}_{l,\lambda}^{-m,m'-q}(\alpha,q)+(-1)^m\mathscr{A}_{l,\lambda}^{m,m'-q}(\alpha,q))K_{l'+1,l',\lambda}^{m_1,m'-q,q}]\\ &-\rho^{l+1} a_{22}r^{l'+1}\dfrac{(2l+1)(l'+1)}{2}[i\cdot (-1)^{m'}(\mathscr{H}_{l,l'}^{-m,-m'}(\alpha)+(-1)^m\mathscr{H}_{l,l'}^{m,-m'}(\alpha))\\ &-i\cdot (\mathscr{H}_{l,l'}^{-m,m'}(\alpha)+(-1)^m\mathscr{H}_{l,l'}^{m,m'}(\alpha))]\\ &+\rho^{l+1} a_{22}r^{l'+1}(2l+1)\dfrac{\sqrt{(l'+1)(2l'+1)}}{2} \sum\limits_{q=-1}^1(-1)^q [i\cdot (-1)^{m'}(\mathscr{A}_{l,l'+1}^{-m,-m'-q}(\alpha,q)\\ &+(-1)^m\mathscr{A}_{l,l'+1}^{m,-m'-q}(\alpha,q))\langle l'+1,-m'-q;1,q\mid l',-m'\rangle -i\cdot (\mathscr{A}_{l,l'+1}^{-m,m'-q}(\alpha,q)\\ &+(-1)^m\mathscr{A}_{l,l'+1}^{m,m'-q}(\alpha,q))\langle l'+1,m'-q;1,q\mid l',m'\rangle] \end{align}\]
When \(m>0\), \(m'=0\), \[\begin{align} &M_{(l',0,1),(lm,2)}(\alpha)\\ =&\sum\limits_{\lambda\in\{l'+1,l'+3\}}\sum\limits_{q=-1}^{1}\sum\limits_{m_1=-1}^{1}\rho^{l+1} (a_{22}-a_{12}) r^\lambda \sqrt{2(l'+1)(2l'+1)} [(\mathscr{A}_{l,\lambda}^{-m,-q}(\alpha,q)+\\ &(-1)^m\mathscr{A}_{l,\lambda}^{m,-q}(\alpha,q))K_{l'+1,l',\lambda}^{m_1,-q,q}]\\ &-\rho^{l+1} a_{22}r^{l'+1}\dfrac{(2l+1)(l'+1)}{\sqrt{2}}[\mathscr{H}_{l,l'}^{-m,0}(\alpha)+(-1)^m\mathscr{H}_{l,l'}^{m,0}(\alpha)]\\ &+\rho^{l+1} a_{22}r^{l'+1}(2l+1)\dfrac{\sqrt{(l'+1)(2l'+1)}}{\sqrt{2}} \sum\limits_{q=-1}^1(-1)^q (\mathscr{A}_{l,l'+1}^{-m,-q}(\alpha,q)+(-1)^m\mathscr{A}_{l,l'+1}^{m,-q}(\alpha,q))\\ &\langle l'+1,-q;1,q\mid l',0\rangle \end{align}\]
When \(m<0\), \(m'>0\), \[\begin{align} &M_{(l'm',1),(lm,2)}(\alpha)\\ =&\sum\limits_{\lambda\in\{l'+1,l'+3\}}\sum\limits_{q=-1}^{1}\sum\limits_{m_1=-1}^{1}\rho^{l+1} (a_{22}-a_{12}) r^\lambda i\cdot\sqrt{(l'+1)(2l'+1)} [(-1)^{m'}(\mathscr{A}_{l,\lambda}^{m,m'-q}(\alpha,q)\\ &-(-1)^m\mathscr{A}_{l,\lambda}^{-m,m'-q}(\alpha,q))K_{l'+1,l',\lambda}^{m_1,m'-q,q} +(\mathscr{A}_{l,\lambda}^{m,-m'-q}(\alpha,q)-(-1)^m\mathscr{A}_{l,\lambda}^{-m,-m'-q}(\alpha,q))K_{l'+1,l',\lambda}^{m_1,-m'-q,q}]\\ &-\rho^{l+1} a_{22}r^{l'+1}i\cdot \dfrac{(2l+1)(l'+1)}{2}[(-1)^{m'}(\mathscr{H}_{l,l'}^{m,m'}(\alpha)-(-1)^m\mathscr{H}_{l,l'}^{-m,m'}(\alpha))\\ &+(\mathscr{H}_{l,l'}^{m,-m'}(\alpha)-(-1)^m\mathscr{H}_{l,l'}^{-m,-m'}(\alpha))]\\ &+\rho^{l+1} a_{22}r^{l'+1}i\cdot (2l+1)\dfrac{\sqrt{(l'+1)(2l'+1)}}{2} \sum\limits_{q=-1}^1(-1)^q [(-1)^{m'}(\mathscr{A}_{l,l'+1}^{m,m'-q}(\alpha,q)-(-1)^m\mathscr{A}_{l,l'+1}^{-m,m'-q}(\alpha,q))\\ &\langle l'+1,m'-q;1,q\mid l',m'\rangle+(\mathscr{A}_{l,l'+1}^{m,-m'-q}(\alpha,q)-(-1)^m\mathscr{A}_{l,l'+1}^{-m,-m'-q}(\alpha,q))\langle l'+1,-m'-q;1,q\mid l',-m'\rangle] \end{align}\]
When \(m<0\), \(m'<0\),
\[\begin{align} &M_{(l'm',1),(lm,2)}(\alpha)\\ =&\sum\limits_{\lambda\in\{l'+1,l'+3\}}\sum\limits_{q=-1}^{1}\sum\limits_{m_1=-1}^{1}\rho^{l+1} (a_{22}-a_{12}) r^\lambda \sqrt{(l'+1)(2l'+1)} [-(-1)^{m'}(\mathscr{A}_{l,\lambda}^{m,-m'-q}(\alpha,q)\\ &-(-1)^m\mathscr{A}_{l,\lambda}^{-m,-m'-q}(\alpha,q))K_{l'+1,l',\lambda}^{m_1,-m'-q,q} + (\mathscr{A}_{l,\lambda}^{m,m'-q}(\alpha,q)-(-1)^m\mathscr{A}_{l,\lambda}^{-m,m'-q}(\alpha,q))K_{l'+1,l',\lambda}^{m_1,m'-q,q}]\\ &-\rho^{l+1} a_{22}r^{l'+1}\dfrac{(2l+1)(l'+1)}{2} [- (-1)^{m'}(\mathscr{H}_{l,l'}^{m,-m'}(\alpha)-(-1)^m\mathscr{H}_{l,l'}^{-m,-m'}(\alpha))\\ &+ (\mathscr{H}_{l,l'}^{m,m'}(\alpha)-(-1)^m\mathscr{H}_{l,l'}^{-m,m'}(\alpha))]\\ &+\rho^{l+1} a_{22}r^{l'+1}(2l+1)\dfrac{\sqrt{(l'+1)(2l'+1)}}{2} \sum\limits_{q=-1}^1(-1)^q [-(-1)^{m'}(\mathscr{A}_{l,l'+1}^{m,-m'-q}(\alpha,q)\\ &-(-1)^m\mathscr{A}_{l,l'+1}^{-m,-m'-q}(\alpha,q)) \langle l'+1,-m'-q;1,q\mid l',-m'\rangle + (\mathscr{A}_{l,l'+1}^{m,m'-q}(\alpha,q)\\ &-(-1)^m\mathscr{A}_{l,l'+1}^{-m,m'-q}(\alpha,q))\langle l'+1,m'-q;1,q\mid l',m'\rangle] \end{align}\]
When \(m<0\), \(m'=0\), \[\begin{align} &M_{(l',0,1),(lm,2)}(\alpha)\\ =&\sum\limits_{\lambda\in\{l'+1,l'+3\}}\sum\limits_{q=-1}^{1}\sum\limits_{m_1=-1}^{1}\rho^{l+1} (a_{22}-a_{12}) r^\lambda i\cdot \sqrt{2(l'+1)(2l'+1)} [(\mathscr{A}_{l,\lambda}^{m,-q}(\alpha,q)\\ &-(-1)^m\mathscr{A}_{l,\lambda}^{-m,-q}(\alpha,q))K_{l'+1,l',\lambda}^{m_1,-q,q}]\\ &-\rho^{l+1} a_{22}r^{l'+1}i\cdot \dfrac{(2l+1)(l'+1)}{\sqrt{2}}[(\mathscr{H}_{l,l'}^{m,0}(\alpha)-(-1)^m\mathscr{H}_{l,l'}^{-m,0}(\alpha)]\\ &+\rho^{l+1} a_{22}r^{l'+1}i\cdot (2l+1)\dfrac{\sqrt{(l'+1)(2l'+1)}}{\sqrt{2}} \sum\limits_{q=-1}^1(-1)^q [ (\mathscr{A}_{l,l'+1}^{m,-q}(\alpha,q)-(-1)^m\mathscr{A}_{l,l'+1}^{-m,-q}(\alpha,q))\\ &\langle l'+1,-q;1,q\mid l',0\rangle] \end{align}\]
When \(m=0\), \(m'>0\), \[\begin{align} &M_{(l'm',1),(l,0,2)}(\alpha)\\ =&\sum\limits_{\lambda\in\{l'+1,l'+3\}}\sum\limits_{q=-1}^{1}\sum\limits_{m_1=-1}^{1}\rho^{l+1} (a_{22}-a_{12}) r^\lambda \sqrt{2(l'+1)(2l'+1)} [(-1)^{m'}\mathscr{A}_{l,\lambda}^{0,m'-q}(\alpha,q) K_{l'+1,l',\lambda}^{m_1,m'-q,q}\\ &+\mathscr{A}_{l,\lambda}^{0,-m'-q}(\alpha,q)K_{l'+1,l',\lambda}^{m_1,-m'-q,q}]\\ &-\rho^{l+1} a_{22}r^{l'+1}\dfrac{(2l+1)(l'+1)}{\sqrt{2}}[(-1)^{m'}\mathscr{H}_{l,l'}^{0,m'}(\alpha)+\mathscr{H}_{l,l'}^{0,-m'}(\alpha)]\\ &+\rho^{l+1} a_{22}r^{l'+1}(2l+1)\dfrac{\sqrt{(l'+1)(2l'+1)}}{\sqrt{2}} \sum\limits_{q=-1}^1(-1)^q [(-1)^{m'}\mathscr{A}_{l,l'+1}^{0,m'-q}(\alpha,q)\\ &\langle l'+1,m'-q;1,q\mid l',m'\rangle+(\mathscr{A}_{l,l'+1}^{0,-m'-q}(\alpha,q)\langle l'+1,-m'-q;1,q\mid l',-m'\rangle] \end{align}\]
When \(m=0\), \(m'<0\),
\[\begin{align} &M_{(l'm',1),(l,0,2)}(\alpha)\\ =&\sum\limits_{\lambda\in\{l'+1,l'+3\}}\sum\limits_{q=-1}^{1}\sum\limits_{m_1=-1}^{1}\rho^{l+1} (a_{22}-a_{12}) r^\lambda \sqrt{2(l'+1)(2l'+1)} [i\cdot (-1)^{m'}\mathscr{A}_{l,\lambda}^{0,-m'-q}(\alpha,q)K_{l'+1,l',\lambda}^{m_1,-m'-q,q}\\ & -i\cdot \mathscr{A}_{l,\lambda}^{0,m'-q}(\alpha,q)K_{l'+1,l',\lambda}^{m_1,m'-q,q}]\\ &-\rho^{l+1} a_{22}r^{l'+1}\dfrac{(2l+1)(l'+1)}{\sqrt{2}}[i\cdot (-1)^{m'}\mathscr{H}_{l,l'}^{0,-m'}(\alpha)-i\cdot \mathscr{H}_{l,l'}^{0,m'}(\alpha)]\\ &+\rho^{l+1} a_{22}r^{l'+1}(2l+1)\dfrac{\sqrt{(l'+1)(2l'+1)}}{\sqrt{2}} \sum\limits_{q=-1}^1(-1)^q [i\cdot (-1)^{m'} \mathscr{A}_{l,l'+1}^{0,-m'-q}(\alpha,q)\\ &\langle l'+1,-m'-q;1,q\mid l',-m'\rangle-i\cdot \mathscr{A}_{l,l'+1}^{0,m'-q}(\alpha,q)\langle l'+1,m'-q;1,q\mid l',m'\rangle] \end{align}\]
When \(m=0\), \(m'=0\), \[\begin{align} &M_{(l',0,1),(l,0,2)}(\alpha)\\ =&\sum\limits_{\lambda\in\{l'+1,l'+3\}}\sum\limits_{q=-1}^{1}\sum\limits_{m_1=-1}^{1}\rho^{l+1} (a_{22}-a_{12}) r^\lambda \sqrt{4(l'+1)(2l'+1)} [\mathscr{A}_{l,\lambda}^{0,-q}(\alpha,q)K_{l'+1,l',\lambda}^{m_1,-q,q}]\\ &-\rho^{l+1} a_{22}r^{l'+1}(2l+1)(l'+1)[\mathscr{H}_{l,l'}^{0,0}(\alpha)]\\ &+\rho^{l+1} a_{22}r^{l'+1}(2l+1)\sqrt{(l'+1)(2l'+1)} \sum\limits_{q=-1}^1(-1)^q [ \mathscr{A}_{l,l'+1}^{0,-q}(\alpha,q)\langle l'+1,-q;1,q\mid l',0\rangle] \end{align}\]
Theorem 20. Entries for \(\boxed{M_{(l'm',3),(lm,2)}(\alpha)}\)(\(l'\geqslant 1,l\geqslant 1\), otherwise this term vanishes):
When \(m>0\), \(m'>0\), \[\begin{align}
&M_{(l'm',3),(lm,2)}(\alpha)\\
=&\sum\limits_{\lambda\in\{l',l'+2\mid l'\geqslant 1\}}\sum\limits_{q=-1}^{1}\sum\limits_{m_1=-1}^{1}-i\cdot \rho^{l+1} (a_{22}-a_{12}) r^\lambda \sqrt{l'(l'+1)}
[(-1)^{m'}(\mathscr{A}_{l,\lambda}^{-m,m'-q}(\alpha,q)\\
&+(-1)^m\mathscr{A}_{l,\lambda}^{m,m'-q}(\alpha,q))K_{l',l',\lambda}^{m_1,m'-q,q}
+(\mathscr{A}_{l,\lambda}^{-m,-m'-q}(\alpha,q)+(-1)^m\mathscr{A}_{l,\lambda}^{m,-m'-q}(\alpha,q))K_{l',l',\lambda}^{m_1,-m'-q,q}]\\
&- i\cdot \rho^{l+1} a_{22}r^{l'}(2l+1)\dfrac{\sqrt{l'(l'+1)}}{2} \sum\limits_{q=-1}^1(-1)^q [(-1)^{m'}(\mathscr{A}_{l,l'}^{-m,m'-q}(\alpha,q)+(-1)^m\mathscr{A}_{l,l'}^{m,m'-q}(\alpha,q))\\
&\langle l',m'-q;1,q\mid l',m'\rangle+(\mathscr{A}_{l,l'}^{-m,-m'-q}(\alpha,q)+(-1)^m\mathscr{A}_{l,l'}^{m,-m'-q}(\alpha,q))\langle l',-m'-q;1,q\mid l',-m'\rangle]
\end{align}\]
When \(m>0\), \(m'<0\), \[\begin{align} &M_{(l'm',3),(lm,2)}(\alpha)\\ =&\sum\limits_{\lambda\in\{l',l'+2\mid l'\geqslant 1\}}\sum\limits_{q=-1}^{1}\sum\limits_{m_1=-1}^{1}\rho^{l+1} (a_{22}-a_{12}) r^\lambda \sqrt{l'(l'+1)} [ (-1)^{m'}(\mathscr{A}_{l,\lambda}^{-m,-m'-q}(\alpha,q)\\ &+(-1)^m\mathscr{A}_{l,\lambda}^{m,-m'-q}(\alpha,q))K_{l',l',\lambda}^{m_1,-m'-q,q} - (\mathscr{A}_{l,\lambda}^{-m,m'-q}(\alpha,q)+(-1)^m\mathscr{A}_{l,\lambda}^{m,m'-q}(\alpha,q))K_{l',l',\lambda}^{m_1,m'-q,q}]\\ &+\rho^{l+1} a_{22}r^{l'}(2l+1)\dfrac{\sqrt{l'(l'+1)}}{2} \sum\limits_{q=-1}^1(-1)^q [ (-1)^{m'}(\mathscr{A}_{l,l'}^{-m,-m'-q}(\alpha,q)+(-1)^m\mathscr{A}_{l,l'}^{m,-m'-q}(\alpha,q))\\ &\langle l',-m'-q;1,q\mid l',-m'\rangle - (\mathscr{A}_{l,l'}^{-m,m'-q}(\alpha,q)+(-1)^m\mathscr{A}_{l,l'}^{m,m'-q}(\alpha,q))\langle l',m'-q;1,q\mid l',m'\rangle]\end{align}\]
When \(m>0\), \(m'=0\), \[\begin{align} &M_{(l',0,3),(lm,2)}(\alpha)\\ =&\sum\limits_{\lambda\in\{l',l'+2\mid l'\geqslant 1\}}\sum\limits_{q=-1}^{1}\sum\limits_{m_1=-1}^{1} -i\cdot \rho^{l+1} (a_{22}-a_{12}) r^\lambda \sqrt{2l'(l'+1)} [(\mathscr{A}_{l,\lambda}^{-m,-q}(\alpha,q)+\\&(-1)^m\mathscr{A}_{l,\lambda}^{m,-q}(\alpha,q))K_{l',l',\lambda}^{m_1,-q,q}]\\ &-i\cdot \rho^{l+1} a_{22}r^{l'}(2l+1)\dfrac{\sqrt{l'(l'+1)}}{\sqrt{2}} \sum\limits_{q=-1}^1(-1)^q [ (\mathscr{A}_{l,l'}^{-m,-q}(\alpha,q)+(-1)^m\mathscr{A}_{l,l'}^{m,-q}(\alpha,q))\\ &\langle l',-q;1,q\mid l',0\rangle] \end{align}\]
When \(m<0\), \(m'>0\), \[\begin{align} &M_{(l'm',3),(lm,2)}(\alpha)\\ =&\sum\limits_{\lambda\in\{l',l'+2\mid l'\geqslant 1\}}\sum\limits_{q=-1}^{1}\sum\limits_{m_1=-1}^{1}\rho^{l+1} (a_{22}-a_{12}) r^\lambda \sqrt{l'(l'+1)} [(-1)^{m'}(\mathscr{A}_{l,\lambda}^{m,m'-q}(\alpha,q)\\ &-(-1)^m\mathscr{A}_{l,\lambda}^{-m,m'-q}(\alpha,q))K_{l',l',\lambda}^{m_1,m'-q,q} +(\mathscr{A}_{l,\lambda}^{m,-m'-q}(\alpha,q)-(-1)^m\mathscr{A}_{l,\lambda}^{-m,-m'-q}(\alpha,q))K_{l',l',\lambda}^{m_1,-m'-q,q}]\\ &+ \rho^{l+1} a_{22}r^{l'} (2l+1)\dfrac{\sqrt{l'(l'+1)}}{2} \sum\limits_{q=-1}^1(-1)^q [(-1)^{m'}(\mathscr{A}_{l,l'}^{m,m'-q}(\alpha,q)-(-1)^m\mathscr{A}_{l,l'}^{-m,m'-q}(\alpha,q))\\ &\langle l',m'-q;1,q\mid l',m'\rangle+(\mathscr{A}_{l,l'}^{m,-m'-q}(\alpha,q)-(-1)^m\mathscr{A}_{l,l'}^{-m,-m'-q}(\alpha,q))\langle l',-m'-q;1,q\mid l',-m'\rangle] \end{align}\]
When \(m<0\), \(m'<0\), \[\begin{align} &M_{(l'm',3),(lm,2)}(\alpha)\\ =&\sum\limits_{\lambda\in\{l',l'+2\mid l'\geqslant 1\}}\sum\limits_{q=-1}^{1}\sum\limits_{m_1=-1}^{1}\rho^{l+1} (a_{22}-a_{12}) r^\lambda i\cdot \sqrt{l'(l'+1)} [(-1)^{m'}(\mathscr{A}_{l,\lambda}^{m,-m'-q}(\alpha,q)\\ &-(-1)^m\mathscr{A}_{l,\lambda}^{-m,-m'-q}(\alpha,q))K_{l',l',\lambda}^{m_1,-m'-q,q} - (\mathscr{A}_{l,\lambda}^{m,m'-q}(\alpha,q)-(-1)^m\mathscr{A}_{l,\lambda}^{-m,m'-q}(\alpha,q))K_{l',l',\lambda}^{m_1,m'-q,q}]\\ &-i\cdot \rho^{l+1} a_{22}r^{l'}(2l+1)\dfrac{\sqrt{l'(l'+1)}}{2} \sum\limits_{q=-1}^1(-1)^q [-(-1)^{m'}(\mathscr{A}_{l,l'}^{m,-m'-q}(\alpha,q)-(-1)^m\mathscr{A}_{l,l'}^{-m,-m'-q}(\alpha,q))\\ &\langle l',-m'-q;1,q\mid l',-m'\rangle + (\mathscr{A}_{l,l'}^{m,m'-q}(\alpha,q)-(-1)^m\mathscr{A}_{l,l'}^{-m,m'-q}(\alpha,q))\langle l',m'-q;1,q\mid l',m'\rangle] \end{align}\]
When \(m<0\), \(m'=0\), \[\begin{align} &M_{(l',0,3),(lm,2)}(\alpha)\\ =&\sum\limits_{\lambda\in\{l',l'+2\mid l'\geqslant 1\}}\sum\limits_{q=-1}^{1}\sum\limits_{m_1=-1}^{1} \rho^{l+1} (a_{22}-a_{12}) r^\lambda \sqrt{2l'(l'+1)} [(\mathscr{A}_{l,\lambda}^{m,-q}(\alpha,q)-(-1)^m\mathscr{A}_{l,\lambda}^{-m,-q}(\alpha,q))K_{l',l',\lambda}^{m_1,-q,q}]\\ &+\rho^{l+1} a_{22}r^{l'} (2l+1)\dfrac{\sqrt{l'(l'+1)}}{\sqrt{2}} \sum\limits_{q=-1}^1(-1)^q [ (\mathscr{A}_{l,l'}^{m,-q}(\alpha,q)-(-1)^m\mathscr{A}_{l,l'}^{-m,-q}(\alpha,q))\\ &\langle l',-q;1,q\mid l',0\rangle] \end{align}\]
When \(m=0\), \(m'>0\), \[\begin{align} &M_{(l'm',3),(l,0,2)}(\alpha)\\ =&\sum\limits_{\lambda\in\{l',l'+2\mid l'\geqslant 1\}}\sum\limits_{q=-1}^{1}\sum\limits_{m_1=-1}^{1} -i\cdot \rho^{l+1} (a_{22}-a_{12}) r^\lambda \sqrt{2l'(l'+1)} [(-1)^{m'}\mathscr{A}_{l,\lambda}^{0,m'-q}(\alpha,q) K_{l',l',\lambda}^{m_1,m'-q,q}\\ &+\mathscr{A}_{l,\lambda}^{0,-m'-q}(\alpha,q)K_{l',l',\lambda}^{m_1,-m'-q,q}]\\ &-i\cdot \rho^{l+1} a_{22}r^{l'}(2l+1)\dfrac{\sqrt{l'(l'+1)}}{\sqrt{2}} \sum\limits_{q=-1}^1(-1)^q [(-1)^{m'}\mathscr{A}_{l,l'}^{0,m'-q}(\alpha,q)\\ &\langle l',m'-q;1,q\mid l',m'\rangle+(\mathscr{A}_{l,l'}^{0,-m'-q}(\alpha,q)\langle l',-m'-q;1,q\mid l',-m'\rangle] \end{align}\]
When \(m=0\), \(m'<0\),
\[\begin{align} &M_{(l'm',3),(l,0,2)}(\alpha)\\ =&\sum\limits_{\lambda\in\{l',l'+2\mid l'\geqslant 1\}}\sum\limits_{q=-1}^{1}\sum\limits_{m_1=-1}^{1}\rho^{l+1} (a_{22}-a_{12}) r^\lambda \sqrt{2l'(l'+1)} [(-1)^{m'}\mathscr{A}_{l,\lambda}^{0,-m'-q}(\alpha,q)K_{l',l',\lambda}^{m_1,-m'-q,q}\\ & - \mathscr{A}_{l,\lambda}^{0,m'-q}(\alpha,q)K_{l',l',\lambda}^{m_1,m'-q,q}]\\ &+\rho^{l+1} a_{22}r^{l'}(2l+1)\dfrac{\sqrt{l'(l'+1)}}{\sqrt{2}} \sum\limits_{q=-1}^1(-1)^q [ (-1)^{m'} \mathscr{A}_{l,l'}^{0,-m'-q}(\alpha,q)\\ &\langle l',-m'-q;1,q\mid l',-m'\rangle- \mathscr{A}_{l,l'}^{0,m'-q}(\alpha,q)\langle l',m'-q;1,q\mid l',m'\rangle] \end{align}\]
When \(m=0\), \(m'=0\), \[\begin{align} &M_{(l',0,3),(l,0,2)}(\alpha)\\ =&\sum\limits_{\lambda\in\{l',l'+2\mid l'\geqslant 1\}}\sum\limits_{q=-1}^{1}\sum\limits_{m_1=-1}^{1} -i\cdot \rho^{l+1} (a_{22}-a_{12}) r^\lambda \sqrt{4l'(l'+1)} [\mathscr{A}_{l,\lambda}^{0,-q}(\alpha,q)K_{l',l',\lambda}^{m_1,-q,q}]\\ &-i\cdot \rho^{l+1} a_{22}r^{l'}(2l+1)\sqrt{l'(l'+1)} \sum\limits_{q=-1}^1(-1)^q [ \mathscr{A}_{l,l'}^{0,-q}(\alpha,q)\langle l',-q;1,q\mid l',0\rangle] \end{align}\]
Theorem 21. Entries for \(\boxed{M_{(l'm',3),(lm,3)}(\alpha)}\)(\(l'\geqslant 1,l\geqslant 1\), otherwise this term vanishes):
When \(m>0,\;m'>0\): \[\begin{align}
&M_{(l'm',3),(lm,3)}(\alpha)\\
=&\;\rho^{l+2}a_{33}r^{l'}\Biggl\{
\frac{l'(l'+1)}{2}\Bigl[(-1)^{m'}\bigl(\mathscr{H}_{l,l'}^{-m,m'}(\alpha)+(-1)^m \mathscr{H}_{l,l'}^{m,m'}(\alpha)\bigr)
+\bigl(\mathscr{H}_{l,l'}^{-m,-m'}(\alpha)+(-1)^m \mathscr{H}_{l,l'}^{m,-m'}(\alpha)\bigr)\Bigr]\\
&-\frac{i\sqrt{l'(l'+1)}}{2}\sum_{q=-1}^{1}\sum_{m_1=-1}^{1}\Bigl[(-1)^{m'}\bigl(\mathscr{L}_{l,l',l'+1}^{-m,m'-q,q,m_1}(\alpha)+(-1)^m \mathscr{L}_{l,l',l'+1}^{m,m'-q,q,m_1}(\alpha)\bigr)\\
&\qquad\qquad\qquad\qquad\quad+\bigl(\mathscr{L}_{l,l',l'+1}^{-m,-m'-q,q,m_1}(\alpha)+(-1)^m \mathscr{L}_{l,l',l'+1}^{m,-m'-q,q,m_1}(\alpha)\bigr)\Bigr]\Biggr\}\\
&+\delta_{ll'}\delta_{mm'}\rho\dfrac{l(l+1)}{(2l+1)\mu} \end{align}\]
When \(m>0,\;m'<0\): \[\begin{align} &M_{(l'm',3),(lm,3)}(\alpha)\\ =&\;\rho^{l+2}a_{33}r^{l'}\Biggl\{ \frac{l'(l'+1)}{2}\Bigl[i(-1)^{m'}\bigl(\mathscr{H}_{l,l'}^{-m,-m'}(\alpha)+(-1)^m \mathscr{H}_{l,l'}^{m,-m'}(\alpha)\bigr) -i\bigl(\mathscr{H}_{l,l'}^{-m,m'}(\alpha)\\ &+(-1)^m \mathscr{H}_{l,l'}^{m,m'}(\alpha)\bigr)\Bigr]\\ &+\frac{\sqrt{l'(l'+1)}}{2}\sum_{q=-1}^{1}\sum_{m_1=-1}^{1}\Bigl[(-1)^{m'}\bigl(\mathscr{L}_{l,l',l'+1}^{-m,-m'-q,q,m_1}(\alpha)+(-1)^m \mathscr{L}_{l,l',l'+1}^{m,-m'-q,q,m_1}(\alpha)\bigr)\\ &\qquad\qquad\qquad\qquad\quad-\bigl(\mathscr{L}_{l,l',l'+1}^{-m,m'-q,q,m_1}(\alpha)+(-1)^m \mathscr{L}_{l,l',l'+1}^{m,m'-q,q,m_1}(\alpha)\bigr)\Bigr]\Biggr\}\\ &+\delta_{ll'}\delta_{mm'}\rho\dfrac{l(l+1)}{(2l+1)\mu} \end{align}\]
When \(m>0,\;m'=0\): \[\begin{align} &M_{(l',0,3),(lm,3)}(\alpha)\\ =&\;\rho^{l+2}a_{33}r^{l'}\Biggl\{ \frac{l'(l'+1)}{\sqrt{2}}\bigl(\mathscr{H}_{l,l'}^{-m,0}(\alpha)+(-1)^m \mathscr{H}_{l,l'}^{m,0}(\alpha)\bigr)\\ &-\frac{i\sqrt{l'(l'+1)}}{\sqrt{2}}\sum_{q=-1}^{1}\sum_{m_1=-1}^{1}\bigl(\mathscr{L}_{l,l',l'+1}^{-m,-q,q,m_1}(\alpha)+(-1)^m \mathscr{L}_{l,l',l'+1}^{m,-q,q,m_1}(\alpha)\bigr)\Biggr\}\\ &+\delta_{ll'}\delta_{mm'}\rho\dfrac{l(l+1)}{(2l+1)\mu} \end{align}\]
