In this paper, we give a direct proof of determinantal formulas for connected \(k\)-point functions for hermitian matrix models. We also give a new proof of KP integrability for them. From the viewpoint of KP hierarchy,
we further give a new proof of the explicit formula for the corresponding affine coordinates. Furthermore, duality for some hermitian matrix models is proved.
Consider a positive measure \(\mathop{}\!\mathrm{d}\mu\) over \(\mathbb{R}\). Let \(\pi_k(\lambda)=\lambda^k+\cdots\), \(k\ge0\), be the associated monic orthogonal polynomials with respect to \(\mathop{}\!\mathrm{d}\mu\). Denote \[h_k:=\int_\mathbb{R}\pi_k(\lambda)^2\mathop{}\!\mathrm{d}\mu(\lambda),\qquad k\ge0. \label{orth}\tag{1}\] The orthogonal polynomials satisfy the following three-term recurrence relations:
\[\lambda\pi_l(\lambda)=\pi_{l+1}(\lambda)+\beta_l\pi_l(\lambda)+\gamma_l\pi_{l-1}(\lambda),\quad l\ge1, \label{three-term}\tag{2}\] where \(\beta_l,\gamma_l\in\mathbb{R}\). In the 19th century, motivated from numerical integration and approximation theory, the Christoffel–Darboux kernel \(K_n(\lambda_1,\lambda_2)\) was introduced:
\[K_n(\lambda_1,\lambda_2):=\sum_{l=0}^{n-1}\frac{\pi_l(\lambda_1)\pi_l(\lambda_2)}{h_l},\quad n\ge0,\] which was shown [1], [2] to satisfy the following identity: for any \(n\ge1\), \[K_n(\lambda_1,\lambda_2)=\frac{1}{h_{n-1}}\frac{\pi_n(\lambda_1)\pi_{n-1}(\lambda_2)-\pi_{n-1}(\lambda_1)\pi_n(\lambda_2)}{\lambda_1-\lambda_2}.\label{c-d}\tag{3}\]
Since the mid-twentieth century, to understand energy level repulsion in complex heavy nuclei, Wigner, Dyson, Mehta and others established the random matrix theory, where the probability distributions of eigenvalues are important subjects. Dyson [3] studied the joint probability density of \(k\) eigenvalues of the hermitian matrix with respect to \(\mathop{}\!\mathrm{d}\mu\)\[\rho_k(n;\lambda_1,\cdots,\lambda_k)=\frac{\int_{\mathbb{R}^{n-k}}\prod_{1\le i<j\le
n}(\lambda_i-\lambda_j)^2\mathop{}\!\mathrm{d}\mu(\lambda_{k+1})\cdots\mathop{}\!\mathrm{d}\mu(\lambda_n)}{\int_{\mathbb{R}^n} \prod_{1\le i<j\le
n}(\lambda_i-\lambda_j)^2\mathop{}\!\mathrm{d}\mu(\lambda_1)\cdots\mathop{}\!\mathrm{d}\mu(\lambda_n)},\] and further showed [4] that it can be
written as a determinant using the Christoffel–Darboux kernel: \[\label{dpp}
\rho_k(n;\lambda_1,\cdots,\lambda_k)=\frac{1}{(n-k+1)_k}\det(K_n(\lambda_i,\lambda_j))_{i,j=1}^{k},\tag{4}\] where \((n)_k:=n(n+1)\cdots(n+k-1)\) denotes the increasing Pochhammer symbol.
The notion of correlators was known [5], [6] to have important relations with field theory of
strong interactions. For the Gaussian unitary ensemble (GUE), where \(\mathop{}\!\mathrm{d}\mu(\lambda)=e^{-\frac{\lambda^2}{2}}\mathop{}\!\mathrm{d}\lambda\), the \(k\)-point correlators
\[\langle\mathop{\mathrm{tr}}M^{i_1}\cdots\mathop{\mathrm{tr}}M^{i_k}\rangle(n):=\frac{\int_{\mathcal{H}_n}\mathop{\mathrm{tr}}M^{i_1}\cdots\mathop{\mathrm{tr}}M^{i_k}e^{-\frac{1}{2}\mathop{\mathrm{tr}}M^2}\mathop{}\!\mathrm{d}M}{\int_{\mathcal{H}_n}e^{-\frac{1}{2}\mathop{\mathrm{tr}}M^2}\mathop{}\!\mathrm{d}M}\]
were considered in [7]–[9] (see also [10], [11]). Here \(\mathcal{H}_n\) denotes the space of \(n\times n\) hermitian matrices. Denote by \(\langle\mathop{\mathrm{tr}}M^{i_1}\cdots\mathop{\mathrm{tr}}M^{i_k}\rangle_c(n)\) the connected \(k\)-point correlators. Explicit formula for \(\langle\mathop{\mathrm{tr}}M^{i}\rangle_c(n)\) was given in [9]. Explicit formula for \(\langle\mathop{\mathrm{tr}}M^{i_1}\mathop{\mathrm{tr}}M^{i_2}\rangle_c(n)\) was given in [12]. For general \(k\ge1\), explicit formula for connected \(k\)-point GUE correlators was given in [13] based on 1D Toda
integrability.
It is known [14] that the partition function of GUE correlators is a KP tau-function. In [15], for any KP tau-function \(\tau\), the \(k\)-point generating series of logarithmic derivatives of \(\tau\) are
expressed by \[\begin{align} \sum_{j\ge1}\left.\frac{\partial\log\tau}{\partial t_j}\right|_{\mathbf{t}=0}\frac{1}{\lambda_1^{j+1}}=&\lim_{\lambda\to\lambda_1}\left(\widehat
A(\lambda,\lambda_1)-\frac{1}{\lambda-\lambda_1}\right),\tag{5}\\ \sum_{j_1,\cdots,j_k\ge1}\left.\frac{\partial^k\log\tau}{\partial t_{j_1}\cdots\partial
t_{j_k}}\right|_{\mathbf{t}=0}\frac{1}{\prod_{i=1}^k\lambda_i^{j_i+1}}=&\frac{(-1)^{k-1}}{k}\sum_{\sigma\in S_k}\prod_{i=1}^k\widehat A(\lambda_{\sigma(i)},\lambda_{\sigma(i+1)})-\frac{\delta_{k,2}}{(\lambda_1-\lambda_2)^2},\,k\ge2,\tag{6}
\end{align}\] where \[\begin{align} \widehat
A(\lambda_1,\lambda_2):=&\sum_{i,j\ge0}\frac{A_{i,j}}{\lambda_1^{j+1}\lambda_2^{i+1}}+\frac{1}{\lambda_1-\lambda_2}.
\end{align}\] Here \(A_{i,j}\) are the affine coordinates (cf. [15]–[19]) of the element in the Sato Grassmannian corresponding to \(\tau\). For GUE, we denote the affine coordinates by \(A_{i,j}^{\text{GUE}}\), whose explicit
expressions are given by [20]\[\label{kpgue} A_{i,j}^{\text{GUE}}=\begin{cases}
(-1)^{i+\left[\frac{i+1}{2}\right]}\frac{(2m-1)!!}{(2m)!}\binom{m-1}{\left[\frac{i}{2}\right]}\prod_{k=-i}^{2m-1-i}(n+k),&i+j=2m-1,\\ 0,&\text{otherwise}. \end{cases}\tag{7}\] In this case, 5 –7 were reproved in [21] using the method of [13] and
the following \[\sum_{i,j\ge0}\frac{A_{i,j}^\text{GUE}}{\lambda_1^{j+1}\lambda_2^{i+1}}+\frac{1}{\lambda_1-\lambda_2}=\frac{1}{\lambda_2\Gamma(n)}\frac{\psi_A^\text{G}(\lambda_1,n)\psi_B^\text{G}(\lambda_2,n-1)-\psi_A^\text{G}(\lambda_1,n-1)\psi_B^\text{G}(\lambda_2,n)}{\lambda_1-\lambda_2}\frac{\lambda_2^n}{\lambda_1^n}\]
with \[\begin{align} \psi_A^\text{G}(\lambda,n)=\sum_{j\ge0}(-1)^j\frac{(n-2j+1)_{2j}}{2^jj!\lambda^{2j}}\lambda^n,\qquad\psi_B^\text{G}(\lambda,n)=\Gamma(n+1)\sum_{j\ge0}\frac{(n+1)_{2j}}{2^jj!\lambda^{2j}}\lambda^{-n}.