When \(m<0,\;m'>0\): \[\begin{align} &M_{(l'm',3),(lm,3)}(\alpha)\\ =&\;\rho^{l+2}a_{33}r^{l'}\Biggl\{ \frac{l'(l'+1)}{2}\Bigl[i(-1)^{m'}\bigl(\mathscr{H}_{l,l'}^{m,m'}(\alpha)-(-1)^m \mathscr{H}_{l,l'}^{-m,m'}(\alpha)\bigr)\\ & +i\bigl(\mathscr{H}_{l,l'}^{m,-m'}(\alpha)-(-1)^m \mathscr{H}_{l,l'}^{-m,-m'}(\alpha)\bigr)\Bigr]\\ &+\frac{\sqrt{l'(l'+1)}}{2}\sum_{q=-1}^{1}\sum_{m_1=-1}^{1}\Bigl[(-1)^{m'}\bigl(\mathscr{L}_{l,l',l'+1}^{m,m'-q,q,m_1}(\alpha)-(-1)^m \mathscr{L}_{l,l',l'+1}^{-m,m'-q,q,m_1}(\alpha)\bigr)\\ &\qquad\qquad\qquad\qquad\quad+\bigl(\mathscr{L}_{l,l',l'+1}^{m,-m'-q,q,m_1}(\alpha)-(-1)^m \mathscr{L}_{l,l',l'+1}^{-m,-m'-q,q,m_1}(\alpha)\bigr)\Bigr]\Biggr\}\\ &+\delta_{ll'}\delta_{mm'}\rho\dfrac{l(l+1)}{(2l+1)\mu}\end{align}\]
When \(m<0,\;m'<0\): \[\begin{align} &M_{(l'm',3),(lm,3)}(\alpha)\\ =&\;\rho^{l+2}a_{33}r^{l'}\Biggl\{ \frac{l'(l'+1)}{2}\Bigl[\bigl(\mathscr{H}_{l,l'}^{m,m'}(\alpha)-(-1)^m \mathscr{H}_{l,l'}^{-m,m'}(\alpha)\bigr) -(-1)^{m'}\bigl(\mathscr{H}_{l,l'}^{m,-m'}(\alpha)-(-1)^m \mathscr{H}_{l,l'}^{-m,-m'}(\alpha)\bigr)\Bigr]\\ &+\frac{i\sqrt{l'(l'+1)}}{2}\sum_{q=-1}^{1}\sum_{m_1=-1}^{1}\Bigl[(-1)^{m'}\bigl(\mathscr{L}_{l,l',l'+1}^{m,-m'-q,q,m_1}(\alpha)-(-1)^m \mathscr{L}_{l,l',l'+1}^{-m,-m'-q,q,m_1}(\alpha)\bigr)\\ &\qquad\qquad\qquad\qquad\quad-\bigl(\mathscr{L}_{l,l',l'+1}^{m,m'-q,q,m_1}(\alpha)-(-1)^m \mathscr{L}_{l,l',l'+1}^{-m,m'-q,q,m_1}(\alpha)\bigr)\Bigr]\Biggr\}\\ &+\delta_{ll'}\delta_{mm'}\rho\dfrac{l(l+1)}{(2l+1)\mu}\end{align}\]
When \(m<0,\;m'=0\): \[\begin{align} &M_{(l',0,3),(lm,3)}(\alpha)\\ =&\;\rho^{l+2}a_{33}r^{l'}\Biggl\{ \frac{il'(l'+1)}{\sqrt{2}}\bigl(\mathscr{H}_{l,l'}^{m,0}(\alpha)-(-1)^m \mathscr{H}_{l,l'}^{-m,0}(\alpha)\bigr)\\ &+\frac{\sqrt{l'(l'+1)}}{\sqrt{2}}\sum_{q=-1}^{1}\sum_{m_1=-1}^{1}\bigl(\mathscr{L}_{l,l',l'+1}^{m,-q,q,m_1}(\alpha)-(-1)^m \mathscr{L}_{l,l',l'+1}^{-m,-q,q,m_1}(\alpha)\bigr)\Biggr\}\\ &+\delta_{ll'}\delta_{mm'}\rho\dfrac{l(l+1)}{(2l+1)\mu} \end{align}\]
When \(m=0,\;m'>0\): \[\begin{align} &M_{(l'm',3),(l,0,3)}(\alpha)\\ =&\;\rho^{l+2}a_{33}r^{l'}\Biggl\{ \frac{l'(l'+1)}{\sqrt{2}}\Bigl[(-1)^{m'}\mathscr{H}_{l,l'}^{0,m'}(\alpha)+\mathscr{H}_{l,l'}^{0,-m'}(\alpha)\Bigr]\\ &-\frac{i\sqrt{l'(l'+1)}}{\sqrt{2}}\sum_{q=-1}^{1}\sum_{m_1=-1}^{1}\Bigl[(-1)^{m'}\mathscr{L}_{l,l',l'+1}^{0,m'-q,q,m_1}(\alpha)+\mathscr{L}_{l,l',l'+1}^{0,-m'-q,q,m_1}(\alpha)\Bigr]\Biggr\}\\ &+\delta_{ll'}\delta_{mm'}\rho\dfrac{l(l+1)}{(2l+1)\mu}\end{align}\]
When \(m=0,\;m'<0\): \[\begin{align} &M_{(l'm',3),(l,0,3)}(\alpha)\\ =&\;\rho^{l+2}a_{33}r^{l'}\Biggl\{ \frac{l'(l'+1)}{\sqrt{2}}\Bigl[i(-1)^{m'}\mathscr{H}_{l,l'}^{0,-m'}(\alpha)-i\mathscr{H}_{l,l'}^{0,m'}(\alpha)\Bigr]\\ &+\frac{\sqrt{l'(l'+1)}}{\sqrt{2}}\sum_{q=-1}^{1}\sum_{m_1=-1}^{1}\Bigl[(-1)^{m'}\mathscr{L}_{l,l',l'+1}^{0,-m'-q,q,m_1}(\alpha)-\mathscr{L}_{l,l',l'+1}^{0,m'-q,q,m_1}(\alpha)\Bigr]\Biggr\}\\ &+\delta_{ll'}\delta_{mm'}\rho\dfrac{l(l+1)}{(2l+1)\mu}\end{align}\]
When \(m=0,\;m'=0\): \[\begin{align} &M_{(l',0,3),(l,0,3)}(\alpha)\\ =&\;\rho^{l+2}a_{33}r^{l'}\Biggl\{ l'(l'+1)\mathscr{H}_{l,l'}^{0,0}(\alpha) -i\sqrt{l'(l'+1)}\sum_{q=-1}^{1}\sum_{m_1=-1}^{1}\mathscr{L}_{l,l',l'+1}^{0,-q,q,m_1}(\alpha)\Biggr\}\\ &+\delta_{ll'}\delta_{mm'}\rho\dfrac{l(l+1)}{(2l+1)\mu}\end{align}\]
Lemma 11. Define \(\mathscr{D}_{l,\lambda}^{m,\mu}(\alpha)=\sum\limits_{n\neq 0} n^2H_{l,\lambda}^{m,\mu}(\mathbf{b})e^{-in\alpha}\). Then \[\begin{align} &\mathscr{D}_{l,\lambda}^{m,\mu}(\alpha)\\ =&(-1)^{\lambda+\mu} \sqrt{\dfrac{2l+1}{2\lambda+1}} \sqrt{{l+\lambda+\mu-m\choose \lambda+\mu}{l+\lambda+m-\mu\choose \lambda-\mu}} \sqrt{\dfrac{4\pi}{2(l+\lambda)+1}}\\ & \big(\mathrm{Li}_{l+\lambda-1}(e^{-i\alpha})Y_{l+\lambda}^{m-\mu}(\dfrac{\pi}{2},\pi)+\mathrm{Li}_{l+\lambda-1}(e^{i\alpha})Y_{l+\lambda}^{m-\mu}(\dfrac{\pi}{2},0)\big) \end{align}\]
Theorem 22. The entries for \(\boxed{M_{(l'm',2),(lm,2)}(\alpha)}\)(\(l'\geqslant 1,l\geqslant 1\),otherwise this term vanishes) are given by:
If \(m>0\) and \(m'>0\), \[\begin{align}&M_{(l'm',2),(lm,2)}(\alpha)\\
&=\dfrac{l'(2l'+1)}{2}[ \rho^{l+3}a_{12}r^{l'-1}+\rho^{l+1}(a_{22}-a_{12})(r^{l'+1} )+\rho^{l+1}a_{22}r^{l'+1}\dfrac{2l+1}{2l'+1}]\\
&[(-1)^{m'}(\mathscr{H}_{l,l'}^{-m,m'}(\alpha)+(-1)^m\mathscr{H}_{l,l'}^{m,m'}(\alpha))+(\mathscr{H}_{l,l'}^{-m,-m'}(\alpha)+(-1)^m\mathscr{H}_{l,l'}^{m,-m'}(\alpha))]\\
&+\dfrac{l'(2l'+1)}{2}[\rho^{l+1}(a_{22}-a_{12})r^{l'-1}][(-1)^{m'}(\mathscr{D}_{l,l'}^{-m,m'}(\alpha)+\\
&(-1)^m\mathscr{D}_{l,l'}^{m,m'}(\alpha))+(\mathscr{D}_{l,l'}^{-m,-m'}(\alpha)+(-1)^m\mathscr{D}_{l,l'}^{m,-m'}(\alpha))]\\
&+\rho^{l+1}(a_{22}-a_{12})\sqrt{l'(2l'+1)}\sum\limits_{\lambda\in\{l'+1,l'-1\}}\sum\limits_{q=-1}^{q=1}\sum\limits_{m_1=-1}^{1}r^\lambda [(-1)^{m'}(\mathscr{A}_{l,\lambda}^{-m,m'-q}(\alpha,q)\\
&+(-1)^m\mathscr{A}_{l,\lambda}^{m,m'-q}(\alpha,q))K_{l'-1,l',\lambda}^{m_1,m'-q,q}+(\mathscr{A}_{l,\lambda}^{-m,-m'-q}(\alpha,q)+(-1)^m\mathscr{A}_{l,\lambda}^{m,-m'-q}(\alpha,q))K_{l'-1,l',\lambda}^{m_1,-m'-q,q}]\\
&+\rho^{l+1}a_{22}(2l+1)\dfrac{\sqrt{l'(2l'+1)}}{2}\sum\limits_{q=-1}^1(-1)^q r^{l'-1}[(-1)^{m'}(\mathscr{A}_{l,l'-1}^{-m,m'-q}(\alpha,q)+(-1)^m\mathscr{A}_{l,l'-1}^{m,m'-q}(\alpha,q))\\
&\langle l'-1,m'-q;1,q\mid l',m' \rangle+(\mathscr{A}_{l,l'-1}^{-m,-m'-q}(\alpha,q)+(-1)^m\mathscr{A}_{l,l'-1}^{m,-m'-q}(\alpha,q))\langle l'-1,-m'-q;1,q\mid l',-m' \rangle]\\
&+\delta_{ll'}\delta_{mm'}l(2l+1)\rho a_{22}.
\end{align}\]
If \(m>0\) and \(m'<0\), \[\begin{align} &M_{(l'm',2),(lm,2)}(\alpha)\\ &=\dfrac{l'(2l'+1)}{2}\Bigl[ \rho^{l+3}a_{12}r^{l'-1}+\rho^{l+1}(a_{22}-a_{12})(r^{l'+1} )+\rho^{l+1}a_{22}r^{l'+1}\dfrac{2l+1}{2l'+1}\Bigr]\\ &\quad\Bigl[i\cdot(-1)^{m'}(\mathscr{H}_{l,l'}^{-m,-m'}(\alpha)+(-1)^m\mathscr{H}_{l,l'}^{m,-m'}(\alpha))-i\cdot(\mathscr{H}_{l,l'}^{-m,m'}(\alpha)+(-1)^m\mathscr{H}_{l,l'}^{m,m'}(\alpha))\Bigr]\\ &+\dfrac{l'(2l'+1)}{2}[\rho^{l+1}(a_{22}-a_{12})r^{l'-1}]\Bigl[i\cdot(-1)^{m'}(\mathscr{D}_{l,l'}^{-m,-m'}(\alpha)+(-1)^m\mathscr{D}_{l,l'}^{m,-m'}(\alpha))\\ &-i\cdot(\mathscr{D}_{l,l'}^{-m,m'}(\alpha)+(-1)^m\mathscr{D}_{l,l'}^{m,m'}(\alpha))\Bigr]\\ &+\rho^{l+1}(a_{22}-a_{12})\sqrt{l'(2l'+1)}\sum_{\lambda\in\{l'+1,l'-1\}}\sum_{q=-1}^{1}\sum_{m_1=-1}^{1}r^\lambda \Bigl[i\cdot(-1)^{m'}(\mathscr{A}_{l,\lambda}^{-m,-m'-q}(\alpha,q)\\ &\quad+(-1)^m\mathscr{A}_{l,\lambda}^{m,-m'-q}(\alpha,q))K_{l'-1,l',\lambda}^{m_1,-m'-q,q}-i\cdot(\mathscr{A}_{l,\lambda}^{-m,m'-q}(\alpha,q)+(-1)^m\mathscr{A}_{l,\lambda}^{m,m'-q}(\alpha,q))K_{l'-1,l',\lambda}^{m_1,m'-q,q}\Bigr]\\ &+\rho^{l+1}a_{22}(2l+1)\dfrac{\sqrt{l'(2l'+1)}}{2}\sum_{q=-1}^{1}(-1)^q r^{l'-1}\Bigl[i\cdot(-1)^{m'}(\mathscr{A}_{l,l'-1}^{-m,-m'-q}(\alpha,q)+(-1)^m\mathscr{A}_{l,l'-1}^{m,-m'-q}(\alpha,q))\\ &\quad\langle l'-1,-m'-q;1,q\mid l',-m' \rangle-i\cdot(\mathscr{A}_{l,l'-1}^{-m,m'-q}(\alpha,q)+\\ &(-1)^m\mathscr{A}_{l,l'-1}^{m,m'-q}(\alpha,q))\langle l'-1,m'-q;1,q\mid l',m' \rangle\Bigr]\\ &+\delta_{ll'}\delta_{mm'}l(2l+1)\rho a_{22}. \end{align}\]
If \(m>0\) and \(m'=0\), \[\begin{align} &M_{(l',0,2),(lm,2)}(\alpha)\\ &=\dfrac{l'(2l'+1)}{\sqrt{2}}\Bigl[ \rho^{l+3}a_{12}r^{l'-1}+\rho^{l+1}(a_{22}-a_{12})(r^{l'+1} )+\rho^{l+1}a_{22}r^{l'+1}\dfrac{2l+1}{2l'+1}\Bigr]\\ &\quad\Bigl[(\mathscr{H}_{l,l'}^{-m,0}(\alpha)+(-1)^m\mathscr{H}_{l,l'}^{m,0}(\alpha))\Bigr]\\ &+\dfrac{l'(2l'+1)}{\sqrt{2}}[\rho^{l+1}(a_{22}-a_{12})r^{l'-1}]\Bigl[(\mathscr{D}_{l,l'}^{-m,0}(\alpha)+(-1)^m\mathscr{D}_{l,l'}^{m,0}(\alpha))\Bigr]\\ &+\rho^{l+1}(a_{22}-a_{12})\sqrt{2l'(2l'+1)}\sum_{\lambda\in\{l'+1,l'-1\}}\sum_{q=-1}^{1}\sum_{m_1=-1}^{1}r^\lambda \Bigl[(\mathscr{A}_{l,\lambda}^{-m,-q}(\alpha,q)+(-1)^m\mathscr{A}_{l,\lambda}^{m,-q}(\alpha,q))K_{l'-1,l',\lambda}^{m_1,-q,q}\Bigr]\\ &+\rho^{l+1}a_{22}(2l+1)\dfrac{\sqrt{l'(2l'+1)}}{\sqrt{2}}\sum_{q=-1}^{1}(-1)^q r^{l'-1}\Bigl[(\mathscr{A}_{l,l'-1}^{-m,-q}(\alpha,q)+\\ &(-1)^m\mathscr{A}_{l,l'-1}^{m,-q}(\alpha,q))\langle l'-1,-q;1,q\mid l',0 \rangle\Bigr]\\ &+\delta_{ll'}\delta_{mm'}l(2l+1)\rho a_{22}. \end{align}\]
If \(m<0\) and \(m'>0\), \[\begin{align} &M_{(l'm',2),(lm,2)}(\alpha)\\ &=\dfrac{l'(2l'+1)}{2}\Bigl[ \rho^{l+3}a_{12}r^{l'-1}+\rho^{l+1}(a_{22}-a_{12})(r^{l'+1} )+\rho^{l+1}a_{22}r^{l'+1}\dfrac{2l+1}{2l'+1}\Bigr]\\ &\quad\cdot i\Bigl[(-1)^{m'}(\mathscr{H}_{l,l'}^{m,m'}(\alpha)-(-1)^m\mathscr{H}_{l,l'}^{-m,m'}(\alpha))+(\mathscr{H}_{l,l'}^{m,-m'}(\alpha)-(-1)^m\mathscr{H}_{l,l'}^{-m,-m'}(\alpha))\Bigr]\\ &+\dfrac{l'(2l'+1)}{2}[\rho^{l+1}(a_{22}-a_{12})r^{l'-1}]\cdot i\Bigl[(-1)^{m'}(\mathscr{D}_{l,l'}^{m,m'}(\alpha)-(-1)^m\mathscr{D}_{l,l'}^{-m,m'}(\alpha))\\ &+(\mathscr{D}_{l,l'}^{m,-m'}(\alpha)-(-1)^m\mathscr{D}_{l,l'}^{-m,-m'}(\alpha))\Bigr]\\ &+\rho^{l+1}(a_{22}-a_{12})\sqrt{l'(2l'+1)}\sum_{\lambda\in\{l'+1,l'-1\}}\sum_{q=-1}^{1}\sum_{m_1=-1}^{1}r^\lambda \;i\Bigl[(-1)^{m'}(\mathscr{A}_{l,\lambda}^{m,m'-q}(\alpha,q)\\ &\quad-(-1)^m\mathscr{A}_{l,\lambda}^{-m,m'-q}(\alpha,q))K_{l'-1,l',\lambda}^{m_1,m'-q,q}+(\mathscr{A}_{l,\lambda}^{m,-m'-q}(\alpha,q)-(-1)^m\mathscr{A}_{l,\lambda}^{-m,-m'-q}(\alpha,q))K_{l'-1,l',\lambda}^{m_1,-m'-q,q}\Bigr]\\ &+\rho^{l+1}a_{22}(2l+1)\dfrac{\sqrt{l'(2l'+1)}}{2}\sum_{q=-1}^{1}(-1)^q r^{l'-1}\;i\Bigl[(-1)^{m'}(\mathscr{A}_{l,l'-1}^{m,m'-q}(\alpha,q)-(-1)^m\mathscr{A}_{l,l'-1}^{-m,m'-q}(\alpha,q))\\ &\quad\langle l'-1,m'-q;1,q\mid l',m' \rangle+(\mathscr{A}_{l,l'-1}^{m,-m'-q}(\alpha,q)\\ &-(-1)^m\mathscr{A}_{l,l'-1}^{-m,-m'-q}(\alpha,q))\langle l'-1,-m'-q;1,q\mid l',-m' \rangle\Bigr]\\ &+\delta_{ll'}\delta_{mm'}l(2l+1)\rho a_{22}. \end{align}\]
If \(m<0\) and \(m'<0\), \[\begin{align} &M_{(l'm',2),(lm,2)}(\alpha)\\ &=\dfrac{l'(2l'+1)}{2}\Bigl[ \rho^{l+3}a_{12}r^{l'-1}+\rho^{l+1}(a_{22}-a_{12})(r^{l'+1} )+\rho^{l+1}a_{22}r^{l'+1}\dfrac{2l+1}{2l'+1}\Bigr]\\ &\quad\Bigl[-(-1)^{m'}(\mathscr{H}_{l,l'}^{m,-m'}(\alpha)-(-1)^m\mathscr{H}_{l,l'}^{-m,-m'}(\alpha))+(\mathscr{H}_{l,l'}^{m,m'}(\alpha)-(-1)^m\mathscr{H}_{l,l'}^{-m,m'}(\alpha))\Bigr]\\ &+\dfrac{l'(2l'+1)}{2}[\rho^{l+1}(a_{22}-a_{12})r^{l'-1}]\Bigl[-(-1)^{m'}(\mathscr{D}_{l,l'}^{m,-m'}(\alpha)-(-1)^m\mathscr{D}_{l,l'}^{-m,-m'}(\alpha))\\ &+(\mathscr{D}_{l,l'}^{m,m'}(\alpha)-(-1)^m\mathscr{D}_{l,l'}^{-m,m'}(\alpha))\Bigr]\\ &+\rho^{l+1}(a_{22}-a_{12})\sqrt{l'(2l'+1)}\sum_{\lambda\in\{l'+1,l'-1\}}\sum_{q=-1}^{1}\sum_{m_1=-1}^{1}r^\lambda \Bigl[-(-1)^{m'}(\mathscr{A}_{l,\lambda}^{m,-m'-q}(\alpha,q)\\ &\quad-(-1)^m\mathscr{A}_{l,\lambda}^{-m,-m'-q}(\alpha,q))K_{l'-1,l',\lambda}^{m_1,-m'-q,q}+(\mathscr{A}_{l,\lambda}^{m,m'-q}(\alpha,q)-(-1)^m\mathscr{A}_{l,\lambda}^{-m,m'-q}(\alpha,q))K_{l'-1,l',\lambda}^{m_1,m'-q,q}\Bigr]\\ &+\rho^{l+1}a_{22}(2l+1)\dfrac{\sqrt{l'(2l'+1)}}{2}\sum_{q=-1}^{1}(-1)^q r^{l'-1}\Bigl[-(-1)^{m'}(\mathscr{A}_{l,l'-1}^{m,-m'-q}(\alpha,q)-(-1)^m\mathscr{A}_{l,l'-1}^{-m,-m'-q}(\alpha,q))\\ &\quad\langle l'-1,-m'-q;1,q\mid l',-m' \rangle+(\mathscr{A}_{l,l'-1}^{m,m'-q}(\alpha,q)-(-1)^m\mathscr{A}_{l,l'-1}^{-m,m'-q}(\alpha,q))\langle l'-1,m'-q;1,q\mid l',m' \rangle\Bigr]\\ &+\delta_{ll'}\delta_{mm'}l(2l+1)\rho a_{22}. \end{align}\]
If \(m<0\) and \(m'=0\), \[\begin{align} &M_{(l',0,2),(lm,2)}(\alpha)\\ &=\dfrac{l'(2l'+1)}{\sqrt{2}}\Bigl[ \rho^{l+3}a_{12}r^{l'-1}+\rho^{l+1}(a_{22}-a_{12})(r^{l'+1} )+\rho^{l+1}a_{22}r^{l'+1}\dfrac{2l+1}{2l'+1}\Bigr]\\ & \cdot i\Bigl[(\mathscr{H}_{l,l'}^{m,0}(\alpha)-(-1)^m\mathscr{H}_{l,l'}^{-m,0}(\alpha))\Bigr]\\ &+\dfrac{l'(2l'+1)}{\sqrt{2}}[\rho^{l+1}(a_{22}-a_{12})r^{l'-1}]\cdot i\Bigl[(\mathscr{D}_{l,l'}^{m,0}(\alpha)-(-1)^m\mathscr{D}_{l,l'}^{-m,0}(\alpha))\Bigr]\\ &+\rho^{l+1}(a_{22}-a_{12})\sqrt{2l'(2l'+1)}\sum_{\lambda\in\{l'+1,l'-1\}}\sum_{q=-1}^{1}\sum_{m_1=-1}^{1}r^\lambda \;i\Bigl[(\mathscr{A}_{l,\lambda}^{m,-q}(\alpha,q)-(-1)^m\mathscr{A}_{l,\lambda}^{-m,-q}(\alpha,q))K_{l'-1,l',\lambda}^{m_1,-q,q}\Bigr]\\ &+\rho^{l+1}a_{22}(2l+1)\dfrac{\sqrt{l'(2l'+1)}}{\sqrt{2}}\sum_{q=-1}^{1}(-1)^q r^{l'-1}\;i\Bigl[(\mathscr{A}_{l,l'-1}^{m,-q}(\alpha,q)\\ &-(-1)^m\mathscr{A}_{l,l'-1}^{-m,-q}(\alpha,q))\langle l'-1,-q;1,q\mid l',0 \rangle\Bigr]\\ &+\delta_{ll'}\delta_{mm'}l(2l+1)\rho a_{22}. \end{align}\]
If \(m=0\) and \(m'>0\), \[\begin{align} &M_{(l'm',2),(l,0,2)}(\alpha)\\ &=\dfrac{l'(2l'+1)}{\sqrt{2}}\Bigl[ \rho^{l+3}a_{12}r^{l'-1}+\rho^{l+1}(a_{22}-a_{12})(r^{l'+1} )+\rho^{l+1}a_{22}r^{l'+1}\dfrac{2l+1}{2l'+1}\Bigr]\\ &\quad\Bigl[(-1)^{m'}\mathscr{H}_{l,l'}^{0,m'}(\alpha)+\mathscr{H}_{l,l'}^{0,-m'}(\alpha)\Bigr]\\ &+\dfrac{l'(2l'+1)}{\sqrt{2}}[\rho^{l+1}(a_{22}-a_{12})r^{l'-1}]\Bigl[(-1)^{m'}\mathscr{D}_{l,l'}^{0,m'}(\alpha)+\mathscr{D}_{l,l'}^{0,-m'}(\alpha)\Bigr]\\ &+\rho^{l+1}(a_{22}-a_{12})\sqrt{2l'(2l'+1)}\sum_{\lambda\in\{l'+1,l'-1\}}\sum_{q=-1}^{1}\sum_{m_1=-1}^{1}r^\lambda \Bigl[(-1)^{m'}\mathscr{A}_{l,\lambda}^{0,m'-q}(\alpha,q)K_{l'-1,l',\lambda}^{m_1,m'-q,q}\\ &+\mathscr{A}_{l,\lambda}^{0,-m'-q}(\alpha,q)K_{l'-1,l',\lambda}^{m_1,-m'-q,q}\Bigr]\\ &+\rho^{l+1}a_{22}(2l+1)\dfrac{\sqrt{l'(2l'+1)}}{\sqrt{2}}\sum_{q=-1}^{1}(-1)^q r^{l'-1}\Bigl[(-1)^{m'}\mathscr{A}_{l,l'-1}^{0,m'-q}(\alpha,q)\langle l'-1,m'-q;1,q\mid l',m' \rangle\\ &+\mathscr{A}_{l,l'-1}^{0,-m'-q}(\alpha,q)\langle l'-1,-m'-q;1,q\mid l',-m' \rangle\Bigr]\\ &+\delta_{ll'}\delta_{mm'}l(2l+1)\rho a_{22}. \end{align}\]
If \(m=0\) and \(m'<0\), \[\begin{align} &M_{(l'm',2),(l,0,2)}(\alpha)\\ &=\dfrac{l'(2l'+1)}{\sqrt{2}}\Bigl[ \rho^{l+3}a_{12}r^{l'-1}+\rho^{l+1}(a_{22}-a_{12})(r^{l'+1} )+\rho^{l+1}a_{22}r^{l'+1}\dfrac{2l+1}{2l'+1}\Bigr]\\ &\quad i\Bigl[(-1)^{m'}\mathscr{H}_{l,l'}^{0,-m'}(\alpha)-\mathscr{H}_{l,l'}^{0,m'}(\alpha)\Bigr]\\ &+\dfrac{l'(2l'+1)}{\sqrt{2}}[\rho^{l+1}(a_{22}-a_{12})r^{l'-1}]\cdot i\Bigl[(-1)^{m'}\mathscr{D}_{l,l'}^{0,-m'}(\alpha)-\mathscr{D}_{l,l'}^{0,m'}(\alpha)\Bigr]\\ &+\rho^{l+1}(a_{22}-a_{12})\sqrt{2l'(2l'+1)}\sum_{\lambda\in\{l'+1,l'-1\}}\sum_{q=-1}^{1}\sum_{m_1=-1}^{1}r^\lambda \;i\Bigl[(-1)^{m'}\mathscr{A}_{l,\lambda}^{0,-m'-q}(\alpha,q)K_{l'-1,l',\lambda}^{m_1,-m'-q,q}\\ &-\mathscr{A}_{l,\lambda}^{0,m'-q}(\alpha,q)K_{l'-1,l',\lambda}^{m_1,m'-q,q}\Bigr]\\ &+\rho^{l+1}a_{22}(2l+1)\dfrac{\sqrt{l'(2l'+1)}}{\sqrt{2}}\sum_{q=-1}^{1}(-1)^q r^{l'-1}\;i\Bigl[(-1)^{m'}\mathscr{A}_{l,l'-1}^{0,-m'-q}(\alpha,q)\langle l'-1,-m'-q;1,q\mid l',-m' \rangle\\ &-\mathscr{A}_{l,l'-1}^{0,m'-q}(\alpha,q)\langle l'-1,m'-q;1,q\mid l',m' \rangle\Bigr]\\ &+\delta_{ll'}\delta_{mm'}l(2l+1)\rho a_{22}. \end{align}\]
If \(m=0\) and \(m'=0\), \[\begin{align} &M_{(l',0,2),(l,0,2)}(\alpha)\\ &=l'(2l'+1)\Bigl[ \rho^{l+3}a_{12}r^{l'-1}+\rho^{l+1}(a_{22}-a_{12})(r^{l'+1} )+\rho^{l+1}a_{22}r^{l'+1}\dfrac{2l+1}{2l'+1}\Bigr]\mathscr{H}_{l,l'}^{0,0}(\alpha)\\ &+{l'(2l'+1)}[\rho^{l+1}(a_{22}-a_{12})r^{l'-1}]\mathscr{D}_{l,l'}^{0,0}(\alpha)\\ &+\rho^{l+1}(a_{22}-a_{12})\cdot 2\sqrt{l'(2l'+1)}\sum_{\lambda\in\{l'+1,l'-1\}}\sum_{q=-1}^{1}\sum_{m_1=-1}^{1}r^\lambda \;\mathscr{A}_{l,\lambda}^{0,-q}(\alpha,q)K_{l'-1,l',\lambda}^{m_1,-q,q}\\ &+\rho^{l+1}a_{22}(2l+1)\sqrt{l'(2l'+1)}\sum_{q=-1}^{1}(-1)^q r^{l'-1}\;\mathscr{A}_{l,l'-1}^{0,-q}(\alpha,q)\langle l'-1,-q;1,q\mid l',0 \rangle\\ &+\delta_{ll'}\delta_{mm'}l(2l+1)\rho a_{22}. \end{align}\]
Theorem 23. The remaining terms are simple: \[M_{(l'm',3),(lm,1)}(\alpha)=M_{(l'm',1),(lm,3)}=0,\] \[M_{(l'm',1),(lm1)}=\delta_{ll'}\delta_{mm'}\rho (l+1)(2l+1) a_{11}.\]
Given a function \(\varphi\in L^2(\partial B_\rho(0))^3\) for \(0<\rho<1/2\), the operator equation is set by \[\label{eq-Sf61phi}\mathcal{S}^{\alpha,0}_D[f]=\varphi,\tag{15}\] where \(D=B_\rho(0)\). The goal is to determine the \(L^2\) function \(f\).