\end{align}\]
For the case when \(\mathop{}\!\mathrm{d}\mu(\lambda)=w(\lambda)\mathop{}\!\mathrm{d}\lambda\) where \(w:\mathbb{R}\to\mathbb{R}\) is a probability density function with finite moments of
all orders, it is known from [13], [21]–[26]
that the partition function of the associated hermitian matrix model is a 1D Toda tau-function. Then from [13], [21]
we know that generating series of connected \(k\)-point correlators can be represented by using the following kernel \[\label{dn}
D_n(\lambda_1,\lambda_2):=\frac{\psi_A(\lambda_1,n)\psi_B(\lambda_2,n-1)-\psi_A(\lambda_1,n-1)\psi_B(\lambda_2,n)}{\lambda_1-\lambda_2},\tag{8}\] where \(\psi_A(\lambda,n),\psi_B(\lambda,n)\) are a pair of
wave functions of the associated Lax operator \(L=\mathcal{T}+\beta_n+\gamma_n\mathcal{T}^{-1}\) with \(\mathcal{T}:f(n)\mapsto f(n+1)\) the shifting operator.
For the case when the measure \(\mathop{}\!\mathrm{d}\mu(\lambda)=e^{V(\lambda)}\mathop{}\!\mathrm{d}\lambda\) with \(V'(\lambda)\) being a rational function, a formula for the
associated connected \(k\)-point correlators was obtained by Ruzza [27] using the Riemann-Hilbert approach, where the
Cauchy–Hilbert–Stieltjes transform (cf. [11], [27]–[29]) of the orthogonal polynomials \(\pi_n(\lambda)\)\[\label{chs}
\widehat{\pi}_n(\xi):=\int_\mathbb{R}\frac{\pi_n(\lambda)}{\lambda-\xi}\mathop{}\!\mathrm{d}\mu(\lambda),\qquad\xi\in\mathbb{C}\backslash\mathbb{R}, \, n\in\mathbb{Z}_{\ge0},\tag{9}\] was considered. Actually, our observation is that the
asymptotic expansions of \(\pi_n(\lambda)\) and of \(-\lambda\widehat\pi_n(\lambda)\) as \(\lambda\to\infty\) within any sector in \(\mathbb{C}\backslash\mathbb{R}\) form a pair of wave functions of the associated Lax operator.
Motivated by the above discussions, for any positive Borel measure \(\mathop{}\!\mathrm{d}\mu\) over \(\mathbb{R}\) supported on infinitely many points with finite moments of all
orders [11], [30], denote by \(\pi_n(\lambda)\) the orthogonal polynomials
with respect to \(\mathop{}\!\mathrm{d}\mu\), by \(\widehat\pi_n(\lambda)\) the Cauchy–Hilbert–Stieltjes transform of \(\pi_n(\lambda)\) (see 9 ), and by \[\psi_A^\star(\lambda,n)=(1+O(\lambda^{-1}))\lambda^n,\quad\psi_B^\star(\lambda,n)=(1+O(\lambda^{-1}))h_n\lambda^{-n}, \qquad n\in\mathbb{Z}_{\ge0},\] the asymptotic expansions of \(\pi_n(\lambda)\), \(-\lambda\widehat\pi_n(\lambda)\) as \(\lambda\to\infty\) within any sector in \(\mathbb{C}\backslash\mathbb{R}\). Introduce the kernel \[\label{dstar}
D_n^\star(\lambda_1,\lambda_2):=\frac{\psi_A^\star(\lambda_1,n)\psi_B^\star(\lambda_2,n-1)-\psi_A^\star(\lambda_1,n-1)\psi_B^\star(\lambda_2,n)}{\lambda_1-\lambda_2},\quad n\ge1.\tag{10}\] We also denote by \(\langle\mathop{\mathrm{tr}}M^{i_1}\cdots\mathop{\mathrm{tr}}M^{i_k}\rangle_c(n)\) the connected \(k\)-point correlators of the associated hermitian matrix model, and let \[C_k(n;\lambda_1,\cdots,\lambda_k):=\sum_{i_1,\cdots,i_k\ge1}\frac{\langle\mathop{\mathrm{tr}}M^{i_1}\cdots\mathop{\mathrm{tr}}M^{i_k}\rangle_c(n)}{\lambda_1^{i_1+1}\cdots\lambda_k^{i_k+1}},\qquad n\in\mathbb{Z}_{\ge0}.\]
Theorem 1. The following formula holds for \(n\in\mathbb{Z}_{\ge0}\): \[\begin{align} \label{maineq}
C_k(n;\lambda_1,\cdots,\lambda_k)=\frac{(-1)^{k-1}}{k}\sum_{\sigma\in S_k}\prod_{i=1}^kB_n(\lambda_{\sigma(i)},\lambda_{\sigma(i+1)})-\frac{\delta_{k,2}}{(\lambda_1-\lambda_2)^2},\qquad k\ge2,
\end{align}\qquad{(1)}\] where \(B_n(\lambda_1,\lambda_2):=D_n^\star(\lambda_1,\lambda_2)/(\lambda_2h_{n-1})\) for \(n\in\mathbb{Z}_{\ge1}\) and \(B_0(\lambda_1,\lambda_2):=1/(\lambda_1-\lambda_2)\). Moreover, for \(k=1\) and \(n\in\mathbb{Z}_{\ge0}\), \[C_1(n;\lambda_1)=\lim_{\lambda\to\lambda_1}\left(B_n(\lambda,\lambda_1)-\frac{1}{\lambda-\lambda_1}\right)-\frac{n}{\lambda_1}.\label{maineq1}\qquad{(2)}\]
The above formula ?? in Theorem 1 was essentially known (perhaps in different generalities) from [13], [21], [25], [27] (see the discussions above). We will give a new proof of
Theorem 1 in Section 3 using 3 , 4 . Our proof may shed some lights on the relationship between the probability
theory of determinantal point processes (cf. [31]) and the theory of integrable hierarchies.
Lemma 1. For \(n\in\mathbb{Z}_{\ge0}\), we have the expansion \[\frac{\lambda_2^n}{\lambda_1^n}B_n(\lambda_1,\lambda_2)=\frac{1}{\lambda_1-\lambda_2}+\sum_{k,l\ge0}\frac{A_{k,l}(n)}{\lambda_1^{l+1}\lambda_2^{k+1}}.\label{affine}\qquad{(3)}\]
Building on Theorem 1, Lemma 1 and Zhou’s theorem [15] (cf. also [32], [33]), we will give a new proof of the following result.
Corollary 1 ([25], [34]–[36]). The
partition function of the hermitian matrix model associated to \(\mathop{}\!\mathrm{d}\mu\) is a KP tau-function.
The \(A_{k,l}(n)\) appeared in ?? are the affine coordinates of the KP tau-function associated to \(\mathop{}\!\mathrm{d}\mu\). Let \(\{m_k\}_{k\ge0}\) be
the moments of \(\mathop{}\!\mathrm{d}\mu\). We introduce the notation \[H_{(i_1,\cdots,i_n;j_1,\cdots,j_n)}:=(m_{i_k+j_l})_{k,l=1}^n.\] By expanding formula 10 , we
will give a new proof of the following
Theorem 2 ([36] (cf. [35])). For
\(j<n\), the affine coordinates \(A_{i,j}(n)\) read \[\label{hk} A_{i,j}(n)=\frac{(-1)^j}{\Delta_{n-1}}\det
H_{(0,1,\cdots,\widehat{n-j-1},\cdots,n-1,n+i;0,1,\cdots,n-1)},\qquad{(4)}\] where \(\Delta_{n-1}:=\det H_{(0,1,\cdots,n-1;0,1,\cdots,n-1)}\).
From the perspective of the KP hierarchy, following [37], [38], for two hermitian matrix models \(M\) and \(\widetilde{M}\), we say that \(\widetilde{M}\) is a KP dual of \(M\) if for any Schur polynomial \(s_\rho\) in \(n\) variables associated to the partition \(\rho\), \[\widetilde{\langle s_\rho\rangle}(n)=(-1)^{|\rho|}\langle
s_{\rho'}\rangle(-n),\qquad |n|\ge\max\{2,|\rho|\},\] where \(\rho'\) is the conjugate partition of \(\rho\). Obviously, \(M\) is also a KP
dual of \(\widetilde{M}\). The following theorem gives a class of hermitian matrix models KP dual to each other.
Theorem 3. For \(\beta\in\mathbb{R}\) and \(\gamma>0\), consider hermitian matrix models \(M,\widetilde{M}\) associated
respectively to \[\mathop{}\!\mathrm{d}\mu(\lambda)=\frac{1}{2\pi\gamma}\sqrt{4\gamma-(\lambda-\beta)^2}\mathbf{1}_I(\lambda)\mathop{}\!\mathrm{d}\lambda,\qquad
\mathop{}\!\mathrm{d}\widetilde{\mu}(\lambda)=\frac{1}{\pi}\frac{1}{\sqrt{4\gamma-(\lambda-\beta)^2}}\mathbf{1}_I(\lambda)\mathop{}\!\mathrm{d}\lambda,\] where \(I:=(\beta-2\sqrt{\gamma},\beta+2\sqrt{\gamma})\). Then
\(\widetilde{M}\) is a KP dual of \(M\).
We will give some preliminaries in Section 2, and prove Theorem 1 in Section 3. In Section 4.1, we
will give the proof of Corollary 1 and Theorem 2. We will also describe the polynomiality of the connected
correlators in terms of coefficients of orthogonal polynomials in Section 4.2. We will prove Theorem 3 in Section 5.
D. Yang would like to thank Giulio Ruzza and Marco Bertola for very helpful discussions. We also thank Don Zagier for helpful discussions. D. Yang and J. Zhao are partially supported by NSFC No. 12371254 and CAS YSBR-032. J. Zhou is partly supported by
NSFC No. 11890662.
For \(n\ge0\), we denote the expected value of a function \(f:\mathcal{H}_n(E)\to\mathbb{R}\) by [10], [39]\[\left\langle f(M)\right\rangle(n):=\frac{1}{Z_n}\int_{\mathcal{H}_n(E)}f(M)\mathop{}\!\mathrm{d}\nu_n(M),\] where \(\mathop{}\!\mathrm{d}\nu_n\) is the unitarily invariant measure on \(\mathcal{H}_n(E)\) defined via the spectral decomposition \(M=U\mathrm{diag}(\lambda_1,\cdots,\lambda_n)U^*\) with respect to the measure \((\mathop{}\!\mathrm{d}\mu)^n\) over \(E^n\) and the Haar measure over \(U(n)\), and \[Z_n:=\int_{\mathcal{H}_n(E)}\mathop{}\!\mathrm{d}\nu_n(M).\] Here \(Z_0\equiv1\). For \(n\ge0\), denote the
disconnected \(k\)-point correlators by \(\langle\mathop{\mathrm{tr}}M^{i_1}\cdots\mathop{\mathrm{tr}}M^{i_k}\rangle(n)\), and denote the connected \(k\)-point correlators \(\langle\mathop{\mathrm{tr}}M^{i_1}\cdots\mathop{\mathrm{tr}}M^{i_k}\rangle_c(n)\) by the Möbius relation [10], [15], [40]: \[\begin{align}
\langle\mathop{\mathrm{tr}}M^{i_1}\cdots\mathop{\mathrm{tr}}M^{i_k}\rangle(n)&=\sum_{\text{partition }\mathcal{P}\text{ of }\mathbf{Z}_k}\prod_{I\in\mathcal{P}}\left\langle\prod_{j\in I}\mathop{\mathrm{tr}}M^{i_j}\right\rangle_c(n),\\
\langle\mathop{\mathrm{tr}}M^{i_1}\cdots\mathop{\mathrm{tr}}M^{i_k}\rangle_c(n)&=\sum_{\text{partition }\mathcal{P}\text{ of }\mathbf{Z}_k}(-1)^{|\mathcal{P}|-1}(|\mathcal{P}|-1)!\prod_{I\in\mathcal{P}}\left\langle\prod_{j\in
I}\mathop{\mathrm{tr}}M^{i_j}\right\rangle(n),
\end{align}\] where \(\mathbf{Z}_k:=\{1,\cdots,k\}\).
We denote for \(\lambda_1,\cdots,\lambda_k\in\mathbb{C}\backslash\mathbb{R}\) and \(n\ge0\), \[\begin{align}
\mathcal{C}_k(n;\lambda_1,\cdots,\lambda_k):=&\sum_{\text{partition }\mathcal{P}\text{ of }\mathbf{Z}_k}(-1)^{|\mathcal{P}|-1}(|\mathcal{P}|-1)!\prod_{I\in\mathcal{P}}\left\langle\prod_{j\in
I}\mathop{\mathrm{tr}}\frac{1}{\lambda_j-M}\right\rangle(n)-\frac{n\delta_{k,1}}{\lambda_1}.
\end{align}\] It is analytic on \((\mathbb{C}\backslash\mathbb{R})^k\). For \(n\ge0\), as \(\lambda_1,\cdots,\lambda_k\to\infty\) within any sector in
\((\mathbb{C}\backslash\mathbb{R})^k\), we have \[\label{asym-func} \mathcal{C}_k(n;\lambda_1,\cdots,\lambda_k)\sim
C_k(n;\lambda_1,\cdots,\lambda_k).\tag{11}\] We note that for the case of \(\mathbb{CP}^1\), the analytic \(k\)-point functions were first discussed in [41].
It is known [27] that for \(n\ge0\), as \(\lambda\to\infty\) within any sector in \(\mathbb{C}\backslash\mathbb{R}\), \[\label{poincare} \psi_B^\star(\lambda,n)=\frac{1}{\lambda^n}\sum_{j=0}^\infty
\frac{1}{\lambda^j}\int_\mathbb{R}y^{j+n}\pi_n(y)\mathop{}\!\mathrm{d}\mu(y).\tag{12}\] In addition, it is known in [27], [39] that for any \(n\ge1\) and \(\lambda\in\mathbb{C}\backslash\mathbb{R}\), \[\pi_n(\lambda)\widehat\pi_{n-1}(\lambda)-\pi_{n-1}(\lambda)\widehat\pi_n(\lambda)=-h_{n-1}.\label{reduced-cd}\tag{13}\] Denote for \(\lambda_1,\lambda_2\in\mathbb{C}\backslash\mathbb{R}\), \[\begin{align}
\mathcal{J}_n(\lambda_1,\lambda_2):=&-\frac{1}{h_{n-1}}\frac{\pi_n(\lambda_1)\widehat\pi_{n-1}(\lambda_2)-\pi_{n-1}(\lambda_1)\widehat\pi_n(\lambda_2)}{\lambda_1-\lambda_2},\qquad n\ge1, \label{jn}\\
\mathcal{J}_0(\lambda_1,\lambda_2):=&\frac{1}{\lambda_1-\lambda_2},\\ \mathcal{J}^\circ_n(\lambda_1,\lambda_2):=&\mathcal{J}_n(\lambda_1,\lambda_2)-\frac{1}{\lambda_1-\lambda_2},\qquad n\ge0.
\end{align}\tag{14}\] It is known that [27], [39]\[\begin{align} \mathcal{J}_n^\circ(\lambda_1,\lambda_2)=-\sum_{l=0}^{n-1}\frac{\pi_l(\lambda_1)\widehat\pi_l(\lambda_2)}{h_l}=-\int_\mathbb{R}\frac{K_n(\lambda_1,\lambda)}{\lambda-\lambda_2}\mathop{}\!\mathrm{d}\mu(\lambda),\qquad
n\ge0. \label{trans-cd}
\end{align}\tag{15}\] Obviously, \(\mathcal{J}_n^\circ(\lambda_1,\lambda_2)\) is analytic on \((\mathbb{C}\backslash\mathbb{R})^2\). Similar to [41], we prove the following
Lemma 2. For \(k\ge2\) and \(n\ge0\), the function \[\label{hkk} \frac{(-1)^{k-1}}{k}\sum_{\sigma\in
S_k}\prod_{i=1}^k\mathcal{J}_n(\lambda_{\sigma(i)},\lambda_{\sigma(i+1)})-\frac{\delta_{k,2}}{(\lambda_1-\lambda_2)^2}\qquad{(5)}\] is analytic along \(\lambda_i=\lambda_j,i\neq j\). Here \(\lambda_1,\cdots,\lambda_k\in\mathbb{C}\backslash\mathbb{R}\).