It can be assumed that the coordinates in the expansion of \(f\) under the vector spherical harmonics are \(\{F_{lm}^k(\alpha)\}\). That means \[\label{eq-f-expression}f(x)=\sum\limits_{l,m,k}F_{lm}^{k}(\alpha)Y_{lm}^k(\hat{x}).\tag{16}\] Starting from the equation 15 , taking inner product over the unit sphere \(\mathbb{S}^2\) yields: \[(Y_{l'm'}^p,\mathcal{S}_D^{\alpha,0}[f])=(Y_{l'm'}^p,\varphi),\] and therefore, \[\sum\limits_{l,m,k}\overline{F_{lm}^{k}(\alpha)}(Y_{l'm'}^p,\mathcal{S}_D^{\alpha,0}[Y_{lm}^k])=(Y_{l'm'}^p,\varphi).\]
Remark 6. Since \(W_{00}=X_{00}=0\), these terms should be excluded from the basis functions.
Now a system of linear equations is obtain: \[\mathbf{M}(\alpha) \overline{\mathbf{F}(\alpha)}=\mathbf{b},\] where \(\overline{\boxed{}}\) represents complex conjugation. The entries of the matrix \(\mathbf{M}(\alpha)\) are given by \(M_{(l'm'p),(lmk)}(\alpha)\) in 14 . The vector \(\mathbf{F}(\alpha)\) is designed as \[\mathbf{F}(\alpha) = \begin{pmatrix} F_{00}^1 \\ F_{00}^2 \\ F_{00}^3 \\ F_{1,-1}^1 \\ F_{1,-1}^2 \\ F_{1,-1}^3 \\ F_{10}^1 \\ F_{10}^2 \\ F_{10}^3 \\ F_{11}^1 \\ F_{11}^2 \\ F_{11}^3 \\ F_{2,-2}^1 \\ \vdots \\ \vdots \\ F_{L_{\max},L_{\max}}^3 \end{pmatrix}\] The components of vector \(\mathbf{b}\in\mathbb{R}^N( N=3\sum_{l=0}^{L_{\max}}(2l+1) )\) are expressed as \(b_{l'm'}^p := \bigl(Y_{l'm'}^p,\varphi\bigr),\) which can be calculated as long as the exact expression of \(\varphi\) is given.
Let \(D=D_1\cup D_2\) with \(D_1=B_\rho(-d,0,0)\) and \(D_2=B_\rho(d,0,0)\). Here the parameter is designed such that these balls are disjoint, say, \(d=0.2\) and \(\rho=0.1\). Given an \(L^2\) function \(\varphi\) on \(\partial D\), it is expected that the function \(f\) defined on \(\partial D\) satisfying \[\mathcal{S}_D^{\alpha,0}[f]=\varphi,\] can be determined. The function \(f\) can be decomposed as \[f=f|_{\partial D_1}\cdot \mathbf{1}_{\partial D_1}+f|_{\partial D_2}\cdot \mathbf{1}_{\partial D_2}.\] Denote these two parts by \(g_1\) and \(g_2\). Therefore, \[\mathcal{S}_D^{\alpha,0}[f](x)=\int_{\partial D_1}G^{\alpha,0}(x-y)g_1(y)d\sigma+\int_{\partial D_2}G^{\alpha,0}(x-y)g_2 (y)d\sigma=\varphi(x),\quad x\in\partial D.\] Denote \(\mathcal{S}^{\alpha,0}_{D_s\to D_t}[g_s](x):=\mathcal{S}^{\alpha,0}_{D_s}[g_s](x)\) with \(x\in \partial D_t\) for \(s,t=1,2\). That is, the notation \(\mathcal{S}_{D_s\to D_t}^{\alpha,0}\) means that the operator maps the function on \(\partial D_s\) to the function on \(\partial D_t\).
For \(x\in\partial D_1\), it holds that \[\mathcal{S}_{D_1\to D_1}^{\alpha,0}[g_1](x)+\mathcal{S}_{D_2\to D_1}^{\alpha,0}[g_2](x)=\varphi |_{\partial D_1}(x).\] Similarly, for \(x\in\partial D_2\), \[\mathcal{S}_{D_1\to D_2}^{\alpha,0}[g_1](x)+\mathcal{S}_{D_2\to D_2}^{\alpha,0}[g_2](x)=\varphi|_{\partial D_2}(x).\] For convenience, it is written as \[\begin{pmatrix} \mathcal{S}_{D_1\to D_1}^{\alpha,0}&\mathcal{S}_{D_2\to D_1}^{\alpha,0}\\ \mathcal{S}_{D_1\to D_2}^{\alpha,0}& \mathcal{S}_{D_2\to D_2}^{\alpha,0}\end{pmatrix}\begin{pmatrix}g_1\\g_2\end{pmatrix}=\begin{pmatrix}\varphi |_{\partial D_1}\\\varphi |_{\partial D_2}\end{pmatrix}.\]
Using the substitution \(x=x_{i'}+\mathbf{c}_{i'}\), \(i'=1,2\), the coordinate \(x\in \partial D_{i'}\) can be pulled back into \(x_{i'}\in \partial B_\rho(0)\), where \(\mathbf{c}_{1}=(-d,0,0)\) and \(\mathbf{c}_2=(d,0,0)\). Also \(c_1=-d\) and \(c_2=d\). Applying this to the function \(\varphi\) and defining the inner product over \(\mathbb{S}^2\), one obtains: \[b_{l'm',(i')}^{p}:=(Y_{l'm'}^p,\varphi|_{\partial D_{i'}}).\] These form a vector called \(\mathbf{b}_{i'}\in\mathbb{R}^{N}\) \((
N=3\sum_{l=0}^{L_{\max}}(2l+1) )\) for \(i'=1,2\).
The function \(g_{i'}(x)=g_{i'}(\mathbf{c}_{i'}+\rho \widehat{x_{i'}})\) can be regarded as a function with respect to \(\widehat{x_{i'}}\in\mathbb{S}^2\). Thus, it is
assumed that \[\boxed{g_{i'}(\mathbf{c}_{i'}+\rho \widehat{x_{i'}})=\sum\limits_{l,m,k}F_{l,m,(i')}^{k}(\alpha) Y_{lm}^k(\widehat{x_{i'}}).}\] For \(x\in \partial D_t\),
\[\begin{align}
\mathcal{S}_{D_s\to D_t}^{\alpha,0}[g_{s}](x)&
=\int_{\partial D_s}\boldsymbol{G}^{\alpha,0}(x-y)g_{s}(y)d\sigma(y)\\
&=\int_{\partial D_s}\boldsymbol{G}^{\alpha,0}(x-y)g_{s}(\mathbf{c}_{s}+\rho \widehat{y_s})d\sigma(y)\\
&=\int_{\partial B_\rho(0)}\boldsymbol{G}^{\alpha,0}(\mathbf{c}_t+\rho \widehat{x_t}-\mathbf{c}_s-\rho\widehat{y_s})g_{s}(\mathbf{c}_{s}+\rho \widehat{y_s})d\sigma(y_s)\\
&=\int_{\partial B_\rho(0)}\boldsymbol{G}^{\alpha,0}(\mathbf{c}_t+\rho \widehat{x_t}-\mathbf{c}_s-\rho\widehat{y_s})\sum\limits_{l,m,k}F_{l,m,(s)}^{k}(\alpha)Y_{lm}^{k}(\widehat{y_s})d\sigma(y_s).
\end{align}\] It is observed that when \(t=s\), \[\mathcal{S}_{D_s\to D_t}^{\alpha,0}[g_{s}](x)=\sum\limits_{l,m,k}F_{l,m,(s)}^{k}(\alpha)\mathcal{S}_D^{\alpha,0}[Y_{lm}^{k}](x_t),\]
where the operator \(\mathcal{S}_D^{\alpha,0}\) is defined in Eq 2 . Therefore, the inner product over \(\mathbb{S}^2\) \[(Y_{l'm'}^p(\widehat{x_t}), \mathcal{S}_{D_s\to D_t}^{\alpha,0}[g_{s}](x))=\sum\limits_{l,m,k}\overline{F_{l,m,(s)}^{k}(\alpha)}(Y_{l'm'}^p(\widehat{x_t}),\mathcal{S}_D^{\alpha,0}[Y_{lm}^{k}](x_t)),\] in which
\((Y_{l'm'}^p(\widehat{x_t}),\mathcal{S}_D^{\alpha,0}[Y_{lm}^{k}](x_t))\) is given by Eq 14 .
When \(s\neq t\), the periodic Green function \[\boldsymbol{G}^{\alpha,0}(\mathbf{c}_t+\rho
\widehat{x_t}-\mathbf{c}_s-\rho\widehat{y_s})=\sum\limits_{n\in\mathbb{Z}}\boldsymbol{G}(x_t-y_s-(\mathbf{c}_s-\mathbf{c}_t+(n,0,0)))e^{-in\alpha}\] So there are only two changes in the previous \(M(\alpha)\):
Delete the diagonal terms \(( Y_{l'm'}^p,\mathcal{S}_D Y_{lm}^q)\) and the summation should involve the case \(n=0\), i.e., \[\label{eq-M-ts-entries}M^{st}_{(l'm'p),(lmq)}(\alpha) = \sum_{n \in\mathbb{Z}}e^{-in\alpha}(Y_{l'm'}^p,\mathcal{S}^{st}_{D+n} Y_{lm}^q).\tag{17}\]
Replace \(\mathbf{b}=(-n,0,0)\) by \(\mathbf{b}_{st}=(-(c_s-c_t+n),0,0)\) in the explicit expression of \(M^{st}_{(l'm'p),(lmq)}(\alpha)\). Meanwhile, the Lerch transcendent is introduced to deal with the infinite lattice sums.
The exact expressions of \(\mathbf{M}^{21}\) and \(\mathbf{M}^{12}\) will be given in 8. Therefore, the system of equations for solving \(g_{i'}\) is given by \[\begin{pmatrix} \mathbf{M}^{11}(\alpha)&\mathbf{M}^{21}(\alpha)\\ \mathbf{M}^{12}(\alpha)&\mathbf{M}^{22}(\alpha)\end{pmatrix}\begin{pmatrix}\overline{\mathbf{F}_{1}}\\\overline{\mathbf{F}_{2}}\end{pmatrix}=\begin{pmatrix}\mathbf{b}_1\\\mathbf{b}_2\end{pmatrix}.\]
Remark 7. Since \(W_{0,0}=X_{0,0}=0\), they are discarded from the basis. So the corresponding components of these vectors and the rows and columns of these matrices should be deleted simultaneously.
This paper has established addition theorems for all three families of real vector spherical harmonics under translation, and used them to compute the full matrix representation of the quasi-periodic elastic single layer potential in closed form for one-dimensional arrays. A key feature of the approach is that the infinite lattice sums, which arise inevitably from the periodic geometry, are handled exactly via polylogarithm functions and the Lerch transcendent rather than by numerical truncation, so the resulting matrix entries carry no discretization error. The framework covers both the single-ball and dimer geometries, and reduces the operator equation to an explicit linear system that can be solved directly. Taken together, these results complete the elastic analogue of the multipole expansion method developed for the acoustic Helmholtz equation in Ammari’s work, and remove what has been a missing analytical ingredient for the rigorous study of elastic subwavelength resonator chains. The explicit matrix representations obtained here are expected to serve as a practical tool in the spectral analysis of periodic elastic structures, including the investigation of band gaps, resonant frequencies, and topological phenomena in the elastic setting.
Definition 2. The Lerch transcendent is defined by a power series in \(z\) generalizing the polylogarithm: \[\mathrm{\Phi}(z,s,D)=\sum\limits_{k=0}^\infty\dfrac{z^k}{(k+D)^s}.\]
Lemma 12. Denote the series \(\sum\limits_{n\in\mathbb{Z}}H_{l,\lambda}^{m,\mu}(\mathbf{b}_{21})e^{-in\alpha}\) by \(\boxed{\mathscr{H}_{l,\lambda}^{m,\mu,(21)}(\alpha)}\). \[\begin{align} \sum\limits_{n\in\mathbb{Z}}H_{l,\lambda}^{m,\mu}(\mathbf{b}_{21})e^{-in\alpha}=&(-1)^{\lambda+\mu} \sqrt{\dfrac{2l+1}{2\lambda+1}} \sqrt{{l+\lambda+\mu-m\choose \lambda+\mu}{l+\lambda+m-\mu\choose \lambda-\mu}} \sqrt{\dfrac{4\pi}{2(l+\lambda)+1}}\\& \big(\mathrm{\Phi}(e^{-i\alpha},l+\lambda+1,2d)Y_{l+\lambda}^{m-\mu}(\dfrac{\pi}{2},\pi)+e^{i\alpha}\cdot \mathrm{\Phi}(e^{i\alpha},l+\lambda+1,1-2d)Y_{l+\lambda}^{m-\mu}(\dfrac{\pi}{2},0)\big). \end{align}.\]
Theorem 24. The expression of \(\boxed{M^{21}_{(l'm'2),(lm1)}(\alpha)}\)(\(l'\geqslant 1\), otherwise this term vanishes) is given by:
If \(m>0\) and \(m'>0\), \[\begin{align}M^{21}_{(l'm'2),(lm1)}(\alpha)=
\rho^{l+3}a_{11}[r^{l'-1}\dfrac{1}{2}(\mathscr{H}_{l,l'}^{-m,-m',(21)}(\alpha)+(-1)^m\mathscr{H}_{l,l'}^{m,-m',(21)}(\alpha))l'(2l'+1)\\
+r^{l'-1}\dfrac{(-1)^{m'}}{2}(\mathscr{H}_{l,l'}^{-m,m',(21)}(\alpha)+(-1)^m\mathscr{H}_{l,l'}^{m,m',(21)}(\alpha))l'(2l'+1)]
\end{align}\] If \(m>0\) and \(m'<0\), \[\begin{align}M^{21}_{(l'm'2),(lm1)}(\alpha)=
\rho^{l+3}a_{11}[r^{l'-1}\dfrac{1}{2}(\mathscr{H}_{l,l'}^{-m,m',(21)}(\alpha)+(-1)^m\mathscr{H}_{l,l'}^{m,m',(21)}(\alpha))(-i \cdot l'(2l'+1))\\
+r^{l'-1}\dfrac{(-1)^{m'}}{2}(\mathscr{H}_{l,l'}^{-m,-m',(21)}(\alpha)+(-1)^m\mathscr{H}_{l,l'}^{m,-m',(21)}(\alpha))(i\cdot l'(2l'+1))]
\end{align}\] If \(m>0\) and \(m'=0\), \[\begin{align}M^{21}_{(l'm'2),(lm1)}(\alpha)=
\rho^{l+3}a_{11}r^{l'-1}\dfrac{1}{\sqrt{2}}(\mathscr{H}_{l,l'}^{-m,0,(21)}(\alpha)+(-1)^m\mathscr{H}_{l,l'}^{m,0,(21)}(\alpha))l'(2l'+1).
\end{align}\]
If \(m<0\) and \(m'>0\), \[\begin{align}M^{21}_{(l'm'2),(lm1)}(\alpha)= \rho^{l+3}a_{11}[r^{l'-1}\dfrac{i}{2}(\mathscr{H}_{l,l'}^{m,-m',(21)}(\alpha)-(-1)^m \mathscr{H}_{l,l'}^{-m,-m',(21)}(\alpha))l'(2l'+1)\\ +r^{l'-1}\dfrac{(-1)^{m'}}{2}\cdot i (\mathscr{H}_{l,l'}^{m,m',(21)}(\alpha)-(-1)^m\mathscr{H}_{l,l'}^{-m,m',(21)}(\alpha))l'(2l'+1)] \end{align}\]
If \(m<0\) and \(m'<0\), \[\begin{align}M^{21}_{(l'm'2),(lm1)}(\alpha)= \rho^{l+3}a_{11}[r^{l'-1}\dfrac{i}{2}(\mathscr{H}_{l,l'}^{m,m',(21)}(\alpha)-(-1)^m \mathscr{H}_{l,l'}^{-m,m',(21)}(\alpha))(-i\cdot l'(2l'+1))\\ +r^{l'-1}\dfrac{(-1)^{m'}}{2}\cdot i (\mathscr{H}_{l,l'}^{m,-m',(21)}(\alpha)-(-1)^m\mathscr{H}_{l,l'}^{-m,-m',(21)}(\alpha))(i\cdot l'(2l'+1))] \end{align}\]
If \(m<0\) and \(m'=0\), \[\begin{align}M^{21}_{(l'm'2),(lm1)}(\alpha)= \rho^{l+3}a_{11}r^{l'-1}\dfrac{i}{\sqrt{2}}(\mathscr{H}_{l,l'}^{m,0,(21)}(\alpha)-(-1)^m\mathscr{H}_{l,l'}^{-m,0,(21)}(\alpha))l'(2l'+1).\end{align}\]
If \(m=0\) and \(m'>0\), \[\begin{align}M^{21}_{(l'm'2),(lm1)}(\alpha)= \rho^{l+3}a_{11}r^{l'-1} [\dfrac{1}{\sqrt{2}} \mathscr{H}_{l,l'}^{0,-m',(21)}(\alpha)l'(2l'+1)+\dfrac{(-1)^{m'}}{\sqrt{2}}\mathscr{H}_{l,l'}^{0,m',(21)}(\alpha)l'(2l'+1)] \end{align}\]
If \(m=0\) and \(m'<0\), \[\begin{align}M^{21}_{(l'm'2),(lm1)}(\alpha)= \rho^{l+3}a_{11}r^{l'-1} [\dfrac{1}{\sqrt{2}} \mathscr{H}_{l,l'}^{0,m',(21)}(\alpha)(-i\cdot l'(2l'+1))+\dfrac{(-1)^{m'}}{\sqrt{2}}\mathscr{H}_{l,l'}^{0,-m',(21)}(\alpha)(i\cdot l'(2l'+1))] \end{align}\]
If \(m=0\) and \(m'=0\), \[\begin{align}M^{21}_{(l'm'2),(lm1)}(\alpha)= \rho^{l+3}a_{11}r^{l'-1} [\mathscr{H}_{l,l'}^{0,0,(21)}(\alpha)l'(2l'+1)] \end{align}\] where \(a_{ij}\) stands for the coefficient of the entries of \(A^{out}_{\mathcal{S},l}(x)\).
Lemma 13. Denote \(\sum\limits_{n\in\mathbb{Z}}L_{l,j,\lambda}^{m,\mu,q,m_1}(\mathbf{b}_{21})e^{-in\alpha}\) by \(\boxed{\mathscr{L}_{l,j,\lambda}^{m,\mu,q,m_1,(21)}(\alpha)}\), then \[\begin{align}&\mathscr{L}_{l,j,\lambda}^{m,\mu,q,m_1,(21)}(\alpha)= i (-1)^{\lambda+\mu+q}\text{sgn}(q-m_1)\epsilon_{q}\sqrt{\lambda(2l+1)} \sqrt{{l+\lambda+\mu-m\choose \lambda+\mu}{l+\lambda+m-\mu\choose \lambda-\mu}}\\ &\langle\lambda-1,\mu-m_1;1,m_1\mid \lambda,\mu\rangle \langle \lambda-1,\mu-m_1;1,q+m_1\mid j,\mu+q \rangle \sqrt{\dfrac{4\pi}{2(l+\lambda)+1}}\\ &\big(\mathrm{\Phi}(e^{-i\alpha},l+\lambda,2d)Y_{l+\lambda}^{m-\mu}(\dfrac{\pi}{2},\pi)-e^{i\alpha}\mathrm{\Phi}(e^{i\alpha},l+\lambda,1-2d)Y_{l+\lambda}^{m-\mu}(\dfrac{\pi}{2},0)\big) \end{align}\] where \[\epsilon_q=\left\{ \begin{align} \dfrac{1}{\sqrt{2}}&,\quad q=-1\\ 0&,\quad q=0\\ -\dfrac{1}{\sqrt{2}}&,\quad,q=1 \end{align}\right.\]
Theorem 25. The expression of \(\boxed{M^{21}_{(l'm',2),(lm,3)}(\alpha)}\)(\(l'\geqslant 1,l\geqslant 1\),otherwise this term vanishes) is given by When \(m>0, m'>0\), \[\begin{align} &M^{21}_{(l'm',2),(lm,3)}(\alpha)\\ =&\;\rho^{l+2}a_{33}\,r^{l'-1}\,\frac{\sqrt{l'(2l'+1)}}{2}\sum\limits_{q=-1}^{1}\sum\limits_{m_1=-1}^{1}\bigl[(-1)^{m'}\bigl(\mathscr{L}_{l,l',l'}^{-m,\,m'-q,\,q,\,m_1,(21)}(\alpha)+(-1)^m\mathscr{L}_{l,l',l'}^{m,\,m'-q,\,q,\,m_1,(21)}(\alpha)\bigr)\\ &\qquad\qquad\qquad\qquad+\bigl(\mathscr{L}_{l,l',l'}^{-m,\,-m'-q,\,q,\,m_1,(21)}(\alpha)+(-1)^m\mathscr{L}_{l,l',l'}^{m,\,-m'-q,\,q,\,m_1,(21)}(\alpha)\bigr)\bigr] \end{align}\] When \(m>0, m'<0\), \[\begin{align} &M^{21}_{(l'm',2),(lm,3)}(\alpha)\\ =&\;\rho^{l+2}a_{33}\,r^{l'-1}\,\frac{\sqrt{l'(2l'+1)}}{2}\sum\limits_{q=-1}^{1}\sum\limits_{m_1=-1}^{1}\bigl[i(-1)^{m'}\bigl(\mathscr{L}_{l,l',l'}^{-m,\,-m'-q,\,q,\,m_1,(21)}(\alpha)+(-1)^m\mathscr{L}_{l,l',l'}^{m,\,-m'-q,\,q,\,m_1,(21)}(\alpha)\bigr)\\ &\qquad\qquad\qquad\qquad-i\bigl(\mathscr{L}_{l,l',l'}^{-m,\,m'-q,\,q,\,m_1,(21)}(\alpha)+(-1)^m\mathscr{L}_{l,l',l'}^{m,\,m'-q,\,q,\,m_1,(21)}(\alpha)\bigr)\bigr] \end{align}\]
When \(m>0, m'=0\), \[\begin{align} &M^{21}_{(l',0,2),(lm,3)}(\alpha)\\ =&\;\rho^{l+2}a_{33}\,r^{l'-1}\,\frac{\sqrt{l'(2l'+1)}}{\sqrt{2}}\sum\limits_{q=-1}^{1}\sum\limits_{m_1=-1}^{1}\bigl(\mathscr{L}_{l,l',l'}^{-m,\,-q,\,q,\,m_1,(21)}(\alpha)+(-1)^m\mathscr{L}_{l,l',l'}^{m,\,-q,\,q,\,m_1,(21)}(\alpha)\bigr) \end{align}\]
When \(m<0, m'>0\), \[\begin{align} &M^{21}_{(l'm',2),(lm,3)}(\alpha)\\ =&\;\rho^{l+2}a_{33}\,r^{l'-1}\,i\cdot\frac{\sqrt{l'(2l'+1)}}{2}\sum\limits_{q=-1}^{1}\sum\limits_{m_1=-1}^{1}\bigl[(-1)^{m'}\bigl(\mathscr{L}_{l,l',l'}^{m,\,m'-q,\,q,\,m_1,(21)}(\alpha)-(-1)^m\mathscr{L}_{l,l',l'}^{-m,\,m'-q,\,q,\,m_1,(21)}(\alpha)\bigr)\\ &\qquad\qquad\qquad\qquad+\bigl(\mathscr{L}_{l,l',l'}^{m,\,-m'-q,\,q,\,m_1,(21)}(\alpha)-(-1)^m\mathscr{L}_{l,l',l'}^{-m,\,-m'-q,\,q,\,m_1,(21)}(\alpha)\bigr)\bigr] \end{align}\] When \(m<0, m'<0\), \[\begin{align} &M^{21}_{(l'm',2),(lm,3)}(\alpha)\\ =&\;\rho^{l+2}a_{33}\,r^{l'-1}\,\frac{\sqrt{l'(2l'+1)}}{2}\sum\limits_{q=-1}^{1}\sum\limits_{m_1=-1}^{1}\bigl[\bigl(\mathscr{L}_{l,l',l'}^{m,\,m'-q,\,q,\,m_1,(21)}(\alpha)-(-1)^m\mathscr{L}_{l,l',l'}^{-m,\,m'-q,\,q,\,m_1,(21)}(\alpha)\bigr)\\ &\qquad\qquad\qquad\qquad-(-1)^{m'}\bigl(\mathscr{L}_{l,l',l'}^{m,\,-m'-q,\,q,\,m_1,(21)}(\alpha)-(-1)^m\mathscr{L}_{l,l',l'}^{-m,\,-m'-q,\,q,\,m_1,(21)}(\alpha)\bigr)\bigr] \end{align}\] When \(m<0, m'=0\), \[\begin{align} &M^{21}_{(l',0,2),(lm,3)}(\alpha)\\ =&\;\rho^{l+2}a_{33}\,r^{l'-1}\,\frac{i\sqrt{l'(2l'+1)}}{\sqrt{2}}\sum\limits_{q=-1}^{1}\sum\limits_{m_1=-1}^{1}\bigl(\mathscr{L}_{l,l',l'}^{m,\,-q,\,q,\,m_1,(21)}(\alpha)-(-1)^m\mathscr{L}_{l,l',l'}^{-m,\,-q,\,q,\,m_1,(21)}(\alpha)\bigr) \end{align}\] When \(m=0, m'>0\), \[\begin{align} &M^{21}_{(l'm',2),(l,0,3)}(\alpha)\\ =&\;\rho^{l+2}a_{33}\,r^{l'-1}\,\frac{\sqrt{l'(2l'+1)}}{\sqrt{2}}\sum\limits_{q=-1}^{1}\sum\limits_{m_1=-1}^{1}\bigl[(-1)^{m'}\mathscr{L}_{l,l',l'}^{0,\,m'-q,\,q,\,m_1,(21)}(\alpha)+\mathscr{L}_{l,l',l'}^{0,\,-m'-q,\,q,\,m_1,(21)}(\alpha)\bigr] \end{align}\]
When \(m=0, m'<0\), \[\begin{align} &M^{21}_{(l'm',2),(l,0,3)}(\alpha)\\ =&\;\rho^{l+2}a_{33}\,r^{l'-1}\,\frac{\sqrt{l'(2l'+1)}}{\sqrt{2}}\sum\limits_{q=-1}^{1}\sum\limits_{m_1=-1}^{1}\bigl[i(-1)^{m'}\mathscr{L}_{l,l',l'}^{0,\,-m'-q,\,q,\,m_1,(21)}(\alpha)-i\,\mathscr{L}_{l,l',l'}^{0,\,m'-q,\,q,\,m_1,(21)}(\alpha)\bigr] \end{align}\]
When \(m=0, m'=0\), \[\begin{align} &M^{21}_{(l',0,2),(l,0,3)}(\alpha)\\ =&\;\rho^{l+2}a_{33}\,r^{l'-1}\,\sqrt{l'(2l'+1)}\sum\limits_{q=-1}^{1}\sum\limits_{m_1=-1}^{1}\mathscr{L}_{l,l',l'}^{0,\,-q,\,q,\,m_1,(21)}(\alpha) \end{align}\]
Lemma 14. Denote \(\sum\limits_{n\in\mathbb{Z}}b^{(21)}_{-q}H_{l,\lambda}^{m,\mu}(\mathbf{b}_{21})e^{-in\alpha}\) by \(\boxed{\mathscr{A}_{l,\lambda}^{m,\mu,(21)}(\alpha,q)}\), then \[\begin{align} \sum\limits_{n\in\mathbb{Z}}b^{(21)}_{-q}H_{l,\lambda}^{m,\mu}(\mathbf{b}_{21})e^{-in\alpha}=&(-1)^{\lambda+\mu}\epsilon_q \sqrt{\dfrac{2l+1}{2\lambda+1}} \sqrt{{l+\lambda+\mu-m\choose \lambda+\mu}{l+\lambda+m-\mu\choose \lambda-\mu}} \sqrt{\dfrac{4\pi}{2(l+\lambda)+1}}\\& \big(\mathrm{\Phi}(e^{-i\alpha},l+\lambda,2d)Y_{l+\lambda}^{m-\mu}(\dfrac{\pi}{2},\pi)-e^{i\alpha}\mathrm{\Phi}(e^{i\alpha},l+\lambda,1-2d)Y_{l+\lambda}^{m-\mu}(\dfrac{\pi}{2},0)\big) \end{align}.\]
Theorem 26. The expression of \(\boxed{M^{21}_{(l'm',1),(lm,2)}(\alpha)}\)(\(l\geqslant 1\), otherwise this term vanishes) is given by
When \(m>0\), \(m'>0\), \[\begin{align}
&M^{21}_{(l'm',1),(lm,2)}(\alpha)\\
=&\sum\limits_{\lambda\in\{l'+1,l'+3\}}\sum\limits_{q=-1}^{1}\sum\limits_{m_1=-1}^{1}\rho^{l+1} (a_{22}-a_{12}) r^\lambda \sqrt{(l'+1)(2l'+1)}
[(-1)^{m'}(\mathscr{A}_{l,\lambda}^{-m,m'-q,(21)}(\alpha,q)\\
&+(-1)^m\mathscr{A}_{l,\lambda}^{m,m'-q,(21)}(\alpha,q))K_{l'+1,l',\lambda}^{m_1,m'-q,q}
+(\mathscr{A}_{l,\lambda}^{-m,-m'-q,(21)}(\alpha,q)+\\
&(-1)^m\mathscr{A}_{l,\lambda}^{m,-m'-q,(21)}(\alpha,q))K_{l'+1,l',\lambda}^{m_1,-m'-q,q}]\\
&-\rho^{l+1} a_{22}r^{l'+1}\dfrac{(2l+1)(l'+1)}{2}[(-1)^{m'}(\mathscr{H}_{l,l'}^{-m,m',(21)}(\alpha)\\
&+(-1)^m\mathscr{H}_{l,l'}^{m,m',(21)}(\alpha))+(\mathscr{H}_{l,l'}^{-m,-m',(21)}(\alpha)+(-1)^m\mathscr{H}_{l,l'}^{m,-m',(21)}(\alpha))]\\
&+\rho^{l+1} a_{22}r^{l'+1}(2l+1)\dfrac{\sqrt{(l'+1)(2l'+1)}}{2} \sum\limits_{q=-1}^1(-1)^q [(-1)^{m'}(\mathscr{A}_{l,l'+1}^{-m,m'-q,(21)}(\alpha,q)\\
&+(-1)^m\mathscr{A}_{l,l'+1}^{m,m'-q,(21)}(\alpha,q))\langle l'+1,m'-q;1,q\mid l',m'\rangle+(\mathscr{A}_{l,l'+1}^{-m,-m'-q,(21)}(\alpha,q)\\
&+(-1)^m\mathscr{A}_{l,l'+1}^{m,-m'-q,(21)}(\alpha,q))\langle l'+1,-m'-q;1,q\mid l',-m'\rangle]
\end{align}\]
When \(m>0\), \(m'<0\),
\[\begin{align} &M^{21}_{(l'm',1),(lm,2)}(\alpha)\\ =&\sum\limits_{\lambda\in\{l'+1,l'+3\}}\sum\limits_{q=-1}^{1}\sum\limits_{m_1=-1}^{1}\rho^{l+1} (a_{22}-a_{12}) r^\lambda \sqrt{(l'+1)(2l'+1)} [i\cdot (-1)^{m'}(\mathscr{A}_{l,\lambda}^{-m,-m'-q,(21)}(\alpha,q)\\ &+(-1)^m\mathscr{A}_{l,\lambda}^{m,-m'-q,(21)}(\alpha,q))K_{l'+1,l',\lambda}^{m_1,-m'-q,q} -i\cdot (\mathscr{A}_{l,\lambda}^{-m,m'-q,(21)}(\alpha,q)\\ &+(-1)^m\mathscr{A}_{l,\lambda}^{m,m'-q,(21)}(\alpha,q))K_{l'+1,l',\lambda}^{m_1,m'-q,q}]\\ &-\rho^{l+1} a_{22}r^{l'+1}\dfrac{(2l+1)(l'+1)}{2}[i\cdot (-1)^{m'}(\mathscr{H}_{l,l'}^{-m,-m',(21)}(\alpha)+(-1)^m\mathscr{H}_{l,l'}^{m,-m',(21)}(\alpha))\\ &-i\cdot (\mathscr{H}_{l,l'}^{-m,m',(21)}(\alpha)+(-1)^m\mathscr{H}_{l,l'}^{m,m',(21)}(\alpha))]\\ &+\rho^{l+1} a_{22}r^{l'+1}(2l+1)\dfrac{\sqrt{(l'+1)(2l'+1)}}{2} \sum\limits_{q=-1}^1(-1)^q [i\cdot (-1)^{m'}(\mathscr{A}_{l,l'+1}^{-m,-m'-q,(21)}(\alpha,q)\\ &+(-1)^m\mathscr{A}_{l,l'+1}^{m,-m'-q,(21)}(\alpha,q))\\ &\langle l'+1,-m'-q;1,q\mid l',-m'\rangle -i\cdot (\mathscr{A}_{l,l'+1}^{-m,m'-q,(21)}(\alpha,q)\\ &+(-1)^m\mathscr{A}_{l,l'+1}^{m,m'-q,(21)}(\alpha,q))\langle l'+1,m'-q;1,q\mid l',m'\rangle] \end{align}\]
When \(m>0\), \(m'=0\), \[\begin{align} &M^{21}_{(l',0,1),(lm,2)}(\alpha)\\ =&\sum\limits_{\lambda\in\{l'+1,l'+3\}}\sum\limits_{q=-1}^{1}\sum\limits_{m_1=-1}^{1}\rho^{l+1} (a_{22}-a_{12}) r^\lambda \sqrt{2(l'+1)(2l'+1)} [(\mathscr{A}_{l,\lambda}^{-m,-q,(21)}(\alpha,q)\\ &+(-1)^m\mathscr{A}_{l,\lambda}^{m,-q,(21)}(\alpha,q))K_{l'+1,l',\lambda}^{m_1,-q,q}]\\ &-\rho^{l+1} a_{22}r^{l'+1}\dfrac{(2l+1)(l'+1)}{\sqrt{2}}[\mathscr{H}_{l,l'}^{-m,0,(21)}(\alpha)+(-1)^m\mathscr{H}_{l,l'}^{m,0,(21)}(\alpha)]\\ &+\rho^{l+1} a_{22}r^{l'+1}(2l+1)\dfrac{\sqrt{(l'+1)(2l'+1)}}{\sqrt{2}} \sum\limits_{q=-1}^1(-1)^q (\mathscr{A}_{l,l'+1}^{-m,-q,(21)}(\alpha,q)+(-1)^m\mathscr{A}_{l,l'+1}^{m,-q,(21)}(\alpha,q))\\ &\langle l'+1,-q;1,q\mid l',0\rangle \end{align}\]
When \(m<0\), \(m'>0\), \[\begin{align} &M^{21}_{(l'm',1),(lm,2)}(\alpha)\\ =&\sum\limits_{\lambda\in\{l'+1,l'+3\}}\sum\limits_{q=-1}^{1}\sum\limits_{m_1=-1}^{1}\rho^{l+1} (a_{22}-a_{12}) r^\lambda i\cdot\sqrt{(l'+1)(2l'+1)} [(-1)^{m'}(\mathscr{A}_{l,\lambda}^{m,m'-q,(21)}(\alpha,q)\\ &-(-1)^m\mathscr{A}_{l,\lambda}^{-m,m'-q,(21)}(\alpha,q))K_{l'+1,l',\lambda}^{m_1,m'-q,q} +(\mathscr{A}_{l,\lambda}^{m,-m'-q,(21)}(\alpha,q)\\ &-(-1)^m\mathscr{A}_{l,\lambda}^{-m,-m'-q,(21)}(\alpha,q))K_{l'+1,l',\lambda}^{m_1,-m'-q,q}]\\ &-\rho^{l+1} a_{22}r^{l'+1}i\cdot \dfrac{(2l+1)(l'+1)}{2}[(-1)^{m'}(\mathscr{H}_{l,l'}^{m,m',(21)}(\alpha)\\ &-(-1)^m\mathscr{H}_{l,l'}^{-m,m',(21)}(\alpha))+(\mathscr{H}_{l,l'}^{m,-m',(21)}(\alpha)-(-1)^m\mathscr{H}_{l,l'}^{-m,-m',(21)}(\alpha))]\\ &+\rho^{l+1} a_{22}r^{l'+1}i\cdot (2l+1)\dfrac{\sqrt{(l'+1)(2l'+1)}}{2} \sum\limits_{q=-1}^1(-1)^q [(-1)^{m'}(\mathscr{A}_{l,l'+1}^{m,m'-q,(21)}(\alpha,q)\\ &-(-1)^m\mathscr{A}_{l,l'+1}^{-m,m'-q,(21)}(\alpha,q))\langle l'+1,m'-q;1,q\mid l',m'\rangle+(\mathscr{A}_{l,l'+1}^{m,-m'-q,(21)}(\alpha,q)\\ &-(-1)^m\mathscr{A}_{l,l'+1}^{-m,-m'-q,(21)}(\alpha,q))\langle l'+1,-m'-q;1,q\mid l',-m'\rangle] \end{align}\]
When \(m<0\), \(m'<0\),
\[\begin{align} &M^{21}_{(l'm',1),(lm,2)}(\alpha)\\ =&\sum\limits_{\lambda\in\{l'+1,l'+3\}}\sum\limits_{q=-1}^{1}\sum\limits_{m_1=-1}^{1}\rho^{l+1} (a_{22}-a_{12}) r^\lambda \sqrt{(l'+1)(2l'+1)} [-(-1)^{m'}(\mathscr{A}_{l,\lambda}^{m,-m'-q,(21)}(\alpha,q)\\ &-(-1)^m\mathscr{A}_{l,\lambda}^{-m,-m'-q,(21)}(\alpha,q))K_{l'+1,l',\lambda}^{m_1,-m'-q,q} + (\mathscr{A}_{l,\lambda}^{m,m'-q,(21)}(\alpha,q)\\ &-(-1)^m\mathscr{A}_{l,\lambda}^{-m,m'-q,(21)}(\alpha,q))K_{l'+1,l',\lambda}^{m_1,m'-q,q}]\\ &-\rho^{l+1} a_{22}r^{l'+1}\dfrac{(2l+1)(l'+1)}{2} [- (-1)^{m'}(\mathscr{H}_{l,l'}^{m,-m',(21)}(\alpha)-(-1)^m\mathscr{H}_{l,l'}^{-m,-m',(21)}(\alpha))\\ &+ (\mathscr{H}_{l,l'}^{m,m',(21)}(\alpha)-(-1)^m\mathscr{H}_{l,l'}^{-m,m',(21)}(\alpha))]\\ &+\rho^{l+1} a_{22}r^{l'+1}(2l+1)\dfrac{\sqrt{(l'+1)(2l'+1)}}{2} \sum\limits_{q=-1}^1(-1)^q [-(-1)^{m'}(\mathscr{A}_{l,l'+1}^{m,-m'-q,(21)}(\alpha,q)\\ &-(-1)^m\mathscr{A}_{l,l'+1}^{-m,-m'-q,(21)}(\alpha,q))\langle l'+1,-m'-q;1,q\mid l',-m'\rangle + (\mathscr{A}_{l,l'+1}^{m,m'-q,(21)}(\alpha,q)\\ &-(-1)^m\mathscr{A}_{l,l'+1}^{-m,m'-q,(21)}(\alpha,q))\langle l'+1,m'-q;1,q\mid l',m'\rangle] \end{align}\]
When \(m<0\), \(m'=0\), \[\begin{align} &M^{21}_{(l',0,1),(lm,2)}(\alpha)\\ =&\sum\limits_{\lambda\in\{l'+1,l'+3\}}\sum\limits_{q=-1}^{1}\sum\limits_{m_1=-1}^{1}\rho^{l+1} (a_{22}-a_{12}) r^\lambda i\cdot \sqrt{2(l'+1)(2l'+1)} [(\mathscr{A}_{l,\lambda}^{m,-q,(21)}(\alpha,q)\\ &-(-1)^m\mathscr{A}_{l,\lambda}^{-m,-q,(21)}(\alpha,q))K_{l'+1,l',\lambda}^{m_1,-q,q}]\\ &-\rho^{l+1} a_{22}r^{l'+1}i\cdot \dfrac{(2l+1)(l'+1)}{\sqrt{2}}[(\mathscr{H}_{l,l'}^{m,0,(21)}(\alpha)-(-1)^m\mathscr{H}_{l,l'}^{-m,0,(21)}(\alpha)]\\ &+\rho^{l+1} a_{22}r^{l'+1}i\cdot (2l+1)\dfrac{\sqrt{(l'+1)(2l'+1)}}{\sqrt{2}} \sum\limits_{q=-1}^1(-1)^q [ (\mathscr{A}_{l,l'+1}^{m,-q,(21)}(\alpha,q)-(-1)^m\mathscr{A}_{l,l'+1}^{-m,-q,(21)}(\alpha,q))\\ &\langle l'+1,-q;1,q\mid l',0\rangle] \end{align}\]
When \(m=0\), \(m'>0\), \[\begin{align} &M^{21}_{(l'm',1),(l,0,2)}(\alpha)\\ =&\sum\limits_{\lambda\in\{l'+1,l'+3\}}\sum\limits_{q=-1}^{1}\sum\limits_{m_1=-1}^{1}\rho^{l+1} (a_{22}-a_{12}) r^\lambda \sqrt{2(l'+1)(2l'+1)} [(-1)^{m'}\mathscr{A}_{l,\lambda}^{0,m'-q,(21)}(\alpha,q) K_{l'+1,l',\lambda}^{m_1,m'-q,q}\\ &+\mathscr{A}_{l,\lambda}^{0,-m'-q,(21)}(\alpha,q)K_{l'+1,l',\lambda}^{m_1,-m'-q,q}]\\ &-\rho^{l+1} a_{22}r^{l'+1}\dfrac{(2l+1)(l'+1)}{\sqrt{2}}[(-1)^{m'}\mathscr{H}_{l,l'}^{0,m',(21)}(\alpha)+\mathscr{H}_{l,l'}^{0,-m',(21)}(\alpha)]\\ &+\rho^{l+1} a_{22}r^{l'+1}(2l+1)\dfrac{\sqrt{(l'+1)(2l'+1)}}{\sqrt{2}} \sum\limits_{q=-1}^1(-1)^q [(-1)^{m'}\mathscr{A}_{l,l'+1}^{0,m'-q,(21)}(\alpha,q)\\ &\langle l'+1,m'-q;1,q\mid l',m'\rangle+(\mathscr{A}_{l,l'+1}^{0,-m'-q,(21)}(\alpha,q)\langle l'+1,-m'-q;1,q\mid l',-m'\rangle] \end{align}\]
When \(m=0\), \(m'<0\),
\[\begin{align} &M^{21}_{(l'm',1),(l,0,2)}(\alpha)\\ =&\sum\limits_{\lambda\in\{l'+1,l'+3\}}\sum\limits_{q=-1}^{1}\sum\limits_{m_1=-1}^{1}\rho^{l+1} (a_{22}-a_{12}) r^\lambda \sqrt{2(l'+1)(2l'+1)} [i\cdot (-1)^{m'}\mathscr{A}_{l,\lambda}^{0,-m'-q,(21)}(\alpha,q)K_{l'+1,l',\lambda}^{m_1,-m'-q,q}\\ & -i\cdot \mathscr{A}_{l,\lambda}^{0,m'-q,(21)}(\alpha,q)K_{l'+1,l',\lambda}^{m_1,m'-q,q}]\\ &-\rho^{l+1} a_{22}r^{l'+1}\dfrac{(2l+1)(l'+1)}{\sqrt{2}}[i\cdot (-1)^{m'}\mathscr{H}_{l,l'}^{0,-m',(21)}(\alpha)-i\cdot \mathscr{H}_{l,l'}^{0,m',(21)}(\alpha)]\\ &+\rho^{l+1} a_{22}r^{l'+1}(2l+1)\dfrac{\sqrt{(l'+1)(2l'+1)}}{\sqrt{2}} \sum\limits_{q=-1}^1(-1)^q [i\cdot (-1)^{m'} \mathscr{A}_{l,l'+1}^{0,-m'-q,(21)}(\alpha,q)\\ &\langle l'+1,-m'-q;1,q\mid l',-m'\rangle-i\cdot \mathscr{A}_{l,l'+1}^{0,m'-q,(21)}(\alpha,q)\langle l'+1,m'-q;1,q\mid l',m'\rangle] \end{align}\]
When \(m=0\), \(m'=0\), \[\begin{align} &M^{21}_{(l',0,1),(l,0,2)}(\alpha)\\ =&\sum\limits_{\lambda\in\{l'+1,l'+3\}}\sum\limits_{q=-1}^{1}\sum\limits_{m_1=-1}^{1}\rho^{l+1} (a_{22}-a_{12}) r^\lambda \sqrt{4(l'+1)(2l'+1)} [\mathscr{A}_{l,\lambda}^{0,-q,(21)}(\alpha,q)K_{l'+1,l',\lambda}^{m_1,-q,q}]\\ &-\rho^{l+1} a_{22}r^{l'+1}(2l+1)(l'+1)[\mathscr{H}_{l,l'}^{0,0,(21)}(\alpha)]\\ &+\rho^{l+1} a_{22}r^{l'+1}(2l+1)\sqrt{(l'+1)(2l'+1)} \sum\limits_{q=-1}^1(-1)^q [ \mathscr{A}_{l,l'+1}^{0,-q,(21)}(\alpha,q)\langle l'+1,-q;1,q\mid l',0\rangle] \end{align}\]
Theorem 27. Entries for \(\boxed{M^{21}_{(l'm',3),(lm,2)}(\alpha)}\)(\(l'\geqslant 1,l\geqslant 1\), otherwise this term vanishes):
When \(m>0\), \(m'>0\), \[\begin{align}
&M^{21}_{(l'm',3),(lm,2)}(\alpha)\\
=&\sum\limits_{\lambda\in\{l',l'+2\mid l'\geqslant 1\}}\sum\limits_{q=-1}^{1}\sum\limits_{m_1=-1}^{1}-i\cdot \rho^{l+1} (a_{22}-a_{12}) r^\lambda \sqrt{l'(l'+1)}
[(-1)^{m'}(\mathscr{A}_{l,\lambda}^{-m,m'-q,(21)}(\alpha,q)\\
&+(-1)^m\mathscr{A}_{l,\lambda}^{m,m'-q,(21)}(\alpha,q))K_{l',l',\lambda}^{m_1,m'-q,q}
+(\mathscr{A}_{l,\lambda}^{-m,-m'-q,(21)}(\alpha,q)\\
&+(-1)^m\mathscr{A}_{l,\lambda}^{m,-m'-q,(21)}(\alpha,q))K_{l',l',\lambda}^{m_1,-m'-q,q}]\\
&- i\cdot \rho^{l+1} a_{22}r^{l'}(2l+1)\dfrac{\sqrt{l'(l'+1)}}{2} \sum\limits_{q=-1}^1(-1)^q [(-1)^{m'}(\mathscr{A}_{l,l'}^{-m,m'-q,(21)}(\alpha,q)+(-1)^m\mathscr{A}_{l,l'}^{m,m'-q,(21)}(\alpha,q))\\
&\langle l',m'-q;1,q\mid l',m'\rangle+(\mathscr{A}_{l,l'}^{-m,-m'-q,(21)}(\alpha,q)+(-1)^m\mathscr{A}_{l,l'}^{m,-m'-q,(21)}(\alpha,q))\langle l',-m'-q;1,q\mid l',-m'\rangle]
\end{align}\]
When \(m>0\), \(m'<0\), \[\begin{align} &M^{21}_{(l'm',3),(lm,2)}(\alpha)\\ =&\sum\limits_{\lambda\in\{l',l'+2\mid l'\geqslant 1\}}\sum\limits_{q=-1}^{1}\sum\limits_{m_1=-1}^{1}\rho^{l+1} (a_{22}-a_{12}) r^\lambda \sqrt{l'(l'+1)} [ (-1)^{m'}(\mathscr{A}_{l,\lambda}^{-m,-m'-q,(21)}(\alpha,q)\\ &+(-1)^m\mathscr{A}_{l,\lambda}^{m,-m'-q,(21)}(\alpha,q))K_{l',l',\lambda}^{m_1,-m'-q,q} - (\mathscr{A}_{l,\lambda}^{-m,m'-q,(21)}(\alpha,q)+(-1)^m\mathscr{A}_{l,\lambda}^{m,m'-q,(21)}(\alpha,q))K_{l',l',\lambda}^{m_1,m'-q,q}]\\ &+\rho^{l+1} a_{22}r^{l'}(2l+1)\dfrac{\sqrt{l'(l'+1)}}{2} \sum\limits_{q=-1}^1(-1)^q [ (-1)^{m'}(\mathscr{A}_{l,l'}^{-m,-m'-q,(21)}(\alpha,q)+(-1)^m\mathscr{A}_{l,l'}^{m,-m'-q,(21)}(\alpha,q))\\ &\langle l',-m'-q;1,q\mid l',-m'\rangle - (\mathscr{A}_{l,l'}^{-m,m'-q,(21)}(\alpha,q)+(-1)^m\mathscr{A}_{l,l'}^{m,m'-q,(21)}(\alpha,q))\langle l',m'-q;1,q\mid l',m'\rangle]\end{align}\]
When \(m>0\), \(m'=0\), \[\begin{align} &M^{21}_{(l',0,3),(lm,2)}(\alpha)\\ =&\sum\limits_{\lambda\in\{l',l'+2\mid l'\geqslant 1\}}\sum\limits_{q=-1}^{1}\sum\limits_{m_1=-1}^{1} -i\cdot \rho^{l+1} (a_{22}-a_{12}) r^\lambda \sqrt{2l'(l'+1)} [(\mathscr{A}_{l,\lambda}^{-m,-q,(21)}(\alpha,q)+\\ &(-1)^m\mathscr{A}_{l,\lambda}^{m,-q,(21)}(\alpha,q))K_{l',l',\lambda}^{m_1,-q,q}]\\ &-i\cdot \rho^{l+1} a_{22}r^{l'}(2l+1)\dfrac{\sqrt{l'(l'+1)}}{\sqrt{2}} \sum\limits_{q=-1}^1(-1)^q [ (\mathscr{A}_{l,l'}^{-m,-q,(21)}(\alpha,q)+(-1)^m\mathscr{A}_{l,l'}^{m,-q,(21)}(\alpha,q))\\ &\langle l',-q;1,q\mid l',0\rangle] \end{align}\]
When \(m<0\), \(m'>0\), \[\begin{align} &M^{21}_{(l'm',3),(lm,2)}(\alpha)\\ =&\sum\limits_{\lambda\in\{l',l'+2\mid l'\geqslant 1\}}\sum\limits_{q=-1}^{1}\sum\limits_{m_1=-1}^{1}\rho^{l+1} (a_{22}-a_{12}) r^\lambda \sqrt{l'(l'+1)} [(-1)^{m'}(\mathscr{A}_{l,\lambda}^{m,m'-q,(21)}(\alpha,q)\\ &-(-1)^m\mathscr{A}_{l,\lambda}^{-m,m'-q,(21)}(\alpha,q))K_{l',l',\lambda}^{m_1,m'-q,q} +(\mathscr{A}_{l,\lambda}^{m,-m'-q,(21)}(\alpha,q)\\ &-(-1)^m\mathscr{A}_{l,\lambda}^{-m,-m'-q,(21)}(\alpha,q))K_{l',l',\lambda}^{m_1,-m'-q,q}]\\ &+\rho^{l+1} a_{22}r^{l'} (2l+1)\dfrac{\sqrt{l'(l'+1)}}{2} \sum\limits_{q=-1}^1(-1)^q [(-1)^{m'}(\mathscr{A}_{l,l'}^{m,m'-q,(21)}(\alpha,q)-(-1)^m\mathscr{A}_{l,l'}^{-m,m'-q,(21)}(\alpha,q))\\ &\langle l',m'-q;1,q\mid l',m'\rangle+(\mathscr{A}_{l,l'}^{m,-m'-q,(21)}(\alpha,q)-(-1)^m\mathscr{A}_{l,l'}^{-m,-m'-q,(21)}(\alpha,q))\langle l',-m'-q;1,q\mid l',-m'\rangle] \end{align}\]
When \(m<0\), \(m'<0\), \[\begin{align} &M^{21}_{(l'm',3),(lm,2)}(\alpha)\\ =&\sum\limits_{\lambda\in\{l',l'+2\mid l'\geqslant 1\}}\sum\limits_{q=-1}^{1}\sum\limits_{m_1=-1}^{1}\rho^{l+1} (a_{22}-a_{12}) r^\lambda i\cdot \sqrt{l'(l'+1)} [(-1)^{m'}(\mathscr{A}_{l,\lambda}^{m,-m'-q,(21)}(\alpha,q)\\ &-(-1)^m\mathscr{A}_{l,\lambda}^{-m,-m'-q,(21)}(\alpha,q))K_{l',l',\lambda}^{m_1,-m'-q,q} - (\mathscr{A}_{l,\lambda}^{m,m'-q,(21)}(\alpha,q)\\ &-(-1)^m\mathscr{A}_{l,\lambda}^{-m,m'-q,(21)}(\alpha,q))K_{l',l',\lambda}^{m_1,m'-q,q}]\\ &-i\cdot \rho^{l+1} a_{22}r^{l'}(2l+1)\dfrac{\sqrt{l'(l'+1)}}{2} \sum\limits_{q=-1}^1(-1)^q [-(-1)^{m'}(\mathscr{A}_{l,l'}^{m,-m'-q,(21)}(\alpha,q)\\ &-(-1)^m\mathscr{A}_{l,l'}^{-m,-m'-q,(21)}(\alpha,q)) \langle l',-m'-q;1,q\mid l',-m'\rangle + (\mathscr{A}_{l,l'}^{m,m'-q,(21)}(\alpha,q)\\ &-(-1)^m\mathscr{A}_{l,l'}^{-m,m'-q,(21)}(\alpha,q))\langle l',m'-q;1,q\mid l',m'\rangle] \end{align}\]
When \(m<0\), \(m'=0\), \[\begin{align} &M^{21}_{(l',0,3),(lm,2)}(\alpha)\\ =&\sum\limits_{\lambda\in\{l',l'+2\mid l'\geqslant 1\}}\sum\limits_{q=-1}^{1}\sum\limits_{m_1=-1}^{1} \rho^{l+1} (a_{22}-a_{12}) r^\lambda \sqrt{2l'(l'+1)} [(\mathscr{A}_{l,\lambda}^{m,-q,(21)}(\alpha,q)\\ &-(-1)^m\mathscr{A}_{l,\lambda}^{-m,-q,(21)}(\alpha,q))K_{l',l',\lambda}^{m_1,-q,q}]\\ &+\rho^{l+1} a_{22}r^{l'} (2l+1)\dfrac{\sqrt{l'(l'+1)}}{\sqrt{2}} \sum\limits_{q=-1}^1(-1)^q [ (\mathscr{A}_{l,l'}^{m,-q,(21)}(\alpha,q)-(-1)^m\mathscr{A}_{l,l'}^{-m,-q,(21)}(\alpha,q))\\ &\langle l',-q;1,q\mid l',0\rangle] \end{align}\]
When \(m=0\), \(m'>0\), \[\begin{align} &M^{21}_{(l'm',3),(l,0,2)}(\alpha)\\ =&\sum\limits_{\lambda\in\{l',l'+2\mid l'\geqslant 1\}}\sum\limits_{q=-1}^{1}\sum\limits_{m_1=-1}^{1} -i\cdot \rho^{l+1} (a_{22}-a_{12}) r^\lambda \sqrt{2l'(l'+1)} [(-1)^{m'}\mathscr{A}_{l,\lambda}^{0,m'-q,(21)}(\alpha,q) K_{l',l',\lambda}^{m_1,m'-q,q}\\ &+\mathscr{A}_{l,\lambda}^{0,-m'-q,(21)}(\alpha,q)K_{l',l',\lambda}^{m_1,-m'-q,q}]\\ &-i\cdot \rho^{l+1} a_{22}r^{l'}(2l+1)\dfrac{\sqrt{l'(l'+1)}}{\sqrt{2}} \sum\limits_{q=-1}^1(-1)^q [(-1)^{m'}\mathscr{A}_{l,l'}^{0,m'-q,(21)}(\alpha,q)\\ &\langle l',m'-q;1,q\mid l',m'\rangle+(\mathscr{A}_{l,l'}^{0,-m'-q,(21)}(\alpha,q)\langle l',-m'-q;1,q\mid l',-m'\rangle] \end{align}\]
When \(m=0\), \(m'<0\),
\[\begin{align} &M^{21}_{(l'm',3),(l,0,2)}(\alpha)\\ =&\sum\limits_{\lambda\in\{l',l'+2\mid l'\geqslant 1\}}\sum\limits_{q=-1}^{1}\sum\limits_{m_1=-1}^{1}\rho^{l+1} (a_{22}-a_{12}) r^\lambda \sqrt{2l'(l'+1)} [(-1)^{m'}\mathscr{A}_{l,\lambda}^{0,-m'-q,(21)}(\alpha,q)K_{l',l',\lambda}^{m_1,-m'-q,q}\\ & - \mathscr{A}_{l,\lambda}^{0,m'-q,(21)}(\alpha,q)K_{l',l',\lambda}^{m_1,m'-q,q}]\\ &+\rho^{l+1} a_{22}r^{l'}(2l+1)\dfrac{\sqrt{l'(l'+1)}}{\sqrt{2}} \sum\limits_{q=-1}^1(-1)^q [ (-1)^{m'} \mathscr{A}_{l,l'}^{0,-m'-q,(21)}(\alpha,q)\\ &\langle l',-m'-q;1,q\mid l',-m'\rangle- \mathscr{A}_{l,l'}^{0,m'-q,(21)}(\alpha,q)\langle l',m'-q;1,q\mid l',m'\rangle] \end{align}\]
When \(m=0\), \(m'=0\), \[\begin{align} &M^{21}_{(l',0,3),(l,0,2)}(\alpha)\\ =&\sum\limits_{\lambda\in\{l',l'+2\mid l'\geqslant 1\}}\sum\limits_{q=-1}^{1}\sum\limits_{m_1=-1}^{1} -i\cdot \rho^{l+1} (a_{22}-a_{12}) r^\lambda \sqrt{4l'(l'+1)} [\mathscr{A}_{l,\lambda}^{0,-q,(21)}(\alpha,q)K_{l',l',\lambda}^{m_1,-q,q}]\\ &-i\cdot \rho^{l+1} a_{22}r^{l'}(2l+1)\sqrt{l'(l'+1)} \sum\limits_{q=-1}^1(-1)^q [ \mathscr{A}_{l,l'}^{0,-q,(21)}(\alpha,q)\langle l',-q;1,q\mid l',0\rangle] \end{align}\]
Theorem 28. Entries for \(\boxed{M^{21}_{(l'm',3),(lm,3)}(\alpha)}\)(\(l'\geqslant 1, l\geqslant 1\), otherwise this term vanishes):
When \(m>0,\;m'>0\): \[\begin{align}
&M^{21}_{(l'm',3),(lm,3)}(\alpha)\\
=&\;\rho^{l+2}a_{33}r^{l'}\Biggl\{
\frac{l'(l'+1)}{2}\Bigl[(-1)^{m'}\bigl(\mathscr{H}_{l,l'}^{-m,m',(21)}(\alpha)+(-1)^m \mathscr{H}_{l,l'}^{m,m',(21)}(\alpha)\bigr)
+\bigl(\mathscr{H}_{l,l'}^{-m,-m',(21)}(\alpha)\\
&+(-1)^m \mathscr{H}_{l,l'}^{m,-m',(21)}(\alpha)\bigr)\Bigr]\\
&-\frac{i\sqrt{l'(l'+1)}}{2}\sum_{q=-1}^{1}\sum_{m_1=-1}^{1}\Bigl[(-1)^{m'}\bigl(\mathscr{L}_{l,l',l'+1}^{-m,m'-q,q,m_1,(21)}(\alpha)+(-1)^m \mathscr{L}_{l,l',l'+1}^{m,m'-q,q,m_1,(21)}(\alpha)\bigr)\\
&\qquad\qquad\qquad\qquad\quad+\bigl(\mathscr{L}_{l,l',l'+1}^{-m,-m'-q,q,m_1,(21)}(\alpha)+(-1)^m \mathscr{L}_{l,l',l'+1}^{m,-m'-q,q,m_1,(21)}(\alpha)\bigr)\Bigr]\Biggr\}\end{align}\]
When \(m>0,\;m'<0\): \[\begin{align} &M^{21}_{(l'm',3),(lm,3)}(\alpha)\\ =&\;\rho^{l+2}a_{33}r^{l'}\Biggl\{ \frac{l'(l'+1)}{2}\Bigl[i(-1)^{m'}\bigl(\mathscr{H}_{l,l'}^{-m,-m',(21)}(\alpha)+(-1)^m \mathscr{H}_{l,l'}^{m,-m',(21)}(\alpha)\bigr) -i\bigl(\mathscr{H}_{l,l'}^{-m,m',(21)}(\alpha)\\ &+(-1)^m \mathscr{H}_{l,l'}^{m,m',(21)}(\alpha)\bigr)\Bigr]\\ &+\frac{\sqrt{l'(l'+1)}}{2}\sum_{q=-1}^{1}\sum_{m_1=-1}^{1}\Bigl[(-1)^{m'}\bigl(\mathscr{L}_{l,l',l'+1}^{-m,-m'-q,q,m_1,(21)}(\alpha)+(-1)^m \mathscr{L}_{l,l',l'+1}^{m,-m'-q,q,m_1,(21)}(\alpha)\bigr)\\ &\qquad\qquad\qquad\qquad\quad-\bigl(\mathscr{L}_{l,l',l'+1}^{-m,m'-q,q,m_1,(21)}(\alpha)+(-1)^m \mathscr{L}_{l,l',l'+1}^{m,m'-q,q,m_1,(21)}(\alpha)\bigr)\Bigr]\Biggr\} \end{align}\]