Proof. Without loss of generality, we show that ?? is analytic along \(\lambda_{k-1}=\lambda_k\). For \(k\ge3\), we have \[\begin{align}
&-\frac{1}{k}\sum_{\sigma\in S_k}\prod_{i=1}^k\mathcal{J}_n(\lambda_{\sigma(i)},\lambda_{\sigma(i+1)})\nonumber\\ =&-\sum_{\sigma\in
S_{k-2}}\left(\prod_{i=1}^{k-3}\mathcal{J}_n(\lambda_{\sigma(i)},\lambda_{\sigma(i+1)})\right)\frac{\mathcal{J}_n(\lambda_{\sigma(k-2)},\lambda_{k-1})\mathcal{J}_n(\lambda_k,\lambda_{\sigma(1)})-\mathcal{J}_n(\lambda_{\sigma(k-2)},\lambda_k)\mathcal{J}_n(\lambda_{k-1},\lambda_{\sigma(1)})}{\lambda_{k-1}-\lambda_k}\nonumber\\
&-\sum_{\sigma\in
S_{k-2}}\left(\prod_{i=1}^{k-3}\mathcal{J}_n(\lambda_{\sigma(i)},\lambda_{\sigma(i+1)})\right)\bigg(\mathcal{J}_n(\lambda_{\sigma(k-2)},\lambda_{k-1})\mathcal{J}_n^\circ(\lambda_{k-1},\lambda_k)\mathcal{J}_n(\lambda_k,\lambda_{\sigma(1)})\nonumber\\
&\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad+\mathcal{J}_n(\lambda_{\sigma(k-2)},\lambda_k)\mathcal{J}_n^\circ(\lambda_k,\lambda_{k-1})\mathcal{J}_n(\lambda_{k-1},\lambda_{\sigma(1)})\bigg)\nonumber\\ &-\sum_{\substack{\sigma\in
S_k\\\sigma(k)=k,\sigma(k-1)\neq k-1,\sigma(1)\neq k-1}}\prod_{i=1}^k\mathcal{J}_n(\lambda_{\sigma(i)},\lambda_{\sigma(i+1)}),
\end{align}\] which is analytic along \(\lambda_{k-1}=\lambda_k\).
For \(k=2\), ?? reads \[\begin{align}
&-\mathcal{J}_n^\circ(\lambda_1,\lambda_2)\mathcal{J}_n^\circ(\lambda_2,\lambda_1)-\frac{\mathcal{J}_n^\circ(\lambda_1,\lambda_2)-\mathcal{J}_n^\circ(\lambda_2,\lambda_1)}{\lambda_1-\lambda_2},
\end{align}\] which is analytic along \(\lambda_1=\lambda_2\). The lemma is proved. ◻
Remark 1. By the three-term recurrence relations of \(\pi_n\), we have \[L(\psi_A^\star(\lambda,n))=\lambda\psi_A^\star(\lambda,n),\qquad\psi_A^\star(\lambda,n)=(1+O(\lambda^{-1}))\lambda^n,\qquad n\in\mathbb{Z}_{\ge0}.\] By 12 for \(n\in\mathbb{Z}_{\ge0}\), \[\psi_B^\star(\lambda,n)=(1+O(\lambda^{-1}))h_n\lambda^{-n}.\] Also for \(n\in\mathbb{Z}_{\ge1}\), \[\begin{align} L(\psi_B^\star(\lambda,n))&=\psi_B^\star(\lambda,n+1)+\beta_n\psi_B^\star(\lambda,n)+\gamma_n\psi_B^\star(\lambda,n-1)\nonumber\\
&\sim-\lambda\int_\mathbb{R}\frac{1}{x-\lambda}(\pi_{n+1}(x)+\beta_n\pi_n(x)+\gamma_n\pi_{n-1}(x))\mathop{}\!\mathrm{d}\mu(x)\nonumber\\ &=-\lambda\int_\mathbb{R}\frac{x\pi_n(x)}{x-\lambda}\mathop{}\!\mathrm{d}\mu(x)\nonumber\\
&=-\lambda(\lambda\widehat\pi_n(\lambda)+h_0\delta_{n,0})\nonumber\\ &\sim\lambda\psi_B^\star(\lambda,n).
\end{align}\] In addition, from 13 for \(n\in\mathbb{Z}_{\ge1}\), \[\begin{align}
\psi_A^\star(\lambda,n)\mathcal{T}^{-1}\psi_B^\star(\lambda,n)-\psi_B^\star(\lambda,n)\mathcal{T}^{-1}\psi_A^\star(\lambda,n)\sim&-\pi_n(\lambda)\lambda\widehat\pi_{n-1}(\lambda)+\lambda\widehat\pi_n(\lambda)\pi_{n-1}(\lambda)\nonumber\\ =&\lambda
h_{n-1}. \label{wronskian}
\end{align}\tag{16}\] Thus for \(n\in\mathbb{Z}_{\ge0}\), \(\psi_A^\star(\lambda,n),\psi_B^\star(\lambda,n)\) satisfy the condition of forming a pair of wave functions.
By 11 and Lemma 2, we only need to prove for pairwise distinct \(\lambda_1,\cdots,\lambda_k\in\mathbb{C}\backslash\mathbb{R}\), \[\begin{align} \mathcal{C}_1(n;\lambda_1)=&\mathcal{J}_n^\circ(\lambda_1,\lambda_1)-\frac{n}{\lambda_1},
\tag{17}\\ \mathcal{C}_k(n;\lambda_1,\cdots,\lambda_k)=&\frac{(-1)^{k-1}}{k}\sum_{\sigma\in S_k}\prod_{i=1}^k\mathcal{J}_n(\lambda_{\sigma(i)},\lambda_{\sigma(i+1)})-\frac{\delta_{k,2}}{(\lambda_1-\lambda_2)^2},\qquad k\ge2, \tag{18}
\end{align}\] (cf. [29], [41]–[45]). The case \(n=0\) is direct. We always let \(n\ge1\) throughout this section. We
first prove a lemma.
Lemma 3. For any \(l\ge k\ge 0\), \[\begin{align} \widehat{\pi_k\pi_l}(\lambda)=\pi_k(\lambda)\widehat\pi_l(\lambda).\label{ax1}
\end{align}\qquad{(6)}\]
Proof.\(k=0\) is direct. For \(k\ge1\), we have \[\begin{align}
\widehat{\pi_k\pi_l}(\lambda)-\pi_k(\lambda)\widehat\pi_l(\lambda)=&\int_\mathbb{R}\frac{\pi_k(y)-\pi_k(\lambda)}{y-\lambda}\pi_l(y)\mathop{}\!\mathrm{d}\mu(y)=0.
\end{align}\] The last equation follows from the fact that \(\frac{\pi_k(y)-\pi_k(\lambda)}{y-\lambda}\) is a polynomial in \(y\) of degree \(k-1<l\). Thus the lemma is proved. ◻
The key method we use in the proof of Theorem 1 will be based on the four fundamental properties 3 , 15 , 13 and ?? of orthogonal polynomials.