When \(m>0,\;m'=0\): \[\begin{align} &M^{21}_{(l',0,3),(lm,3)}(\alpha)\\ =&\;\rho^{l+2}a_{33}r^{l'}\Biggl\{ \frac{l'(l'+1)}{\sqrt{2}}\bigl(\mathscr{H}_{l,l'}^{-m,0,(21)}(\alpha)+(-1)^m \mathscr{H}_{l,l'}^{m,0,(21)}(\alpha)\bigr)\\ &-\frac{i\sqrt{l'(l'+1)}}{\sqrt{2}}\sum_{q=-1}^{1}\sum_{m_1=-1}^{1}\bigl(\mathscr{L}_{l,l',l'+1}^{-m,-q,q,m_1,(21)}(\alpha)+(-1)^m \mathscr{L}_{l,l',l'+1}^{m,-q,q,m_1,(21)}(\alpha)\bigr)\Biggr\} \end{align}\]
When \(m<0,\;m'>0\): \[\begin{align} &M^{21}_{(l'm',3),(lm,3)}(\alpha)\\ =&\;\rho^{l+2}a_{33}r^{l'}\Biggl\{ \frac{l'(l'+1)}{2}\Bigl[i(-1)^{m'}\bigl(\mathscr{H}_{l,l'}^{m,m',(21)}(\alpha)-(-1)^m \mathscr{H}_{l,l'}^{-m,m',(21)}(\alpha)\bigr) +i\bigl(\mathscr{H}_{l,l'}^{m,-m',(21)}(\alpha)\\ &-(-1)^m \mathscr{H}_{l,l'}^{-m,-m',(21)}(\alpha)\bigr)\Bigr]\\ &+\frac{\sqrt{l'(l'+1)}}{2}\sum_{q=-1}^{1}\sum_{m_1=-1}^{1}\Bigl[(-1)^{m'}\bigl(\mathscr{L}_{l,l',l'+1}^{m,m'-q,q,m_1,(21)}(\alpha)-(-1)^m \mathscr{L}_{l,l',l'+1}^{-m,m'-q,q,m_1,(21)}(\alpha)\bigr)\\ &\qquad\qquad\qquad\qquad\quad+\bigl(\mathscr{L}_{l,l',l'+1}^{m,-m'-q,q,m_1,(21)}(\alpha)-(-1)^m \mathscr{L}_{l,l',l'+1}^{-m,-m'-q,q,m_1,(21)}(\alpha)\bigr)\Bigr]\Biggr\} \end{align}\]
When \(m<0,\;m'<0\): \[\begin{align} &M^{21}_{(l'm',3),(lm,3)}(\alpha)\\ =&\;\rho^{l+2}a_{33}r^{l'}\Biggl\{ \frac{l'(l'+1)}{2}\Bigl[\bigl(\mathscr{H}_{l,l'}^{m,m',(21)}(\alpha)-(-1)^m \mathscr{H}_{l,l'}^{-m,m',(21)}(\alpha)\bigr)\\ & -(-1)^{m'}\bigl(\mathscr{H}_{l,l'}^{m,-m',(21)}(\alpha)-(-1)^m \mathscr{H}_{l,l'}^{-m,-m',(21)}(\alpha)\bigr)\Bigr]\\ &+\frac{i\sqrt{l'(l'+1)}}{2}\sum_{q=-1}^{1}\sum_{m_1=-1}^{1}\Bigl[(-1)^{m'}\bigl(\mathscr{L}_{l,l',l'+1}^{m,-m'-q,q,m_1,(21)}(\alpha)-(-1)^m \mathscr{L}_{l,l',l'+1}^{-m,-m'-q,q,m_1,(21)}(\alpha)\bigr)\\ &\qquad\qquad\qquad\qquad\quad-\bigl(\mathscr{L}_{l,l',l'+1}^{m,m'-q,q,m_1,(21)}(\alpha)-(-1)^m \mathscr{L}_{l,l',l'+1}^{-m,m'-q,q,m_1,(21)}(\alpha)\bigr)\Bigr]\Biggr\} \end{align}\]
When \(m<0,\;m'=0\): \[\begin{align} &M^{21}_{(l',0,3),(lm,3)}(\alpha)\\ =&\;\rho^{l+2}a_{33}r^{l'}\Biggl\{ \frac{il'(l'+1)}{\sqrt{2}}\bigl(\mathscr{H}_{l,l'}^{m,0,(21)}(\alpha)-(-1)^m \mathscr{H}_{l,l'}^{-m,0,(21)}(\alpha)\bigr)\\ &+\frac{\sqrt{l'(l'+1)}}{\sqrt{2}}\sum_{q=-1}^{1}\sum_{m_1=-1}^{1}\bigl(\mathscr{L}_{l,l',l'+1}^{m,-q,q,m_1,(21)}(\alpha)-(-1)^m \mathscr{L}_{l,l',l'+1}^{-m,-q,q,m_1,(21)}(\alpha)\bigr)\Biggr\}\end{align}\]
When \(m=0,\;m'>0\): \[\begin{align} &M^{21}_{(l'm',3),(l,0,3)}(\alpha)\\ =&\;\rho^{l+2}a_{33}r^{l'}\Biggl\{ \frac{l'(l'+1)}{\sqrt{2}}\Bigl[(-1)^{m'}\mathscr{H}_{l,l'}^{0,m',(21)}(\alpha)+\mathscr{H}_{l,l'}^{0,-m',(21)}(\alpha)\Bigr]\\ &-\frac{i\sqrt{l'(l'+1)}}{\sqrt{2}}\sum_{q=-1}^{1}\sum_{m_1=-1}^{1}\Bigl[(-1)^{m'}\mathscr{L}_{l,l',l'+1}^{0,m'-q,q,m_1,(21)}(\alpha)+\mathscr{L}_{l,l',l'+1}^{0,-m'-q,q,m_1,(21)}(\alpha)\Bigr]\Biggr\}\end{align}\]
When \(m=0,\;m'<0\): \[\begin{align} &M^{21}_{(l'm',3),(l,0,3)}(\alpha)\\ =&\;\rho^{l+2}a_{33}r^{l'}\Biggl\{ \frac{l'(l'+1)}{\sqrt{2}}\Bigl[i(-1)^{m'}\mathscr{H}_{l,l'}^{0,-m',(21)}(\alpha)-i\mathscr{H}_{l,l'}^{0,m',(21)}(\alpha)\Bigr]\\ &+\frac{\sqrt{l'(l'+1)}}{\sqrt{2}}\sum_{q=-1}^{1}\sum_{m_1=-1}^{1}\Bigl[(-1)^{m'}\mathscr{L}_{l,l',l'+1}^{0,-m'-q,q,m_1,(21)}(\alpha)-\mathscr{L}_{l,l',l'+1}^{0,m'-q,q,m_1,(21)}(\alpha)\Bigr]\Biggr\}\end{align}\]
When \(m=0,\;m'=0\): \[\begin{align} &M^{21}_{(l',0,3),(l,0,3)}(\alpha)\\ =&\;\rho^{l+2}a_{33}r^{l'}\Biggl\{ l'(l'+1)\mathscr{H}_{l,l'}^{0,0,(21)}(\alpha) -i\sqrt{l'(l'+1)}\sum_{q=-1}^{1}\sum_{m_1=-1}^{1}\mathscr{L}_{l,l',l'+1}^{0,-q,q,m_1,(21)}(\alpha)\Biggr\} \end{align}\]
Lemma 15. Define \(\mathscr{D}_{l,\lambda}^{m,\mu,(21)}(\alpha)=\sum\limits_{n\in\mathbb{Z}} (n+2d)^2H_{l,\lambda}^{m,\mu}(\mathbf{b}_{21})e^{-in\alpha}\). Then \[\begin{align} &\mathscr{D}_{l,\lambda}^{m,\mu,(21)}(\alpha)\\ =&(-1)^{\lambda+\mu} \sqrt{\dfrac{2l+1}{2\lambda+1}} \sqrt{{l+\lambda+\mu-m\choose \lambda+\mu}{l+\lambda+m-\mu\choose \lambda-\mu}} \sqrt{\dfrac{4\pi}{2(l+\lambda)+1}}\\ & \big(\mathrm{\Phi}(e^{-i\alpha},l+\lambda-1,2d)Y_{l+\lambda}^{m-\mu}(\dfrac{\pi}{2},\pi)+e^{i\alpha}\mathrm{\Phi}(e^{i\alpha},l+\lambda-1,1-2d)Y_{l+\lambda}^{m-\mu}(\dfrac{\pi}{2},0)\big) \end{align}\]
Theorem 29. The entries \(\boxed{M^{21}_{(l'm',2),(lm,2)}(\alpha)}\)(\(l'\geqslant 1,l\geqslant 1\), otherwise this term vanishes) are given by:
If \(m>0\) and \(m'>0\), \[\begin{align}&M^{21}_{(l'm',2),(lm,2)}(\alpha)\\
&=\dfrac{l'(2l'+1)}{2}[ \rho^{l+3}a_{12}r^{l'-1}+\rho^{l+1}(a_{22}-a_{12})(r^{l'+1} )+\rho^{l+1}a_{22}r^{l'+1}\dfrac{2l+1}{2l'+1}]\\
&[(-1)^{m'}(\mathscr{H}_{l,l'}^{-m,m',(21)}(\alpha)+(-1)^m\mathscr{H}_{l,l'}^{m,m',(21)}(\alpha))+(\mathscr{H}_{l,l'}^{-m,-m',(21)}(\alpha)+(-1)^m\mathscr{H}_{l,l'}^{m,-m',(21)}(\alpha))]\\
&+\dfrac{l'(2l'+1)}{2}[\rho^{l+1}(a_{22}-a_{12})r^{l'-1}][(-1)^{m'}(\mathscr{D}_{l,l'}^{-m,m',(21)}(\alpha)+(-1)^m\mathscr{D}_{l,l'}^{m,m',(21)}(\alpha))\\
&+(\mathscr{D}_{l,l'}^{-m,-m',(21)}(\alpha)+(-1)^m\mathscr{D}_{l,l'}^{m,-m',(21)}(\alpha))]\\
&+\rho^{l+1}(a_{22}-a_{12})\sqrt{l'(2l'+1)}\sum\limits_{\lambda\in\{l'+1,l'-1\}}\sum\limits_{q=-1}^{q=1}\sum\limits_{m_1=-1}^{1}r^\lambda [(-1)^{m'}(\mathscr{A}_{l,\lambda}^{-m,m'-q,(21)}(\alpha,q)\\
&+(-1)^m\mathscr{A}_{l,\lambda}^{m,m'-q,(21)}(\alpha,q))K_{l'-1,l',\lambda}^{m_1,m'-q,q}+(\mathscr{A}_{l,\lambda}^{-m,-m'-q,(21)}(\alpha,q)+(-1)^m\mathscr{A}_{l,\lambda}^{m,-m'-q,(21)}(\alpha,q))K_{l'-1,l',\lambda}^{m_1,-m'-q,q}]\\
&+\rho^{l+1}a_{22}(2l+1)\dfrac{\sqrt{l'(2l'+1)}}{2}\sum\limits_{q=-1}^1(-1)^q r^{l'-1}[(-1)^{m'}(\mathscr{A}_{l,l'-1}^{-m,m'-q,(21)}(\alpha,q)+(-1)^m\mathscr{A}_{l,l'-1}^{m,m'-q,(21)}(\alpha,q))\\
&\langle l'-1,m'-q;1,q\mid l',m' \rangle+(\mathscr{A}_{l,l'-1}^{-m,-m'-q,(21)}(\alpha,q)\\
&+(-1)^m\mathscr{A}_{l,l'-1}^{m,-m'-q,(21)}(\alpha,q))\langle l'-1,-m'-q;1,q\mid l',-m' \rangle]
\end{align}\]
If \(m>0\) and \(m'<0\), \[\begin{align} &M^{21}_{(l'm',2),(lm,2)}(\alpha)\\ &=\dfrac{l'(2l'+1)}{2}\Bigl[ \rho^{l+3}a_{12}r^{l'-1}+\rho^{l+1}(a_{22}-a_{12})(r^{l'+1} )+\rho^{l+1}a_{22}r^{l'+1}\dfrac{2l+1}{2l'+1}\Bigr]\\ &\quad\Bigl[i\cdot(-1)^{m'}(\mathscr{H}_{l,l'}^{-m,-m',(21)}(\alpha)+(-1)^m\mathscr{H}_{l,l'}^{m,-m',(21)}(\alpha))-i\cdot(\mathscr{H}_{l,l'}^{-m,m',(21)}(\alpha)+(-1)^m\mathscr{H}_{l,l'}^{m,m',(21)}(\alpha))\Bigr]\\ &+\dfrac{l'(2l'+1)}{2}[\rho^{l+1}(a_{22}-a_{12})r^{l'-1}]\Bigl[i\cdot(-1)^{m'}(\mathscr{D}_{l,l'}^{-m,-m',(21)}(\alpha)+(-1)^m\mathscr{D}_{l,l'}^{m,-m',(21)}(\alpha))\\ &-i\cdot(\mathscr{D}_{l,l'}^{-m,m',(21)}(\alpha)+(-1)^m\mathscr{D}_{l,l'}^{m,m',(21)}(\alpha))\Bigr] &+\rho^{l+1}(a_{22}-a_{12})\sqrt{l'(2l'+1)}\sum_{\lambda\in\{l'+1,l'-1\}}\sum_{q=-1}^{1}\sum_{m_1=-1}^{1}r^\lambda \Bigl[i\cdot(-1)^{m'}(\mathscr{A}_{l,\lambda}^{-m,-m'-q,(21)}(\alpha,q)\\ &\quad+(-1)^m\mathscr{A}_{l,\lambda}^{m,-m'-q,(21)}(\alpha,q))K_{l'-1,l',\lambda}^{m_1,-m'-q,q}-i\cdot(\mathscr{A}_{l,\lambda}^{-m,m'-q,(21)}(\alpha,q)+\\ &(-1)^m\mathscr{A}_{l,\lambda}^{m,m'-q,(21)}(\alpha,q))K_{l'-1,l',\lambda}^{m_1,m'-q,q}\Bigr]\\ &+\rho^{l+1}a_{22}(2l+1)\dfrac{\sqrt{l'(2l'+1)}}{2}\sum_{q=-1}^{1}(-1)^q r^{l'-1}\Bigl[i\cdot(-1)^{m'}(\mathscr{A}_{l,l'-1}^{-m,-m'-q,(21)}(\alpha,q)\\ &+(-1)^m\mathscr{A}_{l,l'-1}^{m,-m'-q,(21)}(\alpha,q))\langle l'-1,-m'-q;1,q\mid l',-m' \rangle-i\cdot(\mathscr{A}_{l,l'-1}^{-m,m'-q,(21)}(\alpha,q)\\ &+(-1)^m\mathscr{A}_{l,l'-1}^{m,m'-q,(21)}(\alpha,q))\langle l'-1,m'-q;1,q\mid l',m' \rangle\Bigr] \end{align}\]
If \(m>0\) and \(m'=0\), \[\begin{align} &M^{21}_{(l',0,2),(lm,2)}(\alpha)\\ &=\dfrac{l'(2l'+1)}{\sqrt{2}}\Bigl[ \rho^{l+3}a_{12}r^{l'-1}+\rho^{l+1}(a_{22}-a_{12})(r^{l'+1} )+\rho^{l+1}a_{22}r^{l'+1}\dfrac{2l+1}{2l'+1}\Bigr]\\ &\quad\Bigl[(\mathscr{H}_{l,l'}^{-m,0,(21)}(\alpha)+(-1)^m\mathscr{H}_{l,l'}^{m,0,(21)}(\alpha))\Bigr]\\ &+\dfrac{l'(2l'+1)}{\sqrt{2}}[\rho^{l+1}(a_{22}-a_{12})r^{l'-1}]\Bigl[(\mathscr{D}_{l,l'}^{-m,0,(21)}(\alpha)+(-1)^m\mathscr{D}_{l,l'}^{m,0,(21)}(\alpha))\Bigr]\\ &+\rho^{l+1}(a_{22}-a_{12})\sqrt{2l'(2l'+1)}\sum_{\lambda\in\{l'+1,l'-1\}}\sum_{q=-1}^{1}\sum_{m_1=-1}^{1}r^\lambda \Bigl[(\mathscr{A}_{l,\lambda}^{-m,-q,(21)}(\alpha,q)\\ &+(-1)^m\mathscr{A}_{l,\lambda}^{m,-q,(21)}(\alpha,q))K_{l'-1,l',\lambda}^{m_1,-q,q}\Bigr]\\ &+\rho^{l+1}a_{22}(2l+1)\dfrac{\sqrt{l'(2l'+1)}}{\sqrt{2}}\sum_{q=-1}^{1}(-1)^q r^{l'-1}\Bigl[(\mathscr{A}_{l,l'-1}^{-m,-q,(21)}(\alpha,q)\\ &+(-1)^m\mathscr{A}_{l,l'-1}^{m,-q,(21)}(\alpha,q))\langle l'-1,-q;1,q\mid l',0 \rangle\Bigr]. \end{align}\]
If \(m<0\) and \(m'>0\), \[\begin{align} &M^{21}_{(l'm',2),(lm,2)}(\alpha)\\ &=\dfrac{l'(2l'+1)}{2}\Bigl[ \rho^{l+3}a_{12}r^{l'-1}+\rho^{l+1}(a_{22}-a_{12})(r^{l'+1} )+\rho^{l+1}a_{22}r^{l'+1}\dfrac{2l+1}{2l'+1}\Bigr]\\ &\quad\cdot i\Bigl[(-1)^{m'}(\mathscr{H}_{l,l'}^{m,m',(21)}(\alpha)-(-1)^m\mathscr{H}_{l,l'}^{-m,m',(21)}(\alpha))+(\mathscr{H}_{l,l'}^{m,-m',(21)}(\alpha)-(-1)^m\mathscr{H}_{l,l'}^{-m,-m',(21)}(\alpha))\Bigr]\\ &+\dfrac{l'(2l'+1)}{2}[\rho^{l+1}(a_{22}-a_{12})r^{l'-1}]\cdot i\Bigl[(-1)^{m'}(\mathscr{D}_{l,l'}^{m,m',(21)}(\alpha)-(-1)^m\mathscr{D}_{l,l'}^{-m,m',(21)}(\alpha))\\ &+(\mathscr{D}_{l,l'}^{m,-m',(21)}(\alpha)-(-1)^m\mathscr{D}_{l,l'}^{-m,-m',(21)}(\alpha))\Bigr]\\ &+\rho^{l+1}(a_{22}-a_{12})\sqrt{l'(2l'+1)}\sum_{\lambda\in\{l'+1,l'-1\}}\sum_{q=-1}^{1}\sum_{m_1=-1}^{1}r^\lambda \;i\Bigl[(-1)^{m'}(\mathscr{A}_{l,\lambda}^{m,m'-q,(21)}(\alpha,q)\\ &\quad-(-1)^m\mathscr{A}_{l,\lambda}^{-m,m'-q,(21)}(\alpha,q))K_{l'-1,l',\lambda}^{m_1,m'-q,q}+(\mathscr{A}_{l,\lambda}^{m,-m'-q,(21)}(\alpha,q)\\ &-(-1)^m\mathscr{A}_{l,\lambda}^{-m,-m'-q,(21)}(\alpha,q))K_{l'-1,l',\lambda}^{m_1,-m'-q,q}\Bigr]\\ &+\rho^{l+1}a_{22}(2l+1)\dfrac{\sqrt{l'(2l'+1)}}{2}\sum_{q=-1}^{1}(-1)^q r^{l'-1}\;i\Bigl[(-1)^{m'}(\mathscr{A}_{l,l'-1}^{m,m'-q,(21)}(\alpha,q)-(-1)^m\mathscr{A}_{l,l'-1}^{-m,m'-q,(21)}(\alpha,q))\\ &\quad\langle l'-1,m'-q;1,q\mid l',m' \rangle+(\mathscr{A}_{l,l'-1}^{m,-m'-q,(21)}(\alpha,q)\\ &-(-1)^m\mathscr{A}_{l,l'-1}^{-m,-m'-q,(21)}(\alpha,q))\langle l'-1,-m'-q;1,q\mid l',-m' \rangle\Bigr]. \end{align}\]
If \(m<0\) and \(m'<0\), \[\begin{align} &M^{21}_{(l'm',2),(lm,2)}(\alpha)\\ &=\dfrac{l'(2l'+1)}{2}\Bigl[ \rho^{l+3}a_{12}r^{l'-1}+\rho^{l+1}(a_{22}-a_{12})(r^{l'+1} )+\rho^{l+1}a_{22}r^{l'+1}\dfrac{2l+1}{2l'+1}\Bigr]\\ &\quad\Bigl[-(-1)^{m'}(\mathscr{H}_{l,l'}^{m,-m',(21)}(\alpha)-(-1)^m\mathscr{H}_{l,l'}^{-m,-m',(21)}(\alpha))+(\mathscr{H}_{l,l'}^{m,m',(21)}(\alpha)-(-1)^m\mathscr{H}_{l,l'}^{-m,m',(21)}(\alpha))\Bigr]\\ &+\dfrac{l'(2l'+1)}{2}[\rho^{l+1}(a_{22}-a_{12})r^{l'-1}]\Bigl[-(-1)^{m'}(\mathscr{D}_{l,l'}^{m,-m',(21)}(\alpha)-(-1)^m\mathscr{D}_{l,l'}^{-m,-m',(21)}(\alpha))\\ &+(\mathscr{D}_{l,l'}^{m,m',(21)}(\alpha)-(-1)^m\mathscr{D}_{l,l'}^{-m,m',(21)}(\alpha))\Bigr]\\ &+\rho^{l+1}(a_{22}-a_{12})\sqrt{l'(2l'+1)}\sum_{\lambda\in\{l'+1,l'-1\}}\sum_{q=-1}^{1}\sum_{m_1=-1}^{1}r^\lambda \Bigl[-(-1)^{m'}(\mathscr{A}_{l,\lambda}^{m,-m'-q,(21)}(\alpha,q)\\ &\quad-(-1)^m\mathscr{A}_{l,\lambda}^{-m,-m'-q,(21)}(\alpha,q))K_{l'-1,l',\lambda}^{m_1,-m'-q,q}+(\mathscr{A}_{l,\lambda}^{m,m'-q,(21)}(\alpha,q)\\ &-(-1)^m\mathscr{A}_{l,\lambda}^{-m,m'-q,(21)}(\alpha,q))K_{l'-1,l',\lambda}^{m_1,m'-q,q}\Bigr]\\ &+\rho^{l+1}a_{22}(2l+1)\dfrac{\sqrt{l'(2l'+1)}}{2}\sum_{q=-1}^{1}(-1)^q r^{l'-1}\Bigl[-(-1)^{m'}(\mathscr{A}_{l,l'-1}^{m,-m'-q,(21)}(\alpha,q)\\ &-(-1)^m\mathscr{A}_{l,l'-1}^{-m,-m'-q,(21)}(\alpha,q))\\ &\quad\langle l'-1,-m'-q;1,q\mid l',-m' \rangle+(\mathscr{A}_{l,l'-1}^{m,m'-q,(21)}(\alpha,q)\\ &-(-1)^m\mathscr{A}_{l,l'-1}^{-m,m'-q,(21)}(\alpha,q))\langle l'-1,m'-q;1,q\mid l',m' \rangle\Bigr].\end{align}\]
If \(m<0\) and \(m'=0\), \[\begin{align} &M^{21}_{(l',0,2),(lm,2)}(\alpha)\\ &=\dfrac{l'(2l'+1)}{\sqrt{2}}\Bigl[ \rho^{l+3}a_{12}r^{l'-1}+\rho^{l+1}(a_{22}-a_{12})(r^{l'+1} )+\rho^{l+1}a_{22}r^{l'+1}\dfrac{2l+1}{2l'+1}\Bigr] \cdot i\Bigl[(\mathscr{H}_{l,l'}^{m,0,(21)}(\alpha)\\ &-(-1)^m\mathscr{H}_{l,l'}^{-m,0,(21)}(\alpha))\Bigr]\\ &+\dfrac{l'(2l'+1)}{\sqrt{2}}[\rho^{l+1}(a_{22}-a_{12})r^{l'-1}]\cdot i\Bigl[(\mathscr{D}_{l,l'}^{m,0,(21)}(\alpha)-(-1)^m\mathscr{D}_{l,l'}^{-m,0,(21)}(\alpha))\Bigr]\\ &+\rho^{l+1}(a_{22}-a_{12})\sqrt{2l'(2l'+1)}\sum_{\lambda\in\{l'+1,l'-1\}}\sum_{q=-1}^{1}\sum_{m_1=-1}^{1}r^\lambda \;i\Bigl[(\mathscr{A}_{l,\lambda}^{m,-q,(21)}(\alpha,q)\\ &-(-1)^m\mathscr{A}_{l,\lambda}^{-m,-q,(21)}(\alpha,q))K_{l'-1,l',\lambda}^{m_1,-q,q}\Bigr]\\ &+\rho^{l+1}a_{22}(2l+1)\dfrac{\sqrt{l'(2l'+1)}}{\sqrt{2}}\sum_{q=-1}^{1}(-1)^q r^{l'-1}\;i\Bigl[(\mathscr{A}_{l,l'-1}^{m,-q,(21)}(\alpha,q)\\ &-(-1)^m\mathscr{A}_{l,l'-1}^{-m,-q,(21)}(\alpha,q))\langle l'-1,-q;1,q\mid l',0 \rangle\Bigr]. \end{align}\]
If \(m=0\) and \(m'>0\), \[\begin{align} &M^{21}_{(l'm',2),(l,0,2)}(\alpha)\\ &=\dfrac{l'(2l'+1)}{\sqrt{2}}\Bigl[ \rho^{l+3}a_{12}r^{l'-1}+\rho^{l+1}(a_{22}-a_{12})(r^{l'+1} )+\rho^{l+1}a_{22}r^{l'+1}\dfrac{2l+1}{2l'+1}\Bigr]\\ &\quad\Bigl[(-1)^{m'}\mathscr{H}_{l,l'}^{0,m',(21)}(\alpha)+\mathscr{H}_{l,l'}^{0,-m',(21)}(\alpha)\Bigr]\\ &+\dfrac{l'(2l'+1)}{\sqrt{2}}[\rho^{l+1}(a_{22}-a_{12})r^{l'-1}]\Bigl[(-1)^{m'}\mathscr{D}_{l,l'}^{0,m',(21)}(\alpha)+\mathscr{D}_{l,l'}^{0,-m',(21)}(\alpha)\Bigr]\\ &+\rho^{l+1}(a_{22}-a_{12})\sqrt{2l'(2l'+1)}\sum_{\lambda\in\{l'+1,l'-1\}}\sum_{q=-1}^{1}\sum_{m_1=-1}^{1}r^\lambda \Bigl[(-1)^{m'}\mathscr{A}_{l,\lambda}^{0,m'-q,(21)}(\alpha,q)K_{l'-1,l',\lambda}^{m_1,m'-q,q}\\ &+\mathscr{A}_{l,\lambda}^{0,-m'-q,(21)}(\alpha,q)K_{l'-1,l',\lambda}^{m_1,-m'-q,q}\Bigr]\\ &+\rho^{l+1}a_{22}(2l+1)\dfrac{\sqrt{l'(2l'+1)}}{\sqrt{2}}\sum_{q=-1}^{1}(-1)^q r^{l'-1}\Bigl[(-1)^{m'}\mathscr{A}_{l,l'-1}^{0,m'-q,(21)}(\alpha,q)\langle l'-1,m'-q;1,q\mid l',m' \rangle\\ &+\mathscr{A}_{l,l'-1}^{0,-m'-q,(21)}(\alpha,q)\langle l'-1,-m'-q;1,q\mid l',-m' \rangle\Bigr]. \end{align}\]
If \(m=0\) and \(m'<0\), \[\begin{align} &M^{21}_{(l'm',2),(l,0,2)}(\alpha)\\ &=\dfrac{l'(2l'+1)}{\sqrt{2}}\Bigl[ \rho^{l+3}a_{12}r^{l'-1}+\rho^{l+1}(a_{22}-a_{12})(r^{l'+1} )+\rho^{l+1}a_{22}r^{l'+1}\dfrac{2l+1}{2l'+1}\Bigr]\\ &\quad i\Bigl[(-1)^{m'}\mathscr{H}_{l,l'}^{0,-m',(21)}(\alpha)-\mathscr{H}_{l,l'}^{0,m',(21)}(\alpha)\Bigr]\\ &+\dfrac{l'(2l'+1)}{\sqrt{2}}[\rho^{l+1}(a_{22}-a_{12})r^{l'-1}]\cdot i\Bigl[(-1)^{m'}\mathscr{D}_{l,l'}^{0,-m',(21)}(\alpha)-\mathscr{D}_{l,l'}^{0,m',(21)}(\alpha)\Bigr]\\ &+\rho^{l+1}(a_{22}-a_{12})\sqrt{2l'(2l'+1)}\sum_{\lambda\in\{l'+1,l'-1\}}\sum_{q=-1}^{1}\sum_{m_1=-1}^{1}r^\lambda \;i\Bigl[(-1)^{m'}\mathscr{A}_{l,\lambda}^{0,-m'-q,(21)}(\alpha,q)K_{l'-1,l',\lambda}^{m_1,-m'-q,q}\\ &-\mathscr{A}_{l,\lambda}^{0,m'-q,(21)}(\alpha,q)K_{l'-1,l',\lambda}^{m_1,m'-q,q}\Bigr]\\ &+\rho^{l+1}a_{22}(2l+1)\dfrac{\sqrt{l'(2l'+1)}}{\sqrt{2}}\sum_{q=-1}^{1}(-1)^q r^{l'-1}\;i\Bigl[(-1)^{m'}\mathscr{A}_{l,l'-1}^{0,-m'-q,(21)}(\alpha,q)\langle l'-1,-m'-q;1,q\mid l',-m' \rangle\\ &-\mathscr{A}_{l,l'-1}^{0,m'-q,(21)}(\alpha,q)\langle l'-1,m'-q;1,q\mid l',m' \rangle\Bigr]. \end{align}\]
If \(m=0\) and \(m'=0\), \[\begin{align} &M^{21}_{(l',0,2),(l,0,2)}(\alpha)\\ &=l'(2l'+1)\Bigl[ \rho^{l+3}a_{12}r^{l'-1}+\rho^{l+1}(a_{22}-a_{12})(r^{l'+1} )+\rho^{l+1}a_{22}r^{l'+1}\dfrac{2l+1}{2l'+1}\Bigr]\mathscr{H}_{l,l'}^{0,0,(21)}(\alpha)\\ &+{l'(2l'+1)}[\rho^{l+1}(a_{22}-a_{12})r^{l'-1}]\mathscr{D}_{l,l'}^{0,0,(21)}(\alpha)\\ &+\rho^{l+1}(a_{22}-a_{12})\cdot 2\sqrt{l'(2l'+1)}\sum_{\lambda\in\{l'+1,l'-1\}}\sum_{q=-1}^{1}\sum_{m_1=-1}^{1}r^\lambda \;\mathscr{A}_{l,\lambda}^{0,-q,(21)}(\alpha,q)K_{l'-1,l',\lambda}^{m_1,-q,q}\\ &+\rho^{l+1}a_{22}(2l+1)\sqrt{l'(2l'+1)}\sum_{q=-1}^{1}(-1)^q r^{l'-1}\;\mathscr{A}_{l,l'-1}^{0,-q,(21)}(\alpha,q)\langle l'-1,-q;1,q\mid l',0 \rangle. \end{align}\]
Theorem 30. The remaining terms are simple: \[M^{21}_{(l'm',3),(lm,1)}(\alpha)=M^{21}_{(l'm',1),(lm,3)}(\alpha)=0,\] \[M^{21}_{(l'm',1),(lm1)}(\alpha)=0.\]
Lemma 16. Denote the series \(\sum\limits_{n\in\mathbb{Z}}H_{l,\lambda}^{m,\mu}(\mathbf{b}_{12})e^{-in\alpha}\) by \(\boxed{\mathscr{H}_{l,\lambda}^{m,\mu,(12)}(\alpha)}\). Then \[\begin{align} \sum\limits_{n\in\mathbb{Z}}H_{l,\lambda}^{m,\mu}(\mathbf{b}_{12})e^{-in\alpha}=&(-1)^{\lambda+\mu} \sqrt{\dfrac{2l+1}{2\lambda+1}} \sqrt{{l+\lambda+\mu-m\choose \lambda+\mu}{l+\lambda+m-\mu\choose \lambda-\mu}} \sqrt{\dfrac{4\pi}{2(l+\lambda)+1}}\\& \big(e^{-i\alpha}\cdot\mathrm{\Phi}(e^{-i\alpha},l+\lambda+1,1-2d)Y_{l+\lambda}^{m-\mu}(\dfrac{\pi}{2},\pi)+ \mathrm{\Phi}(e^{i\alpha},l+\lambda+1,2d)Y_{l+\lambda}^{m-\mu}(\dfrac{\pi}{2},0)\big) \end{align}.\]
Theorem 31. The expression of \(\boxed{M^{12}_{(l'm'2),(lm1)}(\alpha)}\)(\(l'\geqslant 1\), otherwise this term vanishes ) is given by:
If \(m>0\) and \(m'>0\), \[\begin{align}M^{12}_{(l'm'2),(lm1)}(\alpha)=
\rho^{l+3}a_{11}[r^{l'-1}\dfrac{1}{2}(\mathscr{H}_{l,l'}^{-m,-m',(12)}(\alpha)+(-1)^m\mathscr{H}_{l,l'}^{m,-m',(12)}(\alpha))l'(2l'+1)\\
+r^{l'-1}\dfrac{(-1)^{m'}}{2}(\mathscr{H}_{l,l'}^{-m,m',(12)}(\alpha)+(-1)^m\mathscr{H}_{l,l'}^{m,m',(12)}(\alpha))l'(2l'+1)]
\end{align}\] If \(m>0\) and \(m'<0\), \[\begin{align}M^{12}_{(l'm'2),(lm1)}(\alpha)=
\rho^{l+3}a_{11}[r^{l'-1}\dfrac{1}{2}(\mathscr{H}_{l,l'}^{-m,m',(12)}(\alpha)+(-1)^m\mathscr{H}_{l,l'}^{m,m',(12)}(\alpha))(-i \cdot l'(2l'+1))\\
+r^{l'-1}\dfrac{(-1)^{m'}}{2}(\mathscr{H}_{l,l'}^{-m,-m',(12)}(\alpha)+(-1)^m\mathscr{H}_{l,l'}^{m,-m',(12)}(\alpha))(i\cdot l'(2l'+1))]
\end{align}\] If \(m>0\) and \(m'=0\), \[\begin{align}M^{12}_{(l'm'2),(lm1)}(\alpha)=
\rho^{l+3}a_{11}r^{l'-1}\dfrac{1}{\sqrt{2}}(\mathscr{H}_{l,l'}^{-m,0,(12)}(\alpha)+(-1)^m\mathscr{H}_{l,l'}^{m,0,(12)}(\alpha))l'(2l'+1).
\end{align}\]
If \(m<0\) and \(m'>0\), \[\begin{align}M^{12}_{(l'm'2),(lm1)}(\alpha)= \rho^{l+3}a_{11}[r^{l'-1}\dfrac{i}{2}(\mathscr{H}_{l,l'}^{m,-m',(12)}(\alpha)-(-1)^m \mathscr{H}_{l,l'}^{-m,-m',(12)}(\alpha))l'(2l'+1)\\ +r^{l'-1}\dfrac{(-1)^{m'}}{2}\cdot i (\mathscr{H}_{l,l'}^{m,m',(12)}(\alpha)-(-1)^m\mathscr{H}_{l,l'}^{-m,m',(12)}(\alpha))l'(2l'+1)] \end{align}\]
If \(m<0\) and \(m'<0\), \[\begin{align}M^{12}_{(l'm'2),(lm1)}(\alpha)= \rho^{l+3}a_{11}[r^{l'-1}\dfrac{i}{2}(\mathscr{H}_{l,l'}^{m,m',(12)}(\alpha)-(-1)^m \mathscr{H}_{l,l'}^{-m,m',(12)}(\alpha))(-i\cdot l'(2l'+1))\\ +r^{l'-1}\dfrac{(-1)^{m'}}{2}\cdot i (\mathscr{H}_{l,l'}^{m,-m',(12)}(\alpha)-(-1)^m\mathscr{H}_{l,l'}^{-m,-m',(12)}(\alpha))(i\cdot l'(2l'+1))] \end{align}\]
If \(m<0\) and \(m'=0\), \[\begin{align}M^{12}_{(l'm'2),(lm1)}(\alpha)= \rho^{l+3}a_{11}r^{l'-1}\dfrac{i}{\sqrt{2}}(\mathscr{H}_{l,l'}^{m,0,(12)}(\alpha)-(-1)^m\mathscr{H}_{l,l'}^{-m,0,(12)}(\alpha))l'(2l'+1).\end{align}\]
If \(m=0\) and \(m'>0\), \[\begin{align}M^{12}_{(l'm'2),(lm1)}(\alpha)= \rho^{l+3}a_{11}r^{l'-1} [\dfrac{1}{\sqrt{2}} \mathscr{H}_{l,l'}^{0,-m',(12)}(\alpha)l'(2l'+1)+\dfrac{(-1)^{m'}}{\sqrt{2}}\mathscr{H}_{l,l'}^{0,m',(12)}(\alpha)l'(2l'+1)] \end{align}\]
If \(m=0\) and \(m'<0\), \[\begin{align}M^{12}_{(l'm'2),(lm1)}(\alpha)= \rho^{l+3}a_{11}r^{l'-1} [\dfrac{1}{\sqrt{2}} \mathscr{H}_{l,l'}^{0,m',(12)}(\alpha)(-i\cdot l'(2l'+1))+\dfrac{(-1)^{m'}}{\sqrt{2}}\mathscr{H}_{l,l'}^{0,-m',(12)}(\alpha)(i\cdot l'(2l'+1))] \end{align}\]
If \(m=0\) and \(m'=0\), \[\begin{align}M^{12}_{(l'm'2),(lm1)}(\alpha)= \rho^{l+3}a_{11}r^{l'-1} [\mathscr{H}_{l,l'}^{0,0,(12)}(\alpha)l'(2l'+1)] \end{align}\] where \(a_{ij}\) stands for the coefficient of the entries of \(A^{out}_{\mathcal{S},l}(x)\).
Lemma 17. Denote \(\sum\limits_{n\in\mathbb{Z}}L_{l,j,\lambda}^{m,\mu,q,m_1}(\mathbf{b}_{12})e^{-in\alpha}\) by \(\boxed{\mathscr{L}_{l,j,\lambda}^{m,\mu,q,m_1,(12)}(\alpha)}\), then \[\begin{align}&\mathscr{L}_{l,j,\lambda}^{m,\mu,q,m_1,(12)}(\alpha)= i (-1)^{\lambda+\mu+q}\text{sgn}(q-m_1)\epsilon_{q}\sqrt{\lambda(2l+1)} \sqrt{{l+\lambda+\mu-m\choose \lambda+\mu}{l+\lambda+m-\mu\choose \lambda-\mu}}\\ &\langle\lambda-1,\mu-m_1;1,m_1\mid \lambda,\mu\rangle \langle \lambda-1,\mu-m_1;1,q+m_1\mid j,\mu+q \rangle \sqrt{\dfrac{4\pi}{2(l+\lambda)+1}}\\ &\big(e^{-i\alpha}\cdot \mathrm{\Phi}(e^{-i\alpha},l+\lambda,1-2d)Y_{l+\lambda}^{m-\mu}(\dfrac{\pi}{2},\pi)-\mathrm{\Phi}(e^{i\alpha},l+\lambda,2d)Y_{l+\lambda}^{m-\mu}(\dfrac{\pi}{2},0)\big) \end{align}\] where \[\epsilon_q=\left\{ \begin{align} \dfrac{1}{\sqrt{2}}&,\quad q=-1\\ 0&,\quad q=0\\ -\dfrac{1}{\sqrt{2}}&,\quad,q=1 \end{align}\right.\]
Theorem 32. The expression of \(\boxed{M^{12}_{(l'm',2),(lm,3)}(\alpha)}\)(\(l'\geqslant 1, l\geqslant 1\), otherwise this term vanishes) is given by When \(m>0, m'>0\), \[\begin{align} &M^{12}_{(l'm',2),(lm,3)}(\alpha)\\ =&\;\rho^{l+2}a_{33}\,r^{l'-1}\,\frac{\sqrt{l'(2l'+1)}}{2}\sum\limits_{q=-1}^{1}\sum\limits_{m_1=-1}^{1}\bigl[(-1)^{m'}\bigl(\mathscr{L}_{l,l',l'}^{-m,\,m'-q,\,q,\,m_1,(12)}(\alpha)+(-1)^m\mathscr{L}_{l,l',l'}^{m,\,m'-q,\,q,\,m_1,(12)}(\alpha)\bigr)\\ &\qquad\qquad\qquad\qquad+\bigl(\mathscr{L}_{l,l',l'}^{-m,\,-m'-q,\,q,\,m_1,(12)}(\alpha)+(-1)^m\mathscr{L}_{l,l',l'}^{m,\,-m'-q,\,q,\,m_1,(12)}(\alpha)\bigr)\bigr] \end{align}\] When \(m>0, m'<0\), \[\begin{align} &M^{12}_{(l'm',2),(lm,3)}(\alpha)\\ =&\;\rho^{l+2}a_{33}\,r^{l'-1}\,\frac{\sqrt{l'(2l'+1)}}{2}\sum\limits_{q=-1}^{1}\sum\limits_{m_1=-1}^{1}\bigl[i(-1)^{m'}\bigl(\mathscr{L}_{l,l',l'}^{-m,\,-m'-q,\,q,\,m_1,(12)}(\alpha)+(-1)^m\mathscr{L}_{l,l',l'}^{m,\,-m'-q,\,q,\,m_1,(12)}(\alpha)\bigr)\\ &\qquad\qquad\qquad\qquad-i\bigl(\mathscr{L}_{l,l',l'}^{-m,\,m'-q,\,q,\,m_1,(12)}(\alpha)+(-1)^m\mathscr{L}_{l,l',l'}^{m,\,m'-q,\,q,\,m_1,(12)}(\alpha)\bigr)\bigr] \end{align}\]
When \(m>0, m'=0\), \[\begin{align} &M^{12}_{(l',0,2),(lm,3)}(\alpha)\\ =&\;\rho^{l+2}a_{33}\,r^{l'-1}\,\frac{\sqrt{l'(2l'+1)}}{\sqrt{2}}\sum\limits_{q=-1}^{1}\sum\limits_{m_1=-1}^{1}\bigl(\mathscr{L}_{l,l',l'}^{-m,\,-q,\,q,\,m_1,(12)}(\alpha)+(-1)^m\mathscr{L}_{l,l',l'}^{m,\,-q,\,q,\,m_1,(12)}(\alpha)\bigr) \end{align}\]
When \(m<0, m'>0\), \[\begin{align} &M^{12}_{(l'm',2),(lm,3)}(\alpha)\\ =&\;\rho^{l+2}a_{33}\,r^{l'-1}\,i\cdot\frac{\sqrt{l'(2l'+1)}}{2}\sum\limits_{q=-1}^{1}\sum\limits_{m_1=-1}^{1}\bigl[(-1)^{m'}\bigl(\mathscr{L}_{l,l',l'}^{m,\,m'-q,\,q,\,m_1,(12)}(\alpha)-(-1)^m\mathscr{L}_{l,l',l'}^{-m,\,m'-q,\,q,\,m_1,(12)}(\alpha)\bigr)\\ &\qquad\qquad\qquad\qquad+\bigl(\mathscr{L}_{l,l',l'}^{m,\,-m'-q,\,q,\,m_1,(12)}(\alpha)-(-1)^m\mathscr{L}_{l,l',l'}^{-m,\,-m'-q,\,q,\,m_1,(12)}(\alpha)\bigr)\bigr] \end{align}\] When \(m<0, m'<0\), \[\begin{align} &M^{12}_{(l'm',2),(lm,3)}(\alpha)\\ =&\;\rho^{l+2}a_{33}\,r^{l'-1}\,\frac{\sqrt{l'(2l'+1)}}{2}\sum\limits_{q=-1}^{1}\sum\limits_{m_1=-1}^{1}\bigl[\bigl(\mathscr{L}_{l,l',l'}^{m,\,m'-q,\,q,\,m_1,(12)}(\alpha)-(-1)^m\mathscr{L}_{l,l',l'}^{-m,\,m'-q,\,q,\,m_1,(12)}(\alpha)\bigr)\\ &\qquad\qquad\qquad\qquad-(-1)^{m'}\bigl(\mathscr{L}_{l,l',l'}^{m,\,-m'-q,\,q,\,m_1,(12)}(\alpha)-(-1)^m\mathscr{L}_{l,l',l'}^{-m,\,-m'-q,\,q,\,m_1,(12)}(\alpha)\bigr)\bigr] \end{align}\] When \(m<0, m'=0\), \[\begin{align} &M^{12}_{(l',0,2),(lm,3)}(\alpha)\\ =&\;\rho^{l+2}a_{33}\,r^{l'-1}\,\frac{i\sqrt{l'(2l'+1)}}{\sqrt{2}}\sum\limits_{q=-1}^{1}\sum\limits_{m_1=-1}^{1}\bigl(\mathscr{L}_{l,l',l'}^{m,\,-q,\,q,\,m_1,(12)}(\alpha)-(-1)^m\mathscr{L}_{l,l',l'}^{-m,\,-q,\,q,\,m_1,(12)}(\alpha)\bigr) \end{align}\] When \(m=0, m'>0\), \[\begin{align} &M^{12}_{(l'm',2),(l,0,3)}(\alpha)\\ =&\;\rho^{l+2}a_{33}\,r^{l'-1}\,\frac{\sqrt{l'(2l'+1)}}{\sqrt{2}}\sum\limits_{q=-1}^{1}\sum\limits_{m_1=-1}^{1}\bigl[(-1)^{m'}\mathscr{L}_{l,l',l'}^{0,\,m'-q,\,q,\,m_1,(12)}(\alpha)+\mathscr{L}_{l,l',l'}^{0,\,-m'-q,\,q,\,m_1,(12)}(\alpha)\bigr] \end{align}\]
When \(m=0, m'<0\), \[\begin{align} &M^{12}_{(l'm',2),(l,0,3)}(\alpha)\\ =&\;\rho^{l+2}a_{33}\,r^{l'-1}\,\frac{\sqrt{l'(2l'+1)}}{\sqrt{2}}\sum\limits_{q=-1}^{1}\sum\limits_{m_1=-1}^{1}\bigl[i(-1)^{m'}\mathscr{L}_{l,l',l'}^{0,\,-m'-q,\,q,\,m_1,(12)}(\alpha)-i\,\mathscr{L}_{l,l',l'}^{0,\,m'-q,\,q,\,m_1,(12)}(\alpha)\bigr] \end{align}\]
When \(m=0, m'=0\), \[\begin{align} &M^{12}_{(l',0,2),(l,0,3)}(\alpha)\\ =&\;\rho^{l+2}a_{33}\,r^{l'-1}\,\sqrt{l'(2l'+1)}\sum\limits_{q=-1}^{1}\sum\limits_{m_1=-1}^{1}\mathscr{L}_{l,l',l'}^{0,\,-q,\,q,\,m_1,(12)}(\alpha) \end{align}\]
Lemma 18. Denote \(\sum\limits_{n\in\mathbb{Z}}b^{(12)}_{-q}H_{l,\lambda}^{m,\mu}(\mathbf{b}_{12})e^{-in\alpha}\) by \(\boxed{\mathscr{A}_{l,\lambda}^{m,\mu,(12)}(\alpha,q)}\), then \[\begin{align} \sum\limits_{n\in\mathbb{Z}}b^{(12)}_{-q}H_{l,\lambda}^{m,\mu}(\mathbf{b}_{12})e^{-in\alpha} =&(-1)^{\lambda+\mu}\epsilon_q \sqrt{\dfrac{2l+1}{2\lambda+1}} \sqrt{{l+\lambda+\mu-m\choose \lambda+\mu}{l+\lambda+m-\mu\choose \lambda-\mu}} \sqrt{\dfrac{4\pi}{2(l+\lambda)+1}}\\& \big(e^{-i\alpha}\cdot \mathrm{\Phi}(e^{-i\alpha},l+\lambda,1-2d)Y_{l+\lambda}^{m-\mu}(\dfrac{\pi}{2},\pi)-\mathrm{\Phi}(e^{i\alpha},l+\lambda,2d)Y_{l+\lambda}^{m-\mu}(\dfrac{\pi}{2},0)\big) \end{align}.\]
Theorem 33. The expression of \(\boxed{M^{12}_{(l'm',1),(lm,2)}(\alpha)}\)(\(l\geqslant 1\), otherwise this term vanishes) is given by
When \(m>0\), \(m'>0\), \[\begin{align}
&M^{12}_{(l'm',1),(lm,2)}(\alpha)\\
=&\sum\limits_{\lambda\in\{l'+1,l'+3\}}\sum\limits_{q=-1}^{1}\sum\limits_{m_1=-1}^{1}\rho^{l+1} (a_{22}-a_{12}) r^\lambda \sqrt{(l'+1)(2l'+1)}
[(-1)^{m'}(\mathscr{A}_{l,\lambda}^{-m,m'-q,(12)}(\alpha,q)\\
&+(-1)^m\mathscr{A}_{l,\lambda}^{m,m'-q,(12)}(\alpha,q))K_{l'+1,l',\lambda}^{m_1,m'-q,q}\\
&
+(\mathscr{A}_{l,\lambda}^{-m,-m'-q,(12)}(\alpha,q)+(-1)^m\mathscr{A}_{l,\lambda}^{m,-m'-q,(12)}(\alpha,q))K_{l'+1,l',\lambda}^{m_1,-m'-q,q}]\\
&-\rho^{l+1} a_{22}r^{l'+1}\dfrac{(2l+1)(l'+1)}{2}[(-1)^{m'}(\mathscr{H}_{l,l'}^{-m,m',(12)}(\alpha)\\
&+(-1)^m\mathscr{H}_{l,l'}^{m,m',(12)}(\alpha))+(\mathscr{H}_{l,l'}^{-m,-m',(12)}(\alpha)+(-1)^m\mathscr{H}_{l,l'}^{m,-m',(12)}(\alpha))]\\
&+\rho^{l+1} a_{22}r^{l'+1}(2l+1)\dfrac{\sqrt{(l'+1)(2l'+1)}}{2} \sum\limits_{q=-1}^1(-1)^q [(-1)^{m'}(\mathscr{A}_{l,l'+1}^{-m,m'-q,(12)}(\alpha,q)\\
&+(-1)^m\mathscr{A}_{l,l'+1}^{m,m'-q,(12)}(\alpha,q))\langle l'+1,m'-q;1,q\mid l',m'\rangle+(\mathscr{A}_{l,l'+1}^{-m,-m'-q,(12)}(\alpha,q)\\
&+(-1)^m\mathscr{A}_{l,l'+1}^{m,-m'-q,(12)}(\alpha,q))\langle l'+1,-m'-q;1,q\mid l',-m'\rangle]
\end{align}\]
When \(m>0\), \(m'<0\),
\[\begin{align} &M^{12}_{(l'm',1),(lm,2)}(\alpha)\\ =&\sum\limits_{\lambda\in\{l'+1,l'+3\}}\sum\limits_{q=-1}^{1}\sum\limits_{m_1=-1}^{1}\rho^{l+1} (a_{22}-a_{12}) r^\lambda \sqrt{(l'+1)(2l'+1)} [i\cdot (-1)^{m'}(\mathscr{A}_{l,\lambda}^{-m,-m'-q,(12)}(\alpha,q)\\ &+(-1)^m\mathscr{A}_{l,\lambda}^{m,-m'-q,(12)}(\alpha,q))K_{l'+1,l',\lambda}^{m_1,-m'-q,q} -i\cdot (\mathscr{A}_{l,\lambda}^{-m,m'-q,(12)}(\alpha,q)\\ &+(-1)^m\mathscr{A}_{l,\lambda}^{m,m'-q,(12)}(\alpha,q))K_{l'+1,l',\lambda}^{m_1,m'-q,q}]\\ &-\rho^{l+1} a_{22}r^{l'+1}\dfrac{(2l+1)(l'+1)}{2}[i\cdot (-1)^{m'}(\mathscr{H}_{l,l'}^{-m,-m',(12)}(\alpha)\\ &+(-1)^m\mathscr{H}_{l,l'}^{m,-m',(12)}(\alpha))\\ &-i\cdot (\mathscr{H}_{l,l'}^{-m,m',(12)}(\alpha)+(-1)^m\mathscr{H}_{l,l'}^{m,m',(12)}(\alpha))]\\ &+\rho^{l+1} a_{22}r^{l'+1}(2l+1)\dfrac{\sqrt{(l'+1)(2l'+1)}}{2} \sum\limits_{q=-1}^1(-1)^q [i\cdot (-1)^{m'}(\mathscr{A}_{l,l'+1}^{-m,-m'-q,(12)}(\alpha,q)\\ &+(-1)^m\mathscr{A}_{l,l'+1}^{m,-m'-q,(12)}(\alpha,q))\\ &\langle l'+1,-m'-q;1,q\mid l',-m'\rangle -i\cdot (\mathscr{A}_{l,l'+1}^{-m,m'-q,(12)}(\alpha,q)\\ &+(-1)^m\mathscr{A}_{l,l'+1}^{m,m'-q,(12)}(\alpha,q))\langle l'+1,m'-q;1,q\mid l',m'\rangle] \end{align}\]
When \(m>0\), \(m'=0\), \[\begin{align} &M^{12}_{(l',0,1),(lm,2)}(\alpha)\\ =&\sum\limits_{\lambda\in\{l'+1,l'+3\}}\sum\limits_{q=-1}^{1}\sum\limits_{m_1=-1}^{1}\rho^{l+1} (a_{22}-a_{12}) r^\lambda \sqrt{2(l'+1)(2l'+1)} [(\mathscr{A}_{l,\lambda}^{-m,-q,(12)}(\alpha,q)\\ &+(-1)^m\mathscr{A}_{l,\lambda}^{m,-q,(12)}(\alpha,q))K_{l'+1,l',\lambda}^{m_1,-q,q}]\\ &-\rho^{l+1} a_{22}r^{l'+1}\dfrac{(2l+1)(l'+1)}{\sqrt{2}}[\mathscr{H}_{l,l'}^{-m,0,(12)}(\alpha)+(-1)^m\mathscr{H}_{l,l'}^{m,0,(12)}(\alpha)]\\ &+\rho^{l+1} a_{22}r^{l'+1}(2l+1)\dfrac{\sqrt{(l'+1)(2l'+1)}}{\sqrt{2}} \sum\limits_{q=-1}^1(-1)^q (\mathscr{A}_{l,l'+1}^{-m,-q,(12)}(\alpha,q)+(-1)^m\mathscr{A}_{l,l'+1}^{m,-q,(12)}(\alpha,q))\\ &\langle l'+1,-q;1,q\mid l',0\rangle \end{align}\]
When \(m<0\), \(m'>0\), \[\begin{align} &M^{12}_{(l'm',1),(lm,2)}(\alpha)\\ =&\sum\limits_{\lambda\in\{l'+1,l'+3\}}\sum\limits_{q=-1}^{1}\sum\limits_{m_1=-1}^{1}\rho^{l+1} (a_{22}-a_{12}) r^\lambda i\cdot\sqrt{(l'+1)(2l'+1)} [(-1)^{m'}(\mathscr{A}_{l,\lambda}^{m,m'-q,(12)}(\alpha,q)\\ &-(-1)^m\mathscr{A}_{l,\lambda}^{-m,m'-q,(12)}(\alpha,q))K_{l'+1,l',\lambda}^{m_1,m'-q,q} +(\mathscr{A}_{l,\lambda}^{m,-m'-q,(12)}(\alpha,q)\\ &-(-1)^m\mathscr{A}_{l,\lambda}^{-m,-m'-q,(12)}(\alpha,q))K_{l'+1,l',\lambda}^{m_1,-m'-q,q}]\\ &-\rho^{l+1} a_{22}r^{l'+1}i\cdot \dfrac{(2l+1)(l'+1)}{2}[(-1)^{m'}(\mathscr{H}_{l,l'}^{m,m',(12)}(\alpha)-(-1)^m\mathscr{H}_{l,l'}^{-m,m',(12)}(\alpha))+(\mathscr{H}_{l,l'}^{m,-m',(12)}(\alpha)\\ &-(-1)^m\mathscr{H}_{l,l'}^{-m,-m',(12)}(\alpha))]\\ &+\rho^{l+1} a_{22}r^{l'+1}i\cdot (2l+1)\dfrac{\sqrt{(l'+1)(2l'+1)}}{2} \sum\limits_{q=-1}^1(-1)^q [(-1)^{m'}(\mathscr{A}_{l,l'+1}^{m,m'-q,(12)}(\alpha,q)\\ &-(-1)^m\mathscr{A}_{l,l'+1}^{-m,m'-q,(12)}(\alpha,q))\langle l'+1,m'-q;1,q\mid l',m'\rangle+(\mathscr{A}_{l,l'+1}^{m,-m'-q,(12)}(\alpha,q)\\ &-(-1)^m\mathscr{A}_{l,l'+1}^{-m,-m'-q,(12)}(\alpha,q))\langle l'+1,-m'-q;1,q\mid l',-m'\rangle] \end{align}\]
When \(m<0\), \(m'<0\),
\[\begin{align} &M^{12}_{(l'm',1),(lm,2)}(\alpha)\\ =&\sum\limits_{\lambda\in\{l'+1,l'+3\}}\sum\limits_{q=-1}^{1}\sum\limits_{m_1=-1}^{1}\rho^{l+1} (a_{22}-a_{12}) r^\lambda \sqrt{(l'+1)(2l'+1)} [-(-1)^{m'}(\mathscr{A}_{l,\lambda}^{m,-m'-q,(12)}(\alpha,q)\\ &-(-1)^m\mathscr{A}_{l,\lambda}^{-m,-m'-q,(12)}(\alpha,q))K_{l'+1,l',\lambda}^{m_1,-m'-q,q} + (\mathscr{A}_{l,\lambda}^{m,m'-q,(12)}(\alpha,q)\\ &-(-1)^m\mathscr{A}_{l,\lambda}^{-m,m'-q,(12)}(\alpha,q))K_{l'+1,l',\lambda}^{m_1,m'-q,q}]\\ &-\rho^{l+1} a_{22}r^{l'+1}\dfrac{(2l+1)(l'+1)}{2} [- (-1)^{m'}(\mathscr{H}_{l,l'}^{m,-m',(12)}(\alpha)-(-1)^m\mathscr{H}_{l,l'}^{-m,-m',(12)}(\alpha))\\ &+ (\mathscr{H}_{l,l'}^{m,m',(12)}(\alpha)-(-1)^m\mathscr{H}_{l,l'}^{-m,m',(12)}(\alpha))]\\ &+\rho^{l+1} a_{22}r^{l'+1}(2l+1)\dfrac{\sqrt{(l'+1)(2l'+1)}}{2} \sum\limits_{q=-1}^1(-1)^q [-(-1)^{m'}(\mathscr{A}_{l,l'+1}^{m,-m'-q,(12)}(\alpha,q)\\ &-(-1)^m\mathscr{A}_{l,l'+1}^{-m,-m'-q,(12)}(\alpha,q))\\ &\langle l'+1,-m'-q;1,q\mid l',-m'\rangle + (\mathscr{A}_{l,l'+1}^{m,m'-q,(12)}(\alpha,q)\\ &-(-1)^m\mathscr{A}_{l,l'+1}^{-m,m'-q,(12)}(\alpha,q))\langle l'+1,m'-q;1,q\mid l',m'\rangle] \end{align}\]
When \(m<0\), \(m'=0\), \[\begin{align} &M^{12}_{(l',0,1),(lm,2)}(\alpha)\\ =&\sum\limits_{\lambda\in\{l'+1,l'+3\}}\sum\limits_{q=-1}^{1}\sum\limits_{m_1=-1}^{1}\rho^{l+1} (a_{22}-a_{12}) r^\lambda i\cdot \sqrt{2(l'+1)(2l'+1)} [(\mathscr{A}_{l,\lambda}^{m,-q,(12)}(\alpha,q)\\ &-(-1)^m\mathscr{A}_{l,\lambda}^{-m,-q,(12)}(\alpha,q))K_{l'+1,l',\lambda}^{m_1,-q,q}]\\ &-\rho^{l+1} a_{22}r^{l'+1}i\cdot \dfrac{(2l+1)(l'+1)}{\sqrt{2}}[(\mathscr{H}_{l,l'}^{m,0,(12)}(\alpha)-(-1)^m\mathscr{H}_{l,l'}^{-m,0,(12)}(\alpha)]\\ &+\rho^{l+1} a_{22}r^{l'+1}i\cdot (2l+1)\dfrac{\sqrt{(l'+1)(2l'+1)}}{\sqrt{2}} \sum\limits_{q=-1}^1(-1)^q [ (\mathscr{A}_{l,l'+1}^{m,-q,(12)}(\alpha,q)-(-1)^m\mathscr{A}_{l,l'+1}^{-m,-q,(12)}(\alpha,q))\\ &\langle l'+1,-q;1,q\mid l',0\rangle] \end{align}\]
When \(m=0\), \(m'>0\), \[\begin{align} &M^{12}_{(l'm',1),(l,0,2)}(\alpha)\\ =&\sum\limits_{\lambda\in\{l'+1,l'+3\}}\sum\limits_{q=-1}^{1}\sum\limits_{m_1=-1}^{1}\rho^{l+1} (a_{22}-a_{12}) r^\lambda \sqrt{2(l'+1)(2l'+1)} [(-1)^{m'}\mathscr{A}_{l,\lambda}^{0,m'-q,(12)}(\alpha,q) K_{l'+1,l',\lambda}^{m_1,m'-q,q}\\ &+\mathscr{A}_{l,\lambda}^{0,-m'-q,(12)}(\alpha,q)K_{l'+1,l',\lambda}^{m_1,-m'-q,q}]\\ &-\rho^{l+1} a_{22}r^{l'+1}\dfrac{(2l+1)(l'+1)}{\sqrt{2}}[(-1)^{m'}\mathscr{H}_{l,l'}^{0,m',(12)}(\alpha)+\mathscr{H}_{l,l'}^{0,-m',(12)}(\alpha)]\\ &+\rho^{l+1} a_{22}r^{l'+1}(2l+1)\dfrac{\sqrt{(l'+1)(2l'+1)}}{\sqrt{2}} \sum\limits_{q=-1}^1(-1)^q [(-1)^{m'}\mathscr{A}_{l,l'+1}^{0,m'-q,(12)}(\alpha,q)\\ &\langle l'+1,m'-q;1,q\mid l',m'\rangle+(\mathscr{A}_{l,l'+1}^{0,-m'-q,(12)}(\alpha,q)\langle l'+1,-m'-q;1,q\mid l',-m'\rangle] \end{align}\]