Proof of Theorem 1. We first prove 17 . Using ?? , \[\begin{align}
\mathcal{C}_1(n;\lambda_1)=&\int_{\mathbb{R}^n}\sum_{i=1}^n\frac{1}{\lambda_1-y_i}\rho_n(n;y_1,\cdots,y_n)\mathop{}\!\mathrm{d}\mu(y_1)\cdots\mathop{}\!\mathrm{d}\mu(y_n)-\frac{n}{\lambda_1}\nonumber\\
=&\sum_{i=1}^n\int_{\mathbb{R}}\frac{1}{\lambda_1-y_i}\rho_1(n;y_i)\mathop{}\!\mathrm{d}\mu(y_i)-\frac{n}{\lambda_1}\nonumber\\
=&\int_{\mathbb{R}}\frac{1}{\lambda_1-y}\sum_{l=0}^{n-1}\frac{\pi_l(y)^2}{h_l}\mathop{}\!\mathrm{d}\mu(y)-\frac{n}{\lambda_1}\nonumber\\ =&-\sum_{l=0}^{n-1}\frac{\pi_l(\lambda_1)\widehat\pi_l(\lambda_1)}{h_l}-\frac{n}{\lambda_1}\nonumber\\
=&\mathcal{J}_n^\circ(\lambda_1,\lambda_1)-\frac{n}{\lambda_1}. \label{pf1}
\end{align}\tag{19}\]
Now we prove 18 . For a partition \(\mathcal{P}\) of \(\mathbf{Z}_k\), we write \(\mathcal{P}=\{\mathcal{P}_1,\cdots,\mathcal{P}_{|\mathcal{P}|}\}\). The following formula was given in [45]: for \(k\ge1\), \[\label{ckbar} \left\langle\prod_{j=1}^k\mathop{\mathrm{tr}}\frac{1}{\lambda_j-M}\right\rangle(n)=\sum_{\text{partition }\mathcal{P}\text{ of
}\mathbf{Z}_k}\int_{\mathbb{R}^l}\prod_{i=1}^{|\mathcal{P}|}\left(\prod_{j \in \mathcal{P}_i} \frac{1}{\lambda_{j}-y_{i}}\right)\det(K_{n}(y_{i},y_{j}))_{i,j=1}^{l}
\mathop{}\!\mathrm{d}\mu(y_{1})\cdots\mathop{}\!\mathrm{d}\mu(y_{|\mathcal{P}|}).\tag{20}\] To prove 20 , we consider the following two cases. For the case when \(n\ge k\), \[\begin{align}
\left\langle\prod_{j=1}^k\mathop{\mathrm{tr}}\frac{1}{\lambda_j-M}\right\rangle(n)=&\int_{\mathbb{R}^{n}}\prod_{j=1}^{k}\left(\sum_{i=1}^{n}\frac{1}{\lambda_{j}-y_{i}}\right)\rho_n(n;y_1,\cdots,y_n)\mathop{}\!\mathrm{d}\mu(y_{1})\cdots\mathop{}\!\mathrm{d}\mu(y_{n})
\nonumber\\ =&\sum_{\text{partition }\mathcal{P}\text{ of }\mathbf{Z}_k}\int_{\mathbb{R}^l}\prod_{i=1}^{|\mathcal{P}|}\left(\prod_{j \in \mathcal{P}_i} \frac{1}{\lambda_{j}-y_{i}}\right)\det(K_{n}(y_{i},y_{j}))_{i,j=1}^{l}
\mathop{}\!\mathrm{d}\mu(y_{1})\cdots\mathop{}\!\mathrm{d}\mu(y_{|\mathcal{P}|}).
\end{align}\] For the case when \(n<k\), \[\begin{align} &\left\langle\prod_{j=1}^k\mathop{\mathrm{tr}}\frac{1}{\lambda_j-M}\right\rangle(n)\nonumber\\
=&\sum_{\substack{\text{partition }\mathcal{P}\text{ of }\mathbf{Z}_k\\|\mathcal{P}|\le n}}\int_{\mathbb{R}^l}\prod_{i=1}^{|\mathcal{P}|}\left(\prod_{j \in \mathcal{P}_i} \frac{1}{\lambda_{j}-y_{i}}\right)\det(K_{n}(y_{i},y_{j}))_{i,j=1}^{l}
\mathop{}\!\mathrm{d}\mu(y_{1})\cdots\mathop{}\!\mathrm{d}\mu(y_{|\mathcal{P}|})\nonumber\\ =&\left(\sum_{\text{partition }\mathcal{P}\text{ of }\mathbf{Z}_k}-\sum_{\substack{\text{partition }\mathcal{P}\text{ of
}\mathbf{Z}_k\\|\mathcal{P}|>n}}\right)\int_{\mathbb{R}^l}\prod_{i=1}^{|\mathcal{P}|}\left(\prod_{j \in \mathcal{P}_i} \frac{1}{\lambda_{j}-y_{i}}\right)\det(K_{n}(y_{i},y_{j}))_{i,j=1}^{|\mathcal{P}|}
\mathop{}\!\mathrm{d}\mu(y_{1})\cdots\mathop{}\!\mathrm{d}\mu(y_{|\mathcal{P}|}).
\end{align}\] Note that for any \(m>n\), the matrix \[\begin{align} (K_n(y_i,y_j))_{i,j=1}^m=\left(\frac{\pi_l(y_i)}{\sqrt{h_l}}\right)_{i=1,\cdots,m;l=0,\cdots,n-1}
\left(\frac{\pi_l(y_j)}{\sqrt{h_l}}\right)^T_{j=1,\cdots,m;l=0,\cdots,n-1}
\end{align}\] has rank \(\le n<m\). Thus \(\det(K_n(y_i,y_j))_{i,j=1}^m=0\). Therefore we obtain 20 .
Now we claim that for \(k\ge1\), \[\mathcal{C}_{k}(n;\lambda_{1},\cdots,\lambda_{k})=\sum_{\text{partition }\Pi\text{ of
}\mathbf{Z}_k}C_{\mathbf{Z}_k,\Pi}-\frac{n\delta_{k,1}}{\lambda_1}, \label{ck39}\tag{21}\] where for \(I\subset\mathbf{Z}_k\) and a partition \(\mathcal{P}=\{P_1,\cdots,P_{|\mathcal{P}|}\}\) of \(I\), we denote \[C_{I,\mathcal{P}}:=\int_{\mathbb{R}^{|\mathcal{P}|}}\prod_{i=1}^{|\mathcal{P}|}\left(\frac{\mathop{}\!\mathrm{d}\mu(y_i)}{\prod_{j\in P_i}(\lambda_j-y_i)}\right)\frac{(-1)^{|\mathcal{P}|-1}}{|\mathcal{P}|}\sum_{\sigma\in
S_{|\mathcal{P}|}}\prod_{i=1}^{|\mathcal{P}|}K_{n}(y_{\sigma(i)},y_{\sigma(i+1)}).\] The case \(k=1\) is straightforward by definition. We prove by induction on \(k\). Assume that 21 holds for all \(m<k\). For \(I\subset\mathbf{Z}_k\) and a partition \(\mathcal{P}\) of \(\mathbf{Z}_k\), we write \(\mathcal{P}(I)\) for the partition of \(I\) induced by \(\mathcal{P}\). For two paritions \(\Pi,\mathcal{P}\) of \(\mathbf{Z}_k\), we write \(\Pi\prec\mathcal{P}\) if \(\Pi\) is finer than \(\mathcal{P}\). Then for case \(k\), \[\begin{align} &\mathcal{C}_{k}(n;\lambda_{1},\cdots,\lambda_{k})\nonumber\\
=&\left\langle\prod_{j=1}^k\mathop{\mathrm{tr}}\frac{1}{\lambda_j-M}\right\rangle(n)-\sum_{\substack{\text{partition }\mathcal{P}\text{ of
}\mathbf{Z}_k\\|\mathcal{P}|>1}}\prod_{\substack{I\in\mathcal{P}\\I:=\{i_1,\cdots,i_{|I|}\}}}\left(\mathcal{C}_{|I|}(n;\lambda_{i_1},\cdots,\lambda_{i_{|I|}})+\frac{n\delta_{|I|,1}}{\lambda_{i_1}}\right) \nonumber\\ =&\sum_{\text{partition
}\Pi\text{ of }\mathbf{Z}_k}\int_{\mathbb{R}^l}\prod_{i=1}^{|\mathcal{P}|}\left(\prod_{j \in \mathcal{P}_i} \frac{1}{\lambda_{j}-y_{i}}\right)\det(K_{n}(y_{i},y_{j}))_{i,j=1}^{l}
\mathop{}\!\mathrm{d}\mu(y_{1})\cdots\mathop{}\!\mathrm{d}\mu(y_{|\mathcal{P}|})\nonumber\\ &-\sum_{\substack{\text{partition }\mathcal{P}\text{ of
}\mathbf{Z}_k\\|\mathcal{P}|>1}}\prod_{\substack{I\in\mathcal{P}\\I:=\{i_1,\cdots,i_{|I|}\}}}\left(\mathcal{C}_{|I|}(n;\lambda_{i_1},\cdots,\lambda_{i_{|I|}})+\frac{n\delta_{|I|,1}}{\lambda_{i_1}}\right)\nonumber\\ =&\sum_{\text{partition }\Pi\text{
of }\mathbf{Z}_k}\sum_{\substack{\text{partition }\mathcal{P}\text{ of }\mathbf{Z}_k \\ \Pi\prec\mathcal{P}}}\prod_{I\in\mathcal{P}}C_{I,\Pi(I)}-\sum_{\substack{\text{partition }\mathcal{P}\text{ of }\mathbf{Z}_k \\
|\mathcal{P}|>1}}\prod_{I\in\mathcal{P}}\sum_{\text{partition }\Pi(I)\text{ of }I}C_{I,\Pi(I)} \nonumber\\ =&\sum_{\text{partition }\Pi\text{ of }\mathbf{Z}_k}C_{\mathbf{Z}_k,\Pi}. \label{n62k}
\end{align}\tag{22}\]
Now by 21 , \[\begin{align} &\mathcal{C}_{k}(n;\lambda_{1},\cdots,\lambda_{k})\nonumber\\ =&\sum_{\text{partition }\Pi\text{ of }\mathbf{Z}_k}\int_{\mathbb{R}^{|\Pi|}}
\prod_{i=1}^{|\Pi|}\left(\frac{\mathop{}\!\mathrm{d}\mu(y_i)}{\prod_{j\in\Pi_i}(\lambda_j-y_i)}\right)\frac{(-1)^{|\Pi|-1}}{|\Pi|}\sum_{\sigma\in S_{|\Pi|}}\prod_{i=1}^{|\Pi|}K_n(y_{\sigma(i)},y_{\sigma(i+1)}) \nonumber\\ =&-\sum_{\text{partition
}\Pi\text{ of }\mathbf{Z}_k}\sum_{(a_1,\cdots,a_{|\Pi|})\in\Pi_1\times\cdots\times\Pi_{|\Pi|}} \left(\prod_{i=1}^{|\Pi|} Q_{a_i,\Pi_i\backslash\{a_i\}}(\lambda_1,\cdots,\lambda_k)\right)\frac{1}{|\Pi|}\sum_{\sigma\in
S_{|\Pi|}}\ell_{|\Pi|}(n;\lambda_{a_{\sigma(1)}},\cdots,\lambda_{a_{\sigma(|\Pi|)}}),
\end{align}\] where we denote the following rational functions in \(\lambda_1,\cdots,\lambda_k\) for \(I\subset \mathbf{Z}_k\) and \(a\in
\mathbf{Z}_k\backslash I\): \[Q_{a,I}(\lambda_1,\cdots,\lambda_k):=\prod_{b\in I}\frac{1}{\lambda_b-\lambda_a},\] and denote the cyclic integrals \[\begin{align} \ell_0:=&1,\\
\ell_k(n;\lambda_1,\cdots,\lambda_k):=&\int_{\mathbb{R}^k}\left(\prod_{i=1}^{k}\frac{K_n(y_i,y_{i+1})}{y_i-\lambda_i}\mathop{}\!\mathrm{d}\mu(y_i)\right),\;k\ge1,
\end{align}\] where \(y_{k+1}:=y_1\). To compute \(\ell_k\), we first define \[\begin{align} E_1(n;\lambda_1,\lambda_2):=&K_n(\lambda_1,\lambda_2),\\
E_k(n;\lambda_1,\cdots,\lambda_{k+1}):=&\int_{\mathbb{R}^{k-1}}K_n(\lambda_1,y_2)\left(\prod_{i=2}^{k-1}\frac{K_n(y_i,y_{i+1})}{y_i-\lambda_i}\mathop{}\!\mathrm{d}\mu(y_i)\right)\frac{K_n(y_k,\lambda_{k+1})}{y_k-\lambda_k}\mathop{}\!\mathrm{d}y_k,\qquad
k\ge2.
\end{align}\] Then \[\begin{align} E_k(n;\lambda_1,\cdots,\lambda_{k+1})=&\int_\mathbb{R}\frac{1}{y_k-\lambda_k}E_{k-1}(n;\lambda_1,\cdots,\lambda_{k-1},y_k)K_n(y_k,\lambda_{k+1})\mathop{}\!\mathrm{d}\mu(y_k),\\
\ell_k(n;\lambda_1,\cdots,\lambda_k)=&\int_\mathbb{R}\frac{1}{y-\lambda_1}E_k(n;y,\lambda_2,\cdots,\lambda_k,y)\mathop{}\!\mathrm{d}\mu(y).
\end{align}\]
Lemma 4. For \(k\ge2\), \[\label{ek}
E_k(n;\lambda_1,\cdots,\lambda_{k+1})=-E_{k-1}(n;\lambda_1,\cdots,\lambda_k)\mathcal{J}_n(\lambda_{k+1},\lambda_k)-\frac{1}{\lambda_k-\lambda_{k+1}}E_{k-1}(n;\lambda_1,\cdots,\lambda_{k-1},\lambda_{k+1}).\qquad{(7)}\]
Proof. For \(k=2\), using 3 , 15 , 13 and ?? , \[\begin{align}
&E_2(n;\lambda_1,\lambda_2,\lambda_3)+K_n(\lambda_1,\lambda_2)\mathcal{J}_n^\circ(\lambda_3,\lambda_2)\nonumber\\
=&\sum_{l,m=0}^{n-1}\frac{1}{h_lh_m}(\pi_l(\lambda_1)\widehat{\pi_l\pi_m}(\lambda_2)\pi_m(\lambda_3)-\pi_l(\lambda_1)\pi_l(\lambda_2)\widehat\pi_m(\lambda_2)\pi_m(\lambda_3))\nonumber\\
=&\sum_{l>m}\frac{1}{h_lh_m}\pi_l(\lambda_1)\pi_m(\lambda_3)(\widehat\pi_l(\lambda_2)\pi_m(\lambda_2)-\pi_l(\lambda_2)\widehat\pi_m(\lambda_2))\nonumber\\
=&\sum_{l=1}^{n-1}\frac{\pi_l(\lambda_1)\widehat\pi_l(\lambda_2)}{h_lh_{l-1}}\frac{\pi_l(\lambda_2)\pi_{l-1}(\lambda_3)-\pi_{l-1}(\lambda_2)\pi_l(\lambda_3)}{\lambda_2-\lambda_3}\nonumber\\
&-\sum_{l=1}^{n-1}\frac{\pi_l(\lambda_1)\pi_l(\lambda_2)}{h_l}\left(\frac{1}{h_{l-1}}\frac{\pi_l(\lambda_3)\widehat\pi_{l-1}(\lambda_2)-\pi_{l-1}(\lambda_3)\widehat\pi_l(\lambda_2)}{\lambda_3-\lambda_2} -\frac{1}{\lambda_2-\lambda_3}\right) \nonumber\\
=&\sum_{l=1}^{n-1}\frac{\pi_l(\lambda_1)\pi_l(\lambda_3)}{(\lambda_2-\lambda_3)h_l}\frac{1}{h_{l-1}}(-\widehat\pi_l(\lambda_2)\pi_{l-1}(\lambda_2)+\widehat\pi_{l-1}(\lambda_2)\pi_l(\lambda_2))+\sum_{l=1}^{n-1}\frac{\pi_l(\lambda_1)\pi_l(\lambda_2)}{(\lambda_2-\lambda_3)h_l}
\nonumber\\ =&\frac{1}{\lambda_3-\lambda_2}(K_n(\lambda_1,\lambda_3)-K_n(\lambda_1,\lambda_2)). \label{method}
\end{align}\tag{23}\] Similarly, \[\label{dk}
\int_\mathbb{R}\frac{1}{y_2-\lambda_2}\mathcal{J}_n(y_2,\lambda_1)K_n(y_2,\lambda_3)\mathop{}\!\mathrm{d}\mu(y_2)=-\mathcal{J}_n(\lambda_2,\lambda_1)\mathcal{J}_n(\lambda_3,\lambda_2)-\left(\frac{1}{\lambda_1-\lambda_2}+\frac{1}{\lambda_2-\lambda_3}\right)\mathcal{J}_n(\lambda_3,\lambda_1).\tag{24}\]
We prove ?? by induction on \(k\). From now on, we will omit the parameter \(n\) in \(E_k\). Assume ?? holds for \(k\). Now
for \(k+1\), using 24 , \[\begin{align} &E_{k+1}(\lambda_1,\cdots,\lambda_{k+2})\nonumber\\
=&E_{k-1}(\lambda_1,\cdots,\lambda_k)\left(\mathcal{J}_n(\lambda_{k+1},\lambda_k)\mathcal{J}_n(\lambda_{k+2},\lambda_{k+1})+\left(\frac{1}{\lambda_k-\lambda_{k+1}}+\frac{1}{\lambda_{k+1}-\lambda_{k+2}}\right)\mathcal{J}_n(\lambda_{k+2},\lambda_k)\right)\nonumber\\