When \(m=0\), \(m'<0\),
\[\begin{align} &M^{12}_{(l'm',1),(l,0,2)}(\alpha)\\ =&\sum\limits_{\lambda\in\{l'+1,l'+3\}}\sum\limits_{q=-1}^{1}\sum\limits_{m_1=-1}^{1}\rho^{l+1} (a_{22}-a_{12}) r^\lambda \sqrt{2(l'+1)(2l'+1)} [i\cdot (-1)^{m'}\mathscr{A}_{l,\lambda}^{0,-m'-q,(12)}(\alpha,q)K_{l'+1,l',\lambda}^{m_1,-m'-q,q}\\ & -i\cdot \mathscr{A}_{l,\lambda}^{0,m'-q,(12)}(\alpha,q)K_{l'+1,l',\lambda}^{m_1,m'-q,q}]\\ &-\rho^{l+1} a_{22}r^{l'+1}\dfrac{(2l+1)(l'+1)}{\sqrt{2}}[i\cdot (-1)^{m'}\mathscr{H}_{l,l'}^{0,-m',(12)}(\alpha)-i\cdot \mathscr{H}_{l,l'}^{0,m',(12)}(\alpha)]\\ &+\rho^{l+1} a_{22}r^{l'+1}(2l+1)\dfrac{\sqrt{(l'+1)(2l'+1)}}{\sqrt{2}} \sum\limits_{q=-1}^1(-1)^q [i\cdot (-1)^{m'} \mathscr{A}_{l,l'+1}^{0,-m'-q,(12)}(\alpha,q)\\ &\langle l'+1,-m'-q;1,q\mid l',-m'\rangle-i\cdot \mathscr{A}_{l,l'+1}^{0,m'-q,(12)}(\alpha,q)\langle l'+1,m'-q;1,q\mid l',m'\rangle] \end{align}\]
When \(m=0\), \(m'=0\), \[\begin{align} &M^{12}_{(l',0,1),(l,0,2)}(\alpha)\\ =&\sum\limits_{\lambda\in\{l'+1,l'+3\}}\sum\limits_{q=-1}^{1}\sum\limits_{m_1=-1}^{1}\rho^{l+1} (a_{22}-a_{12}) r^\lambda \sqrt{4(l'+1)(2l'+1)} [\mathscr{A}_{l,\lambda}^{0,-q,(12)}(\alpha,q)K_{l'+1,l',\lambda}^{m_1,-q,q}]\\ &-\rho^{l+1} a_{22}r^{l'+1}(2l+1)(l'+1)[\mathscr{H}_{l,l'}^{0,0,(12)}(\alpha)]\\ &+\rho^{l+1} a_{22}r^{l'+1}(2l+1)\sqrt{(l'+1)(2l'+1)} \sum\limits_{q=-1}^1(-1)^q [ \mathscr{A}_{l,l'+1}^{0,-q,(12)}(\alpha,q)\langle l'+1,-q;1,q\mid l',0\rangle] \end{align}\]
Theorem 34. Entries for \(\boxed{M^{12}_{(l'm',3),(lm,2)}(\alpha)}\)(\(l'\geqslant 1,l\geqslant 1\), otherwise this term vanishes):
When \(m>0\), \(m'>0\), \[\begin{align}
&M^{12}_{(l'm',3),(lm,2)}(\alpha)\\
=&\sum\limits_{\lambda\in\{l',l'+2\mid l'\geqslant 1\}}\sum\limits_{q=-1}^{1}\sum\limits_{m_1=-1}^{1}-i\cdot \rho^{l+1} (a_{22}-a_{12}) r^\lambda \sqrt{l'(l'+1)}
[(-1)^{m'}(\mathscr{A}_{l,\lambda}^{-m,m'-q,(12)}(\alpha,q)\\
&+(-1)^m\mathscr{A}_{l,\lambda}^{m,m'-q,(12)}(\alpha,q))K_{l',l',\lambda}^{m_1,m'-q,q}
+(\mathscr{A}_{l,\lambda}^{-m,-m'-q,(12)}(\alpha,q)\\
&+(-1)^m\mathscr{A}_{l,\lambda}^{m,-m'-q,(12)}(\alpha,q))K_{l',l',\lambda}^{m_1,-m'-q,q}]\\
&- i\cdot \rho^{l+1} a_{22}r^{l'}(2l+1)\dfrac{\sqrt{l'(l'+1)}}{2} \sum\limits_{q=-1}^1(-1)^q [(-1)^{m'}(\mathscr{A}_{l,l'}^{-m,m'-q,(12)}(\alpha,q)+(-1)^m\mathscr{A}_{l,l'}^{m,m'-q,(12)}(\alpha,q))\\
&\langle l',m'-q;1,q\mid l',m'\rangle+(\mathscr{A}_{l,l'}^{-m,-m'-q,(12)}(\alpha,q)+(-1)^m\mathscr{A}_{l,l'}^{m,-m'-q,(12)}(\alpha,q))\langle l',-m'-q;1,q\mid l',-m'\rangle]
\end{align}\]
When \(m>0\), \(m'<0\), \[\begin{align} &M^{12}_{(l'm',3),(lm,2)}(\alpha)\\ =&\sum\limits_{\lambda\in\{l',l'+2\mid l'\geqslant 1\}}\sum\limits_{q=-1}^{1}\sum\limits_{m_1=-1}^{1}\rho^{l+1} (a_{22}-a_{12}) r^\lambda \sqrt{l'(l'+1)} [ (-1)^{m'}(\mathscr{A}_{l,\lambda}^{-m,-m'-q,(12)}(\alpha,q)\\ &+(-1)^m\mathscr{A}_{l,\lambda}^{m,-m'-q,(12)}(\alpha,q))K_{l',l',\lambda}^{m_1,-m'-q,q} - (\mathscr{A}_{l,\lambda}^{-m,m'-q,(12)}(\alpha,q)+(-1)^m\mathscr{A}_{l,\lambda}^{m,m'-q,(12)}(\alpha,q))K_{l',l',\lambda}^{m_1,m'-q,q}]\\ &+\rho^{l+1} a_{22}r^{l'}(2l+1)\dfrac{\sqrt{l'(l'+1)}}{2} \sum\limits_{q=-1}^1(-1)^q [ (-1)^{m'}(\mathscr{A}_{l,l'}^{-m,-m'-q,(12)}(\alpha,q)+(-1)^m\mathscr{A}_{l,l'}^{m,-m'-q,(12)}(\alpha,q))\\ &\langle l',-m'-q;1,q\mid l',-m'\rangle - (\mathscr{A}_{l,l'}^{-m,m'-q,(12)}(\alpha,q)+(-1)^m\mathscr{A}_{l,l'}^{m,m'-q,(12)}(\alpha,q))\langle l',m'-q;1,q\mid l',m'\rangle]\end{align}\]
When \(m>0\), \(m'=0\), \[\begin{align} &M^{12}_{(l',0,3),(lm,2)}(\alpha)\\ =&\sum\limits_{\lambda\in\{l',l'+2\mid l'\geqslant 1\}}\sum\limits_{q=-1}^{1}\sum\limits_{m_1=-1}^{1} -i\cdot \rho^{l+1} (a_{22}-a_{12}) r^\lambda \sqrt{2l'(l'+1)} [(\mathscr{A}_{l,\lambda}^{-m,-q,(12)}(\alpha,q)\\ &+(-1)^m\mathscr{A}_{l,\lambda}^{m,-q,(12)}(\alpha,q))K_{l',l',\lambda}^{m_1,-q,q}]\\ &-i\cdot \rho^{l+1} a_{22}r^{l'}(2l+1)\dfrac{\sqrt{l'(l'+1)}}{\sqrt{2}} \sum\limits_{q=-1}^1(-1)^q [ (\mathscr{A}_{l,l'}^{-m,-q,(12)}(\alpha,q)+(-1)^m\mathscr{A}_{l,l'}^{m,-q,(12)}(\alpha,q))\\ &\langle l',-q;1,q\mid l',0\rangle] \end{align}\]
When \(m<0\), \(m'>0\), \[\begin{align} &M^{12}_{(l'm',3),(lm,2)}(\alpha)\\ =&\sum\limits_{\lambda\in\{l',l'+2\mid l'\geqslant 1\}}\sum\limits_{q=-1}^{1}\sum\limits_{m_1=-1}^{1}\rho^{l+1} (a_{22}-a_{12}) r^\lambda \sqrt{l'(l'+1)} [(-1)^{m'}(\mathscr{A}_{l,\lambda}^{m,m'-q,(12)}(\alpha,q)\\ &-(-1)^m\mathscr{A}_{l,\lambda}^{-m,m'-q,(12)}(\alpha,q))K_{l',l',\lambda}^{m_1,m'-q,q} +(\mathscr{A}_{l,\lambda}^{m,-m'-q,(12)}(\alpha,q)\\ &-(-1)^m\mathscr{A}_{l,\lambda}^{-m,-m'-q,(12)}(\alpha,q))K_{l',l',\lambda}^{m_1,-m'-q,q}]\\ &+\rho^{l+1} a_{22}r^{l'} (2l+1)\dfrac{\sqrt{l'(l'+1)}}{2} \sum\limits_{q=-1}^1(-1)^q [(-1)^{m'}(\mathscr{A}_{l,l'}^{m,m'-q,(12)}(\alpha,q)-(-1)^m\mathscr{A}_{l,l'}^{-m,m'-q,(12)}(\alpha,q))\\ &\langle l',m'-q;1,q\mid l',m'\rangle+(\mathscr{A}_{l,l'}^{m,-m'-q,(12)}(\alpha,q)-(-1)^m\mathscr{A}_{l,l'}^{-m,-m'-q,(12)}(\alpha,q))\langle l',-m'-q;1,q\mid l',-m'\rangle] \end{align}\]
When \(m<0\), \(m'<0\), \[\begin{align} &M^{12}_{(l'm',3),(lm,2)}(\alpha)\\ =&\sum\limits_{\lambda\in\{l',l'+2\mid l'\geqslant 1\}}\sum\limits_{q=-1}^{1}\sum\limits_{m_1=-1}^{1}\rho^{l+1} (a_{22}-a_{12}) r^\lambda i\cdot \sqrt{l'(l'+1)} [(-1)^{m'}(\mathscr{A}_{l,\lambda}^{m,-m'-q,(12)}(\alpha,q)\\ &-(-1)^m\mathscr{A}_{l,\lambda}^{-m,-m'-q,(12)}(\alpha,q))K_{l',l',\lambda}^{m_1,-m'-q,q} - (\mathscr{A}_{l,\lambda}^{m,m'-q,(12)}(\alpha,q)\\ &-(-1)^m\mathscr{A}_{l,\lambda}^{-m,m'-q,(12)}(\alpha,q))K_{l',l',\lambda}^{m_1,m'-q,q}]\\ &-i\cdot \rho^{l+1} a_{22}r^{l'}(2l+1)\dfrac{\sqrt{l'(l'+1)}}{2} \sum\limits_{q=-1}^1(-1)^q [-(-1)^{m'}(\mathscr{A}_{l,l'}^{m,-m'-q,(12)}(\alpha,q)\\ &-(-1)^m\mathscr{A}_{l,l'}^{-m,-m'-q,(12)}(\alpha,q))\\ &\langle l',-m'-q;1,q\mid l',-m'\rangle + (\mathscr{A}_{l,l'}^{m,m'-q,(12)}(\alpha,q)-(-1)^m\mathscr{A}_{l,l'}^{-m,m'-q,(12)}(\alpha,q))\langle l',m'-q;1,q\mid l',m'\rangle] \end{align}\]
When \(m<0\), \(m'=0\), \[\begin{align} &M^{12}_{(l',0,3),(lm,2)}(\alpha)\\ =&\sum\limits_{\lambda\in\{l',l'+2\mid l'\geqslant 1\}}\sum\limits_{q=-1}^{1}\sum\limits_{m_1=-1}^{1} \rho^{l+1} (a_{22}-a_{12}) r^\lambda \sqrt{2l'(l'+1)} [(\mathscr{A}_{l,\lambda}^{m,-q,(12)}(\alpha,q)\\ &-(-1)^m\mathscr{A}_{l,\lambda}^{-m,-q,(12)}(\alpha,q))K_{l',l',\lambda}^{m_1,-q,q}]\\ &+\rho^{l+1} a_{22}r^{l'} (2l+1)\dfrac{\sqrt{l'(l'+1)}}{\sqrt{2}} \sum\limits_{q=-1}^1(-1)^q [ (\mathscr{A}_{l,l'}^{m,-q,(12)}(\alpha,q)-(-1)^m\mathscr{A}_{l,l'}^{-m,-q,(12)}(\alpha,q))\\ &\langle l',-q;1,q\mid l',0\rangle] \end{align}\]
When \(m=0\), \(m'>0\), \[\begin{align} &M^{12}_{(l'm',3),(l,0,2)}(\alpha)\\ =&\sum\limits_{\lambda\in\{l',l'+2\mid l'\geqslant 1\}}\sum\limits_{q=-1}^{1}\sum\limits_{m_1=-1}^{1} -i\cdot \rho^{l+1} (a_{22}-a_{12}) r^\lambda \sqrt{2l'(l'+1)} [(-1)^{m'}\mathscr{A}_{l,\lambda}^{0,m'-q,(12)}(\alpha,q) K_{l',l',\lambda}^{m_1,m'-q,q}\\ &+\mathscr{A}_{l,\lambda}^{0,-m'-q,(12)}(\alpha,q)K_{l',l',\lambda}^{m_1,-m'-q,q}]\\ &-i\cdot \rho^{l+1} a_{22}r^{l'}(2l+1)\dfrac{\sqrt{l'(l'+1)}}{\sqrt{2}} \sum\limits_{q=-1}^1(-1)^q [(-1)^{m'}\mathscr{A}_{l,l'}^{0,m'-q,(12)}(\alpha,q)\\ &\langle l',m'-q;1,q\mid l',m'\rangle+(\mathscr{A}_{l,l'}^{0,-m'-q,(12)}(\alpha,q)\langle l',-m'-q;1,q\mid l',-m'\rangle] \end{align}\]
When \(m=0\), \(m'<0\),
\[\begin{align} &M^{12}_{(l'm',3),(l,0,2)}(\alpha)\\ =&\sum\limits_{\lambda\in\{l',l'+2\mid l'\geqslant 1\}}\sum\limits_{q=-1}^{1}\sum\limits_{m_1=-1}^{1}\rho^{l+1} (a_{22}-a_{12}) r^\lambda \sqrt{2l'(l'+1)} [(-1)^{m'}\mathscr{A}_{l,\lambda}^{0,-m'-q,(12)}(\alpha,q)K_{l',l',\lambda}^{m_1,-m'-q,q}\\ & - \mathscr{A}_{l,\lambda}^{0,m'-q,(12)}(\alpha,q)K_{l',l',\lambda}^{m_1,m'-q,q}]\\ &+\rho^{l+1} a_{22}r^{l'}(2l+1)\dfrac{\sqrt{l'(l'+1)}}{\sqrt{2}} \sum\limits_{q=-1}^1(-1)^q [ (-1)^{m'} \mathscr{A}_{l,l'}^{0,-m'-q,(12)}(\alpha,q)\\ &\langle l',-m'-q;1,q\mid l',-m'\rangle- \mathscr{A}_{l,l'}^{0,m'-q,(12)}(\alpha,q)\langle l',m'-q;1,q\mid l',m'\rangle] \end{align}\]
When \(m=0\), \(m'=0\), \[\begin{align} &M^{12}_{(l',0,3),(l,0,2)}(\alpha)\\ =&\sum\limits_{\lambda\in\{l',l'+2\mid l'\geqslant 1\}}\sum\limits_{q=-1}^{1}\sum\limits_{m_1=-1}^{1} -i\cdot \rho^{l+1} (a_{22}-a_{12}) r^\lambda \sqrt{4l'(l'+1)} [\mathscr{A}_{l,\lambda}^{0,-q,(12)}(\alpha,q)K_{l',l',\lambda}^{m_1,-q,q}]\\ &-i\cdot \rho^{l+1} a_{22}r^{l'}(2l+1)\sqrt{l'(l'+1)} \sum\limits_{q=-1}^1(-1)^q [ \mathscr{A}_{l,l'}^{0,-q,(12)}(\alpha,q)\langle l',-q;1,q\mid l',0\rangle] \end{align}\]
Theorem 35. Entries for \(\boxed{M^{12}_{(l'm',3),(lm,3)}(\alpha)}\)(\(l'\geqslant 1,l\geqslant 1\), otherwise this term vanishes):
When \(m>0,\;m'>0\): \[\begin{align}
&M^{12}_{(l'm',3),(lm,3)}(\alpha)\\
=&\;\rho^{l+2}a_{33}r^{l'}\Biggl\{
\frac{l'(l'+1)}{2}\Bigl[(-1)^{m'}\bigl(\mathscr{H}_{l,l'}^{-m,m',(12)}(\alpha)+(-1)^m \mathscr{H}_{l,l'}^{m,m',(12)}(\alpha)\bigr)\\
&
+\bigl(\mathscr{H}_{l,l'}^{-m,-m',(12)}(\alpha)+(-1)^m \mathscr{H}_{l,l'}^{m,-m',(12)}(\alpha)\bigr)\Bigr]\\
&-\frac{i\sqrt{l'(l'+1)}}{2}\sum_{q=-1}^{1}\sum_{m_1=-1}^{1}\Bigl[(-1)^{m'}\bigl(\mathscr{L}_{l,l',l'+1}^{-m,m'-q,q,m_1,(12)}(\alpha)+(-1)^m \mathscr{L}_{l,l',l'+1}^{m,m'-q,q,m_1,(12)}(\alpha)\bigr)\\
&\qquad\qquad\qquad\qquad\quad+\bigl(\mathscr{L}_{l,l',l'+1}^{-m,-m'-q,q,m_1,(12)}(\alpha)+(-1)^m \mathscr{L}_{l,l',l'+1}^{m,-m'-q,q,m_1,(12)}(\alpha)\bigr)\Bigr]\Biggr\}\end{align}\]
When \(m>0,\;m'<0\): \[\begin{align} &M^{12}_{(l'm',3),(lm,3)}(\alpha)\\ =&\;\rho^{l+2}a_{33}r^{l'}\Biggl\{ \frac{l'(l'+1)}{2}\Bigl[i(-1)^{m'}\bigl(\mathscr{H}_{l,l'}^{-m,-m',(12)}(\alpha)+(-1)^m \mathscr{H}_{l,l'}^{m,-m',(12)}(\alpha)\bigr)\\ & -i\bigl(\mathscr{H}_{l,l'}^{-m,m',(12)}(\alpha)+(-1)^m \mathscr{H}_{l,l'}^{m,m',(12)}(\alpha)\bigr)\Bigr]\\ &+\frac{\sqrt{l'(l'+1)}}{2}\sum_{q=-1}^{1}\sum_{m_1=-1}^{1}\Bigl[(-1)^{m'}\bigl(\mathscr{L}_{l,l',l'+1}^{-m,-m'-q,q,m_1,(12)}(\alpha)+(-1)^m \mathscr{L}_{l,l',l'+1}^{m,-m'-q,q,m_1,(12)}(\alpha)\bigr)\\ &\qquad\qquad\qquad\qquad\quad-\bigl(\mathscr{L}_{l,l',l'+1}^{-m,m'-q,q,m_1,(12)}(\alpha)+(-1)^m \mathscr{L}_{l,l',l'+1}^{m,m'-q,q,m_1,(12)}(\alpha)\bigr)\Bigr]\Biggr\} \end{align}\]
When \(m>0,\;m'=0\): \[\begin{align} &M^{12}_{(l',0,3),(lm,3)}(\alpha)\\ =&\;\rho^{l+2}a_{33}r^{l'}\Biggl\{ \frac{l'(l'+1)}{\sqrt{2}}\bigl(\mathscr{H}_{l,l'}^{-m,0,(12)}(\alpha)+(-1)^m \mathscr{H}_{l,l'}^{m,0,(12)}(\alpha)\bigr)\\ &-\frac{i\sqrt{l'(l'+1)}}{\sqrt{2}}\sum_{q=-1}^{1}\sum_{m_1=-1}^{1}\bigl(\mathscr{L}_{l,l',l'+1}^{-m,-q,q,m_1,(12)}(\alpha)+(-1)^m \mathscr{L}_{l,l',l'+1}^{m,-q,q,m_1,(12)}(\alpha)\bigr)\Biggr\} \end{align}\]
When \(m<0,\;m'>0\): \[\begin{align} &M^{12}_{(l'm',3),(lm,3)}(\alpha)\\ =&\;\rho^{l+2}a_{33}r^{l'}\Biggl\{ \frac{l'(l'+1)}{2}\Bigl[i(-1)^{m'}\bigl(\mathscr{H}_{l,l'}^{m,m',(12)}(\alpha)-(-1)^m \mathscr{H}_{l,l'}^{-m,m',(12)}(\alpha)\bigr) +i\bigl(\mathscr{H}_{l,l'}^{m,-m',(12)}(\alpha)\\ &-(-1)^m \mathscr{H}_{l,l'}^{-m,-m',(12)}(\alpha)\bigr)\Bigr]\\ &+\frac{\sqrt{l'(l'+1)}}{2}\sum_{q=-1}^{1}\sum_{m_1=-1}^{1}\Bigl[(-1)^{m'}\bigl(\mathscr{L}_{l,l',l'+1}^{m,m'-q,q,m_1,(12)}(\alpha)-(-1)^m \mathscr{L}_{l,l',l'+1}^{-m,m'-q,q,m_1,(12)}(\alpha)\bigr)\\ &\qquad\qquad\qquad\qquad\quad+\bigl(\mathscr{L}_{l,l',l'+1}^{m,-m'-q,q,m_1,(12)}(\alpha)-(-1)^m \mathscr{L}_{l,l',l'+1}^{-m,-m'-q,q,m_1,(12)}(\alpha)\bigr)\Bigr]\Biggr\} \end{align}\]
When \(m<0,\;m'<0\): \[\begin{align} &M^{12}_{(l'm',3),(lm,3)}(\alpha)\\ =&\;\rho^{l+2}a_{33}r^{l'}\Biggl\{ \frac{l'(l'+1)}{2}\Bigl[\bigl(\mathscr{H}_{l,l'}^{m,m',(12)}(\alpha)-(-1)^m \mathscr{H}_{l,l'}^{-m,m',(12)}(\alpha)\bigr) -(-1)^{m'}\bigl(\mathscr{H}_{l,l'}^{m,-m',(12)}(\alpha)\\ &-(-1)^m \mathscr{H}_{l,l'}^{-m,-m',(12)}(\alpha)\bigr)\Bigr]\\ &+\frac{i\sqrt{l'(l'+1)}}{2}\sum_{q=-1}^{1}\sum_{m_1=-1}^{1}\Bigl[(-1)^{m'}\bigl(\mathscr{L}_{l,l',l'+1}^{m,-m'-q,q,m_1,(12)}(\alpha)-(-1)^m \mathscr{L}_{l,l',l'+1}^{-m,-m'-q,q,m_1,(12)}(\alpha)\bigr)\\ &\qquad\qquad\qquad\qquad\quad-\bigl(\mathscr{L}_{l,l',l'+1}^{m,m'-q,q,m_1,(12)}(\alpha)-(-1)^m \mathscr{L}_{l,l',l'+1}^{-m,m'-q,q,m_1,(12)}(\alpha)\bigr)\Bigr]\Biggr\} \end{align}\]
When \(m<0,\;m'=0\): \[\begin{align} &M^{12}_{(l',0,3),(lm,3)}(\alpha)\\ =&\;\rho^{l+2}a_{33}r^{l'}\Biggl\{ \frac{il'(l'+1)}{\sqrt{2}}\bigl(\mathscr{H}_{l,l'}^{m,0,(12)}(\alpha)-(-1)^m \mathscr{H}_{l,l'}^{-m,0,(12)}(\alpha)\bigr)\\ &+\frac{\sqrt{l'(l'+1)}}{\sqrt{2}}\sum_{q=-1}^{1}\sum_{m_1=-1}^{1}\bigl(\mathscr{L}_{l,l',l'+1}^{m,-q,q,m_1,(12)}(\alpha)-(-1)^m \mathscr{L}_{l,l',l'+1}^{-m,-q,q,m_1,(12)}(\alpha)\bigr)\Biggr\}\end{align}\]
When \(m=0,\;m'>0\): \[\begin{align} &M^{12}_{(l'm',3),(l,0,3)}(\alpha)\\ =&\;\rho^{l+2}a_{33}r^{l'}\Biggl\{ \frac{l'(l'+1)}{\sqrt{2}}\Bigl[(-1)^{m'}\mathscr{H}_{l,l'}^{0,m',(12)}(\alpha)+\mathscr{H}_{l,l'}^{0,-m',(12)}(\alpha)\Bigr]\\ &-\frac{i\sqrt{l'(l'+1)}}{\sqrt{2}}\sum_{q=-1}^{1}\sum_{m_1=-1}^{1}\Bigl[(-1)^{m'}\mathscr{L}_{l,l',l'+1}^{0,m'-q,q,m_1,(12)}(\alpha)+\mathscr{L}_{l,l',l'+1}^{0,-m'-q,q,m_1,(12)}(\alpha)\Bigr]\Biggr\}\end{align}\]
When \(m=0,\;m'<0\): \[\begin{align} &M^{12}_{(l'm',3),(l,0,3)}(\alpha)\\ =&\;\rho^{l+2}a_{33}r^{l'}\Biggl\{ \frac{l'(l'+1)}{\sqrt{2}}\Bigl[i(-1)^{m'}\mathscr{H}_{l,l'}^{0,-m',(12)}(\alpha)-i\mathscr{H}_{l,l'}^{0,m',(12)}(\alpha)\Bigr]\\ &+\frac{\sqrt{l'(l'+1)}}{\sqrt{2}}\sum_{q=-1}^{1}\sum_{m_1=-1}^{1}\Bigl[(-1)^{m'}\mathscr{L}_{l,l',l'+1}^{0,-m'-q,q,m_1,(12)}(\alpha)-\mathscr{L}_{l,l',l'+1}^{0,m'-q,q,m_1,(12)}(\alpha)\Bigr]\Biggr\}\end{align}\]
When \(m=0,\;m'=0\): \[\begin{align} &M^{12}_{(l',0,3),(l,0,3)}(\alpha)\\ =&\;\rho^{l+2}a_{33}r^{l'}\Biggl\{ l'(l'+1)\mathscr{H}_{l,l'}^{0,0,(12)}(\alpha) -i\sqrt{l'(l'+1)}\sum_{q=-1}^{1}\sum_{m_1=-1}^{1}\mathscr{L}_{l,l',l'+1}^{0,-q,q,m_1,(12)}(\alpha)\Biggr\} \end{align}\]
Lemma 19. Define \(\mathscr{D}_{l,\lambda}^{m,\mu,(12)}(\alpha)=\sum\limits_{n\in\mathbb{Z}} (n+2d)^2H_{l,\lambda}^{m,\mu}(\mathbf{b}_{12})e^{-in\alpha}\). Then \[\begin{align} &\mathscr{D}_{l,\lambda}^{m,\mu,(12)}(\alpha)\\ =&(-1)^{\lambda+\mu} \sqrt{\dfrac{2l+1}{2\lambda+1}} \sqrt{{l+\lambda+\mu-m\choose \lambda+\mu}{l+\lambda+m-\mu\choose \lambda-\mu}} \sqrt{\dfrac{4\pi}{2(l+\lambda)+1}}\\ & \big(e^{-i\alpha}\cdot \mathrm{\Phi}(e^{-i\alpha},l+\lambda-1,1-2d)Y_{l+\lambda}^{m-\mu}(\dfrac{\pi}{2},\pi)+\mathrm{\Phi}(e^{i\alpha},l+\lambda-1,2d)Y_{l+\lambda}^{m-\mu}(\dfrac{\pi}{2},0)\big) \end{align}\]