&+\frac{1}{\lambda_k-\lambda_{k+1}}(E_k(\lambda_1,\cdots,\lambda_k,\lambda_{k+2})-E_k(\lambda_1,\cdots,\lambda_{k-1},\lambda_{k+1},\lambda_{k+2}))\nonumber\\
=&-\left(E_k(\lambda_1,\cdots,\lambda_{k+1})+\frac{1}{\lambda_k-\lambda_{k+1}}E_{k-1}(\lambda_1,\cdots,\lambda_{k-1},\lambda_{k+1})\right)\mathcal{J}_n(\lambda_{k+2},\lambda_{k+1})\nonumber\\
&+\frac{1}{\lambda_k-\lambda_{k+1}}E_{k-1}(\lambda_1,\cdots,\lambda_k)\mathcal{J}_n(\lambda_{k+2},\lambda_k)\nonumber\\
&-\frac{1}{\lambda_{k+1}-\lambda_{k+2}}\left(E_k(\lambda_1,\cdots,\lambda_k,\lambda_{k+2})+\frac{1}{\lambda_k-\lambda_{k+2}}E_{k-1}(\lambda_1,\cdots,\lambda_{k-1},\lambda_{k+2})\right)\nonumber\\
&-\frac{1}{\lambda_k-\lambda_{k+1}}\left(E_{k-1}(\lambda_1,\cdots,\lambda_k)\mathcal{J}_n(\lambda_{k+2},\lambda_k)+\frac{1}{\lambda_k-\lambda_{k+2}}E_{k-1}(\lambda_1,\cdots,\lambda_{k-1},\lambda_{k+2})\right)\nonumber\\
&+\frac{1}{\lambda_k-\lambda_{k+1}}\left(E_{k-1}(\lambda_1,\cdots,\lambda_{k-1},\lambda_{k+1})\mathcal{J}_n(\lambda_{k+2},\lambda_{k+1})+\frac{1}{\lambda_{k+1}-\lambda_{k+2}}E_{k-1}(\lambda_1,\cdots,\lambda_{k-1},\lambda_{k+2})\right)\nonumber\\
=&-E_k(\lambda_1,\cdots,\lambda_{k+1})\mathcal{J}_n(\lambda_{k+2},\lambda_{k+1})-\frac{1}{\lambda_{k+1}-\lambda_{k+2}}E_k(\lambda_1,\cdots,\lambda_k,\lambda_{k+2}).
\end{align}\] The lemma is proved. ◻
Having established a recurrence for \(E_k\), we now turn to the cyclic integral \(\ell_k\). To solve this recursively, we introduce an auxiliary double sequence of functions, \(\ell_{k,(m)}\), defined as follows for \(k\ge2\) and \(1\le m\le k-2\): \[\begin{align}
\ell_{1,(1)}(n;\lambda_1):=&\ell_1(n;\lambda_1)=-\mathcal{J}_n^\circ(\lambda_1,\lambda_1),\\
\ell_{k,(1)}(n;\lambda_1,\cdots,\lambda_k):=&\ell_k(n;\lambda_1,\cdots,\lambda_k)+\frac{1}{\lambda_k-\lambda_1}(\ell_{k-1}(n;\lambda_1,\cdots,\lambda_{k-1})-\ell_{k-1}(n;\lambda_2,\cdots,\lambda_k)),\\
\ell_{k,(m+1)}(n;\lambda_1,\cdots,\lambda_k):=&\ell_{k,(m)}(n;\lambda_1,\cdots,\lambda_k)\nonumber\\ &+\frac{1}{\lambda_{k-m}-\lambda_{k-m+1}}\ell_{k-1,(m)}(n;\lambda_1,\cdots,\lambda_{k-m-1},\lambda_{k-m+1},\cdots,\lambda_k),\\
\ell_{k,(k)}(n;\lambda_1,\cdots,\lambda_k):=&\ell_{k,(k-1)}(n;\lambda_1,\cdots,\lambda_k)+\left(\frac{1}{\lambda_1-\lambda_2}+\frac{1}{\lambda_k-\lambda_1}\right)\ell_{k-1,(k-1)}(n;\lambda_2,\cdots,\lambda_k).
\end{align}\] This construction allows us to relate the desired term \(\ell_k\) to a more tractable term \(\ell_{k,(k)}\) which we can compute directly.
Lemma 5. For \(k\ge2\), \[\ell_{k,(k)}(n;\lambda_1,\cdots,\lambda_k)=(-1)^k\prod_{i=1}^k\mathcal{J}_n(\lambda_{i+1},\lambda_i)-\frac{1}{\prod_{i=1}^k(\lambda_i-\lambda_{i+1})}. \label{lkk}\qquad{(8)}\]
Proof. Throughout this proof we will omit the parameter \(n\) in \(\ell_{k,(m)}\). We first claim that for \(1\le m\le k-1\),
\[\ell_{k,(m)}(\lambda_1,\cdots,\lambda_k)=\int_\mathbb{R}\frac{(-1)^m}{y-\lambda_1}E_{k-m}(y,\lambda_2,\cdots,\lambda_{k-m+1})\left(\prod_{i=k-m+1}^{k-1}\mathcal{J}_n(\lambda_{i+1},\lambda_i)\right)\mathcal{J}_n(y,\lambda_k)\mathop{}\!\mathrm{d}\mu(y).
\label{claim}\tag{25}\] We prove this claim by induction on \(m\). For \(m=1\), \[\begin{align} &\ell_k(\lambda_1,\cdots,\lambda_k)\nonumber\\
=&\int_\mathbb{R}\frac{1}{y-\lambda_1}\left(-E_{k-1}(y,\lambda_2,\cdots,\lambda_k)\mathcal{J}_n(y,\lambda_k)-\frac{1}{\lambda_k-y}E_{k-1}(y,\lambda_2,\cdots,\lambda_{k-1},y)\right)\mathop{}\!\mathrm{d}\mu(y)\nonumber\\
=&-\int_\mathbb{R}\frac{1}{y-\lambda_1}E_{k-1}(y,\lambda_2,\cdots,\lambda_k)\mathcal{J}_n(y,\lambda_k)\mathop{}\!\mathrm{d}\mu(y)+\frac{1}{\lambda_1-\lambda_k}(\ell_{k-1}(\lambda_1,\cdots,\lambda_{k-1})-\ell_{k-1}(\lambda_2,\cdots,\lambda_k)),
\end{align}\] where in the last equation we use the fact that by definition \(\ell_{k-1}(\lambda_k,\lambda_2,\cdots,\lambda_{k-1})=\ell_{k-1}(\lambda_2,\cdots,\lambda_k)\). Now assume that 25
holds for \(m\). Then for \(m+1\), \[\begin{align} &\ell_{k,(m)}(\lambda_1,\cdots,\lambda_k)\nonumber\\
=&\int_\mathbb{R}\frac{(-1)^m}{y-\lambda_1}E_{k-m}(y,\lambda_2,\cdots,\lambda_{k-m+1})\left(\prod_{i=k-m+1}^{k-1}\mathcal{J}_n(\lambda_{i+1},\lambda_i)\right)\mathcal{J}_n(y,\lambda_k)\mathop{}\!\mathrm{d}\mu(y)\nonumber\\
=&(-1)^m\left(\prod_{i=k-m+1}^{k-1}\mathcal{J}_n(\lambda_{i+1},\lambda_i)\right)\int_\mathbb{R}\frac{\mathcal{J}_n(y,\lambda_k)}{y-\lambda_1}\bigg(-E_{k-m-1}(y,\lambda_2,\cdots,\lambda_{k-m})\mathcal{J}_n(\lambda_{k-m+1},\lambda_{k-m})\nonumber\\
&\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad-\frac{1}{\lambda_{k-m}-\lambda_{k-m+1}}E_{k-m-1}(y,\lambda_2,\cdots,\lambda_{k-m-1},\lambda_{k-m+1})\bigg)\mathop{}\!\mathrm{d}\mu(y)\nonumber\\
=&\int_\mathbb{R}\frac{(-1)^{m+1}}{y-\lambda_1}E_{k-m-1}(y,\lambda_2,\cdots,\lambda_{k-m})\left(\prod_{i=k-m}^{k-1}\mathcal{J}_n(\lambda_{i+1},\lambda_i)\right)\mathcal{J}_n(y,\lambda_k)\mathop{}\!\mathrm{d}\mu(y)\nonumber\\
&-\frac{1}{\lambda_{k-m}-\lambda_{k-m+1}}\ell_{k-1,(m)}(\lambda_1,\cdots,\lambda_{k-m-1},\lambda_{k-m+1},\cdots,\lambda_k).