Theorem 36. The entries for \(\boxed{M^{12}_{(l'm',2),(lm,2)}(\alpha)}\)(\(l'\geqslant 1,l\geqslant 1\), otherwise this term vanishes) are given by:
If \(m>0\) and \(m'>0\), \[\begin{align}&M^{12}_{(l'm',2),(lm,2)}(\alpha)\\
&=\dfrac{l'(2l'+1)}{2}[ \rho^{l+3}a_{12}r^{l'-1}+\rho^{l+1}(a_{22}-a_{12})(r^{l'+1} )+\rho^{l+1}a_{22}r^{l'+1}\dfrac{2l+1}{2l'+1}]\\
&[(-1)^{m'}(\mathscr{H}_{l,l'}^{-m,m',(12)}(\alpha)+(-1)^m\mathscr{H}_{l,l'}^{m,m',(12)}(\alpha))+(\mathscr{H}_{l,l'}^{-m,-m',(12)}(\alpha)+(-1)^m\mathscr{H}_{l,l'}^{m,-m',(12)}(\alpha))]\\
&+\dfrac{l'(2l'+1)}{2}[\rho^{l+1}(a_{22}-a_{12})r^{l'-1}][(-1)^{m'}(\mathscr{D}_{l,l'}^{-m,m',(12)}(\alpha)+(-1)^m\mathscr{D}_{l,l'}^{m,m',(12)}(\alpha))+(\mathscr{D}_{l,l'}^{-m,-m',(12)}(\alpha)\\
&+(-1)^m\mathscr{D}_{l,l'}^{m,-m',(12)}(\alpha))]\\
&+\rho^{l+1}(a_{22}-a_{12})\sqrt{l'(2l'+1)}\sum\limits_{\lambda\in\{l'+1,l'-1\}}\sum\limits_{q=-1}^{q=1}\sum\limits_{m_1=-1}^{1}r^\lambda [(-1)^{m'}(\mathscr{A}_{l,\lambda}^{-m,m'-q,(12)}(\alpha,q)\\
&+(-1)^m\mathscr{A}_{l,\lambda}^{m,m'-q,(12)}(\alpha,q))K_{l'-1,l',\lambda}^{m_1,m'-q,q}+(\mathscr{A}_{l,\lambda}^{-m,-m'-q,(12)}(\alpha,q)+(-1)^m\mathscr{A}_{l,\lambda}^{m,-m'-q,(12)}(\alpha,q))K_{l'-1,l',\lambda}^{m_1,-m'-q,q}]\\
&+\rho^{l+1}a_{22}(2l+1)\dfrac{\sqrt{l'(2l'+1)}}{2}\sum\limits_{q=-1}^1(-1)^q r^{l'-1}[(-1)^{m'}(\mathscr{A}_{l,l'-1}^{-m,m'-q,(12)}(\alpha,q)+(-1)^m\mathscr{A}_{l,l'-1}^{m,m'-q,(12)}(\alpha,q))\\
&\langle l'-1,m'-q;1,q\mid l',m' \rangle+(\mathscr{A}_{l,l'-1}^{-m,-m'-q,(12)}(\alpha,q)\\
&+(-1)^m\mathscr{A}_{l,l'-1}^{m,-m'-q,(12)}(\alpha,q))\langle l'-1,-m'-q;1,q\mid l',-m' \rangle]
\end{align}\]
If \(m>0\) and \(m'<0\), \[\begin{align} &M^{12}_{(l'm',2),(lm,2)}(\alpha)\\ &=\dfrac{l'(2l'+1)}{2}\Bigl[ \rho^{l+3}a_{12}r^{l'-1}+\rho^{l+1}(a_{22}-a_{12})(r^{l'+1} )+\rho^{l+1}a_{22}r^{l'+1}\dfrac{2l+1}{2l'+1}\Bigr]\\ &\quad\Bigl[i\cdot(-1)^{m'}(\mathscr{H}_{l,l'}^{-m,-m',(12)}(\alpha)+(-1)^m\mathscr{H}_{l,l'}^{m,-m',(12)}(\alpha))-i\cdot(\mathscr{H}_{l,l'}^{-m,m',(12)}(\alpha)+(-1)^m\mathscr{H}_{l,l'}^{m,m',(12)}(\alpha))\Bigr]\\ &+\dfrac{l'(2l'+1)}{2}[\rho^{l+1}(a_{22}-a_{12})r^{l'-1}]\Bigl[i\cdot(-1)^{m'}(\mathscr{D}_{l,l'}^{-m,-m',(12)}(\alpha)+(-1)^m\mathscr{D}_{l,l'}^{m,-m',(12)}(\alpha))\\ &-i\cdot(\mathscr{D}_{l,l'}^{-m,m',(12)}(\alpha)+(-1)^m\mathscr{D}_{l,l'}^{m,m',(12)}(\alpha))\Bigr]\\ &+\rho^{l+1}(a_{22}-a_{12})\sqrt{l'(2l'+1)}\sum_{\lambda\in\{l'+1,l'-1\}}\sum_{q=-1}^{1}\sum_{m_1=-1}^{1}r^\lambda \Bigl[i\cdot(-1)^{m'}(\mathscr{A}_{l,\lambda}^{-m,-m'-q,(12)}(\alpha,q)\\ &+(-1)^m\mathscr{A}_{l,\lambda}^{m,-m'-q,(12)}(\alpha,q))K_{l'-1,l',\lambda}^{m_1,-m'-q,q}-i\cdot(\mathscr{A}_{l,\lambda}^{-m,m'-q,(12)}(\alpha,q)\\ &+(-1)^m\mathscr{A}_{l,\lambda}^{m,m'-q,(12)}(\alpha,q))K_{l'-1,l',\lambda}^{m_1,m'-q,q}\Bigr]\\ &+\rho^{l+1}a_{22}(2l+1)\dfrac{\sqrt{l'(2l'+1)}}{2}\sum_{q=-1}^{1}(-1)^q r^{l'-1}\Bigl[i\cdot(-1)^{m'}(\mathscr{A}_{l,l'-1}^{-m,-m'-q,(12)}(\alpha,q)\\ &+(-1)^m\mathscr{A}_{l,l'-1}^{m,-m'-q,(12)}(\alpha,q))\langle l'-1,-m'-q;1,q\mid l',-m' \rangle-i\cdot(\mathscr{A}_{l,l'-1}^{-m,m'-q,(12)}(\alpha,q)\\ &+(-1)^m\mathscr{A}_{l,l'-1}^{m,m'-q,(12)}(\alpha,q))\langle l'-1,m'-q;1,q\mid l',m' \rangle\Bigr] \end{align}\]
If \(m>0\) and \(m'=0\), \[\begin{align} &M^{12}_{(l',0,2),(lm,2)}(\alpha)\\ &=\dfrac{l'(2l'+1)}{\sqrt{2}}\Bigl[ \rho^{l+3}a_{12}r^{l'-1}+\rho^{l+1}(a_{22}-a_{12})(r^{l'+1} )+\rho^{l+1}a_{22}r^{l'+1}\dfrac{2l+1}{2l'+1}\Bigr]\\ &\quad\Bigl[(\mathscr{H}_{l,l'}^{-m,0,(12)}(\alpha)+(-1)^m\mathscr{H}_{l,l'}^{m,0,(12)}(\alpha))\Bigr]\\ &+\dfrac{l'(2l'+1)}{\sqrt{2}}[\rho^{l+1}(a_{22}-a_{12})r^{l'-1}]\Bigl[(\mathscr{D}_{l,l'}^{-m,0,(12)}(\alpha)+(-1)^m\mathscr{D}_{l,l'}^{m,0,(12)}(\alpha))\Bigr]\\ &+\rho^{l+1}(a_{22}-a_{12})\sqrt{2l'(2l'+1)}\sum_{\lambda\in\{l'+1,l'-1\}}\sum_{q=-1}^{1}\sum_{m_1=-1}^{1}r^\lambda \Bigl[(\mathscr{A}_{l,\lambda}^{-m,-q,(12)}(\alpha,q)\\ &+(-1)^m\mathscr{A}_{l,\lambda}^{m,-q,(12)}(\alpha,q))K_{l'-1,l',\lambda}^{m_1,-q,q}\Bigr]\\ &+\rho^{l+1}a_{22}(2l+1)\dfrac{\sqrt{l'(2l'+1)}}{\sqrt{2}}\sum_{q=-1}^{1}(-1)^q r^{l'-1}\Bigl[(\mathscr{A}_{l,l'-1}^{-m,-q,(12)}(\alpha,q)\\ &+(-1)^m\mathscr{A}_{l,l'-1}^{m,-q,(12)}(\alpha,q))\langle l'-1,-q;1,q\mid l',0 \rangle\Bigr]. \end{align}\]
If \(m<0\) and \(m'>0\), \[\begin{align} &M^{12}_{(l'm',2),(lm,2)}(\alpha)\\ &=\dfrac{l'(2l'+1)}{2}\Bigl[ \rho^{l+3}a_{12}r^{l'-1}+\rho^{l+1}(a_{22}-a_{12})(r^{l'+1} )+\rho^{l+1}a_{22}r^{l'+1}\dfrac{2l+1}{2l'+1}\Bigr]\\ &\quad\cdot i\Bigl[(-1)^{m'}(\mathscr{H}_{l,l'}^{m,m',(12)}(\alpha)-(-1)^m\mathscr{H}_{l,l'}^{-m,m',(12)}(\alpha))+(\mathscr{H}_{l,l'}^{m,-m',(12)}(\alpha)-(-1)^m\mathscr{H}_{l,l'}^{-m,-m',(12)}(\alpha))\Bigr]\\ &+\dfrac{l'(2l'+1)}{2}[\rho^{l+1}(a_{22}-a_{12})r^{l'-1}]\cdot i\Bigl[(-1)^{m'}(\mathscr{D}_{l,l'}^{m,m',(12)}(\alpha)-(-1)^m\mathscr{D}_{l,l'}^{-m,m',(12)}(\alpha))\\ &+(\mathscr{D}_{l,l'}^{m,-m',(12)}(\alpha)-(-1)^m\mathscr{D}_{l,l'}^{-m,-m',(12)}(\alpha))\Bigr]\\ &+\rho^{l+1}(a_{22}-a_{12})\sqrt{l'(2l'+1)}\sum_{\lambda\in\{l'+1,l'-1\}}\sum_{q=-1}^{1}\sum_{m_1=-1}^{1}r^\lambda \;i\Bigl[(-1)^{m'}(\mathscr{A}_{l,\lambda}^{m,m'-q,(12)}(\alpha,q)\\ &\quad-(-1)^m\mathscr{A}_{l,\lambda}^{-m,m'-q,(12)}(\alpha,q))K_{l'-1,l',\lambda}^{m_1,m'-q,q}+(\mathscr{A}_{l,\lambda}^{m,-m'-q,(12)}(\alpha,q)\\ &-(-1)^m\mathscr{A}_{l,\lambda}^{-m,-m'-q,(12)}(\alpha,q))K_{l'-1,l',\lambda}^{m_1,-m'-q,q}\Bigr]\\ &+\rho^{l+1}a_{22}(2l+1)\dfrac{\sqrt{l'(2l'+1)}}{2}\sum_{q=-1}^{1}(-1)^q r^{l'-1}\;i\Bigl[(-1)^{m'}(\mathscr{A}_{l,l'-1}^{m,m'-q,(12)}(\alpha,q)-(-1)^m\mathscr{A}_{l,l'-1}^{-m,m'-q,(12)}(\alpha,q))\\ &\quad\langle l'-1,m'-q;1,q\mid l',m' \rangle+(\mathscr{A}_{l,l'-1}^{m,-m'-q,(12)}(\alpha,q)\\ &-(-1)^m\mathscr{A}_{l,l'-1}^{-m,-m'-q,(12)}(\alpha,q))\langle l'-1,-m'-q;1,q\mid l',-m' \rangle\Bigr]. \end{align}\]
If \(m<0\) and \(m'<0\), \[\begin{align} &M^{12}_{(l'm',2),(lm,2)}(\alpha)\\ &=\dfrac{l'(2l'+1)}{2}\Bigl[ \rho^{l+3}a_{12}r^{l'-1}+\rho^{l+1}(a_{22}-a_{12})(r^{l'+1} )+\rho^{l+1}a_{22}r^{l'+1}\dfrac{2l+1}{2l'+1}\Bigr]\\ &\quad\Bigl[-(-1)^{m'}(\mathscr{H}_{l,l'}^{m,-m',(12)}(\alpha)-(-1)^m\mathscr{H}_{l,l'}^{-m,-m',(12)}(\alpha))+(\mathscr{H}_{l,l'}^{m,m',(12)}(\alpha)-(-1)^m\mathscr{H}_{l,l'}^{-m,m',(12)}(\alpha))\Bigr]\\ &+\dfrac{l'(2l'+1)}{2}[\rho^{l+1}(a_{22}-a_{12})r^{l'-1}]\Bigl[-(-1)^{m'}(\mathscr{D}_{l,l'}^{m,-m',(12)}(\alpha)\\ &-(-1)^m\mathscr{D}_{l,l'}^{-m,-m',(12)}(\alpha))+(\mathscr{D}_{l,l'}^{m,m',(12)}(\alpha)-(-1)^m\mathscr{D}_{l,l'}^{-m,m',(12)}(\alpha))\Bigr]\\ &+\rho^{l+1}(a_{22}-a_{12})\sqrt{l'(2l'+1)}\sum_{\lambda\in\{l'+1,l'-1\}}\sum_{q=-1}^{1}\sum_{m_1=-1}^{1}r^\lambda \Bigl[-(-1)^{m'}(\mathscr{A}_{l,\lambda}^{m,-m'-q,(12)}(\alpha,q)\\ &\quad-(-1)^m\mathscr{A}_{l,\lambda}^{-m,-m'-q,(12)}(\alpha,q))K_{l'-1,l',\lambda}^{m_1,-m'-q,q}+(\mathscr{A}_{l,\lambda}^{m,m'-q,(12)}(\alpha,q)\\ &-(-1)^m\mathscr{A}_{l,\lambda}^{-m,m'-q,(12)}(\alpha,q))K_{l'-1,l',\lambda}^{m_1,m'-q,q}\Bigr]\\ &+\rho^{l+1}a_{22}(2l+1)\dfrac{\sqrt{l'(2l'+1)}}{2}\sum_{q=-1}^{1}(-1)^q r^{l'-1}\Bigl[-(-1)^{m'}(\mathscr{A}_{l,l'-1}^{m,-m'-q,(12)}(\alpha,q)\\ &-(-1)^m\mathscr{A}_{l,l'-1}^{-m,-m'-q,(12)}(\alpha,q))\\ &\quad\langle l'-1,-m'-q;1,q\mid l',-m' \rangle+(\mathscr{A}_{l,l'-1}^{m,m'-q,(12)}(\alpha,q)\\ &-(-1)^m\mathscr{A}_{l,l'-1}^{-m,m'-q,(12)}(\alpha,q))\langle l'-1,m'-q;1,q\mid l',m' \rangle\Bigr].\end{align}\]
If \(m<0\) and \(m'=0\), \[\begin{align} &M^{12}_{(l',0,2),(lm,2)}(\alpha)\\ &=\dfrac{l'(2l'+1)}{\sqrt{2}}\Bigl[ \rho^{l+3}a_{12}r^{l'-1}+\rho^{l+1}(a_{22}-a_{12})(r^{l'+1} )+\rho^{l+1}a_{22}r^{l'+1}\dfrac{2l+1}{2l'+1}\Bigr] \cdot i\Bigl[(\mathscr{H}_{l,l'}^{m,0,(12)}(\alpha)\\ &-(-1)^m\mathscr{H}_{l,l'}^{-m,0,(12)}(\alpha))\Bigr]\\ &+\dfrac{l'(2l'+1)}{\sqrt{2}}[\rho^{l+1}(a_{22}-a_{12})r^{l'-1}]\cdot i\Bigl[(\mathscr{D}_{l,l'}^{m,0,(12)}(\alpha)-(-1)^m\mathscr{D}_{l,l'}^{-m,0,(12)}(\alpha))\Bigr]\\ &+\rho^{l+1}(a_{22}-a_{12})\sqrt{2l'(2l'+1)}\sum_{\lambda\in\{l'+1,l'-1\}}\sum_{q=-1}^{1}\sum_{m_1=-1}^{1}r^\lambda \;i\Bigl[(\mathscr{A}_{l,\lambda}^{m,-q,(12)}(\alpha,q)\\ &-(-1)^m\mathscr{A}_{l,\lambda}^{-m,-q,(12)}(\alpha,q))K_{l'-1,l',\lambda}^{m_1,-q,q}\Bigr]\\ &+\rho^{l+1}a_{22}(2l+1)\dfrac{\sqrt{l'(2l'+1)}}{\sqrt{2}}\sum_{q=-1}^{1}(-1)^q r^{l'-1}\;i\Bigl[(\mathscr{A}_{l,l'-1}^{m,-q,(12)}(\alpha,q)\\ &-(-1)^m\mathscr{A}_{l,l'-1}^{-m,-q,(12)}(\alpha,q))\langle l'-1,-q;1,q\mid l',0 \rangle\Bigr]. \end{align}\]
If \(m=0\) and \(m'>0\), \[\begin{align} &M^{12}_{(l'm',2),(l,0,2)}(\alpha)\\ &=\dfrac{l'(2l'+1)}{\sqrt{2}}\Bigl[ \rho^{l+3}a_{12}r^{l'-1}+\rho^{l+1}(a_{22}-a_{12})(r^{l'+1} )+\rho^{l+1}a_{22}r^{l'+1}\dfrac{2l+1}{2l'+1}\Bigr]\\ &\quad\Bigl[(-1)^{m'}\mathscr{H}_{l,l'}^{0,m',(12)}(\alpha)+\mathscr{H}_{l,l'}^{0,-m',(12)}(\alpha)\Bigr]\\ &+\dfrac{l'(2l'+1)}{\sqrt{2}}[\rho^{l+1}(a_{22}-a_{12})r^{l'-1}]\Bigl[(-1)^{m'}\mathscr{D}_{l,l'}^{0,m',(12)}(\alpha)+\mathscr{D}_{l,l'}^{0,-m',(12)}(\alpha)\Bigr]\\ &+\rho^{l+1}(a_{22}-a_{12})\sqrt{2l'(2l'+1)}\sum_{\lambda\in\{l'+1,l'-1\}}\sum_{q=-1}^{1}\sum_{m_1=-1}^{1}r^\lambda \Bigl[(-1)^{m'}\mathscr{A}_{l,\lambda}^{0,m'-q,(12)}(\alpha,q)K_{l'-1,l',\lambda}^{m_1,m'-q,q}\\ &+\mathscr{A}_{l,\lambda}^{0,-m'-q,(12)}(\alpha,q)K_{l'-1,l',\lambda}^{m_1,-m'-q,q}\Bigr]\\ &+\rho^{l+1}a_{22}(2l+1)\dfrac{\sqrt{l'(2l'+1)}}{\sqrt{2}}\sum_{q=-1}^{1}(-1)^q r^{l'-1}\Bigl[(-1)^{m'}\mathscr{A}_{l,l'-1}^{0,m'-q,(12)}(\alpha,q)\langle l'-1,m'-q;1,q\mid l',m' \rangle\\ &+\mathscr{A}_{l,l'-1}^{0,-m'-q,(12)}(\alpha,q)\langle l'-1,-m'-q;1,q\mid l',-m' \rangle\Bigr]. \end{align}\]
If \(m=0\) and \(m'<0\), \[\begin{align} &M^{12}_{(l'm',2),(l,0,2)}(\alpha)\\ &=\dfrac{l'(2l'+1)}{\sqrt{2}}\Bigl[ \rho^{l+3}a_{12}r^{l'-1}+\rho^{l+1}(a_{22}-a_{12})(r^{l'+1} )+\rho^{l+1}a_{22}r^{l'+1}\dfrac{2l+1}{2l'+1}\Bigr]\\ &\quad i\Bigl[(-1)^{m'}\mathscr{H}_{l,l'}^{0,-m',(12)}(\alpha)-\mathscr{H}_{l,l'}^{0,m',(12)}(\alpha)\Bigr]\\ &+\dfrac{l'(2l'+1)}{\sqrt{2}}[\rho^{l+1}(a_{22}-a_{12})r^{l'-1}]\cdot i\Bigl[(-1)^{m'}\mathscr{D}_{l,l'}^{0,-m',(12)}(\alpha)-\mathscr{D}_{l,l'}^{0,m',(12)}(\alpha)\Bigr]\\ &+\rho^{l+1}(a_{22}-a_{12})\sqrt{2l'(2l'+1)}\sum_{\lambda\in\{l'+1,l'-1\}}\sum_{q=-1}^{1}\sum_{m_1=-1}^{1}r^\lambda \;i\Bigl[(-1)^{m'}\mathscr{A}_{l,\lambda}^{0,-m'-q,(12)}(\alpha,q)K_{l'-1,l',\lambda}^{m_1,-m'-q,q}\\ &-\mathscr{A}_{l,\lambda}^{0,m'-q,(12)}(\alpha,q)K_{l'-1,l',\lambda}^{m_1,m'-q,q}\Bigr]\\ &+\rho^{l+1}a_{22}(2l+1)\dfrac{\sqrt{l'(2l'+1)}}{\sqrt{2}}\sum_{q=-1}^{1}(-1)^q r^{l'-1}\;i\Bigl[(-1)^{m'}\mathscr{A}_{l,l'-1}^{0,-m'-q,(12)}(\alpha,q)\langle l'-1,-m'-q;1,q\mid l',-m' \rangle\\ &-\mathscr{A}_{l,l'-1}^{0,m'-q,(12)}(\alpha,q)\langle l'-1,m'-q;1,q\mid l',m' \rangle\Bigr]. \end{align}\]
If \(m=0\) and \(m'=0\), \[\begin{align} &M^{12}_{(l',0,2),(l,0,2)}(\alpha)\\ &=l'(2l'+1)\Bigl[ \rho^{l+3}a_{12}r^{l'-1}+\rho^{l+1}(a_{22}-a_{12})(r^{l'+1} )+\rho^{l+1}a_{22}r^{l'+1}\dfrac{2l+1}{2l'+1}\Bigr]\mathscr{H}_{l,l'}^{0,0,(12)}(\alpha)\\ &+{l'(2l'+1)}[\rho^{l+1}(a_{22}-a_{12})r^{l'-1}]\mathscr{D}_{l,l'}^{0,0,(12)}(\alpha)\\ &+\rho^{l+1}(a_{22}-a_{12})\cdot 2\sqrt{l'(2l'+1)}\sum_{\lambda\in\{l'+1,l'-1\}}\sum_{q=-1}^{1}\sum_{m_1=-1}^{1}r^\lambda \;\mathscr{A}_{l,\lambda}^{0,-q,(12)}(\alpha,q)K_{l'-1,l',\lambda}^{m_1,-q,q}\\ &+\rho^{l+1}a_{22}(2l+1)\sqrt{l'(2l'+1)}\sum_{q=-1}^{1}(-1)^q r^{l'-1}\;\mathscr{A}_{l,l'-1}^{0,-q,(12)}(\alpha,q)\langle l'-1,-q;1,q\mid l',0 \rangle. \end{align}\]
Theorem 37. The remaining terms are simple: \[M^{12}_{(l'm',3),(lm,1)}(\alpha)=M^{12}_{(l'm',1),(lm,3)}(\alpha)=0,\] \[M^{12}_{(l'm',1),(lm,1)}(\alpha)=0.\]
First, consider the table of vector spherical harmonics up to the second order as listed below:
\[l = 0: \quad Y_{0,0} = \frac{1}{2} \sqrt{\frac{1}{\pi}}, \quad l = 1: \quad Y_{1,-1} = \sqrt{\frac{3}{4\pi}} y, \quad Y_{1,0} = \sqrt{\frac{3}{4\pi}} z, \quad Y_{1,1} = \sqrt{\frac{3}{4\pi}} x,\]
\[l = 2: \quad Y_{2,-2} = \frac{1}{2} \sqrt{\frac{15}{\pi}} xy, \quad Y_{2,-1} = \frac{1}{2} \sqrt{\frac{15}{\pi}} yz, \quad Y_{2,0} = \frac{1}{4} \sqrt{\frac{5}{\pi}} (-x^2 - y^2 + 2z^2),\]
\[Y_{2,1} = \frac{1}{2} \sqrt{\frac{15}{\pi}} xz, \quad Y_{2,2} = \frac{1}{4} \sqrt{\frac{15}{\pi}} (x^2 - y^2).\]
This gives first the obvious result that
\[l = 0: \quad V_{0,0} = -\frac{1}{2} \sqrt{\frac{1}{\pi}} (x, y, z)^\top \quad \text{and} \quad W_{00} = X_{00} = 0.\]
The spherical harmonics \(V_{l m}\) up to order 2 are then given as follows
\[l = 0: \quad V_{0,0} = -\frac{1}{2} \sqrt{\frac{1}{\pi}} (x, y, z)^\top,\]
\[l = 1: \quad V_{1,-1} = \sqrt{\frac{3}{4\pi}} ((0, 1, 0) - 2y(x, y, z))^\top,\]
\[V_{1,0} = \sqrt{\frac{3}{4\pi}} ((0, 0, 1) - 2z(x, y, z))^\top,\]
\[V_{1,1} = \sqrt{\frac{3}{4\pi}} ((1, 0, 0) - 2x(x, y, z))^\top,\]
\[l = 2: \quad V_{2,-2} = \frac{1}{2} \sqrt{\frac{15}{\pi}} ((y, x, 0) - 5xy(x, y, z))^\top,\]
\[V_{2,-1} = \frac{1}{2} \sqrt{\frac{15}{\pi}} ((0, z, y) - 5yz(x, y, z))^\top,\]
\[V_{2,0} = \frac{1}{2} \sqrt{\frac{5}{\pi}} ((-x, -y, 2z) - \frac{5}{2}(-x^2 - y^2 + 2z^2)(x, y, z))^\top,\]
\[V_{2,1} = \frac{1}{2} \sqrt{\frac{15}{\pi}} ((z, 0, x) - 5xz(x, y, z))^\top,\]
\[V_{2,2} = \frac{1}{2} \sqrt{\frac{15}{\pi}} ((x, -y, 0) - \frac{5}{2}(x^2 - y^2)(x, y, z))^\top.\]
The spherical harmonics \(W_{l m}\) up to order 2 are given by
\[l = 0: \quad W_{0,0} = 0,\]
\[l = 1: \quad W_{1,-1} = \sqrt{\frac{3}{4\pi}} (0, 1, 0)^\top, \quad W_{1,0} = \sqrt{\frac{3}{4\pi}} (0, 0, 1)^\top, \quad W_{1,1} = \sqrt{\frac{3}{4\pi}} (1, 0, 0)^\top,\]
\[l = 2: \quad W_{2,-2} = \frac{1}{2} \sqrt{\frac{15}{\pi}} (y, x, 0)^\top, \quad W_{2,-1} = \frac{1}{2} \sqrt{\frac{15}{\pi}} (0, z, y)^\top, \quad W_{2,0} = \frac{1}{2} \sqrt{\frac{5}{\pi}} (-x, -y, 2z)^\top,\]
\[W_{2,1} = \frac{1}{2} \sqrt{\frac{15}{\pi}} (z, 0, x)^\top, \quad W_{2,2} = \frac{1}{2} \sqrt{\frac{15}{\pi}} (x, -y, 0)^\top.\]
And finally, the spherical harmonics \(X_{l m}\) up to order 2 are given by
\[l = 0: \quad X_{0,0} = 0,\]
\[l = 1: \quad X_{1,-1} = \sqrt{\frac{3}{4\pi}} (-z, 0, x)^\top, \quad X_{1,0} = \sqrt{\frac{3}{4\pi}} (y, -x, 0)^\top,\]
\[X_{1,1} = \sqrt{\frac{3}{4\pi}} (0, z, -y)^\top,\]
\[l = 2: \quad X_{2,-2} = \frac{1}{2} \sqrt{\frac{15}{\pi}} (-xz, yz, x^2 - y^2)^\top, \quad X_{2,-1} = \frac{1}{2} \sqrt{\frac{15}{\pi}} (y^2 - z^2, -xy, xz)^\top,\]
\[X_{2,0} = \frac{1}{2} \sqrt{\frac{5}{\pi}} (3yz, -3xz, 0)^\top, \quad X_{2,1} = \frac{1}{2} \sqrt{\frac{15}{\pi}} (xy, z^2 - x^2, -yz)^\top,\]
\[X_{2,2} = \frac{1}{2} \sqrt{\frac{15}{\pi}} (yz, xz, -2xy)^\top.\]