\end{align}\] Thus we have proved 25 .
Now we prove ?? by induction on \(k\). For \(k=2\), using an argument similar to that of 23 , \[\begin{align}
\ell_{2,(1)}(\lambda_1,\lambda_2)=&-\int_\mathbb{R}\frac{1}{y-\lambda_1}E_1(y,\lambda_2)\mathcal{J}_n(y,\lambda_2)\mathop{}\!\mathrm{d}\mu(y)\nonumber\\
=&\mathcal{J}_n(\lambda_2,\lambda_1)\mathcal{J}_n(\lambda_1,\lambda_2)+\frac{1}{(\lambda_1-\lambda_2)^2}.
\end{align}\] Then \[\begin{align} \ell_{2,(2)}(\lambda_1,\lambda_2)=&\ell_{2,(1)}(\lambda_1,\lambda_2)+\left(\frac{1}{\lambda_1-\lambda_2}+\frac{1}{\lambda_2-\lambda_1}\right)\ell_{1,(1)}(\lambda_2)\nonumber\\
=&\mathcal{J}_n(\lambda_2,\lambda_1)\mathcal{J}_n(\lambda_1,\lambda_2)-\frac{1}{(\lambda_1-\lambda_2)(\lambda_2-\lambda_1)}.
\end{align}\] Assume that ?? holds for \(k-1\), then for \(k\), following 24 , \[\begin{align}
&\ell_{k,(k-1)}(\lambda_1,\cdots,\lambda_k)\nonumber\\
=&(-1)^{k-1}\left(\prod_{i=2}^{k-1}\mathcal{J}_n(\lambda_{i+1},\lambda_i)\right)\int_\mathbb{R}\frac{1}{y-\lambda_1}\mathcal{J}_n(y,\lambda_k)K_n(y,\lambda_2)\mathop{}\!\mathrm{d}\mu(y)\nonumber\\
=&(-1)^k\left(\prod_{i=2}^{k-1}\mathcal{J}_n(\lambda_{i+1},\lambda_i)\right)\left(\mathcal{J}_n(\lambda_1,\lambda_k)\mathcal{J}_n(\lambda_2,\lambda_1)+\left(\frac{1}{\lambda_k-\lambda_1}+\frac{1}{\lambda_1-\lambda_2}\right)\mathcal{J}_n(\lambda_2,\lambda_k)\right)\nonumber\\
=&(-1)^k\left(\prod_{i=1}^k\mathcal{J}_n(\lambda_{i+1},\lambda_i)+\left(\frac{1}{\lambda_1-\lambda_2}+\frac{1}{\lambda_k-\lambda_1}\right)\mathcal{J}_n(\lambda_2,\lambda_k)\prod_{i=2}^{k-1}\mathcal{J}_n(\lambda_{i+1},\lambda_i)\right).
\end{align}\] Thus \[\begin{align} &\ell_{k,(k)}(\lambda_1,\cdots,\lambda_k)\nonumber\\
=&\ell_{k,(k-1)}(\lambda_1,\cdots,\lambda_k)+\left(\frac{1}{\lambda_1-\lambda_2}+\frac{1}{\lambda_k-\lambda_1}\right)\ell_{k-1,(k-1)}(\lambda_2,\cdots,\lambda_k)\nonumber\\
=&(-1)^k\left(\prod_{i=1}^k\mathcal{J}_n(\lambda_{i+1},\lambda_i)+\left(\frac{1}{\lambda_1-\lambda_2}+\frac{1}{\lambda_k-\lambda_1}\right)\mathcal{J}_n(\lambda_2,\lambda_k)\prod_{i=2}^{k-1}\mathcal{J}_n(\lambda_{i+1},\lambda_i)\right)\nonumber\\
&+\left(\frac{1}{\lambda_1-\lambda_2}+\frac{1}{\lambda_k-\lambda_1}\right)\left((-1)^{k-1}\mathcal{J}_n(\lambda_2,\lambda_k)\prod_{i=2}^{k-1}\mathcal{J}_n(\lambda_{i+1},\lambda_i)-\frac{1}{(\lambda_k-\lambda_2)\prod_{i=1}^{k-1}(\lambda_i-\lambda_{i+1})}\right)\nonumber\\
=&(-1)^k\prod_{i=1}^k\mathcal{J}_n(\lambda_{i+1},\lambda_i)-\frac{1}{\prod_{i=1}^k(\lambda_i-\lambda_{i+1})}.
\end{align}\] The lemma is proved. ◻
The double sequence \(\{\ell_{k,(m)}\}\) forms the following table: \[\label{table} \begin{array}{c|cccccc} \text{row}\backslash\text{column} & 0 & 1
& 2 & \dots & k-1 & k \\ \hline 0 & \ell_0 & \ell_1 & \ell_2 & \dots & \ell_{k-1} & \ell_k \\ 1 & & \ell_{1,(1)} & \ell_{2,(1)} & \dots & \ell_{k-1,(1)} & \ell_{k,(1)} \\ 2 & & &
\ell_{2,(2)} & \dots & \ell_{k-1,(2)} & \ell_{k,(2)} \\ \vdots & & & & \ddots & \vdots & \vdots \\ k-1 & & & & & \ell_{k-1,(k-1)} & \ell_{k,(k-1)} \\ k & & & & & &
\ell_{k,(k)} \end{array}\tag{26}\] Each entry in this table is a sum of the terms located directly above and to the upper-left. Given the diagonal entries, we can uniquely determine the entire table, specifically the 0th-row entries.
Denote the following rational functions in \(\lambda_1,\cdots,\lambda_k\) for \(\{a_1,\cdots,a_t\}\subset \mathbf{Z}_k\): \[R_{a_1,\cdots,a_t}(\lambda_1,\cdots,\lambda_k):=\prod_{b\in \mathbf{Z}_k\backslash\{a_1,\cdots,a_t\}}\frac{1}{\lambda_b-\lambda_{b+1}}.\]
Proof. We first outline the strategy of the proof. In the context of Table 26 , equation ?? can be interpreted as expressing the entry at \((0,k)\) in terms of the entries at \((0,m)\), \(0\le m\le k-1\) and the diagonal entry at \((k,k)\). To achieve this, we initially express the entry at \((0,k)\) in
terms of the entries at the \((k-1)\)th column and at the diagonal term \((k,k)\). Subsequently, in each step, we iteratively reduce the dependence on the \(m\)-th column to the \((m-1)\)-th column and the boundary entry at \((0,m)\). This procedure ultimately yields ?? . The process is illustrated as follows: \[\begin{array}{c|c} & k \\ \hline 0 & * \end{array} \Rightarrow \begin{array}{c} \begin{array}{c|cc} & k-1 & k \\ \hline 0 & * & * \\ 1 & * & 0 \\ \vdots & \vdots & \vdots \\ k-1 & * & 0
\\ k & & * \end{array} \end{array} \Rightarrow \begin{array}{c} \begin{array}{c|ccc} & k-2 & k-1 & k \\ \hline 0 & * & * & * \\ 1 & * & 0 & 0 \\ \vdots & \vdots & \vdots & \vdots \\ k-2 & * &
0 & 0 \\ k-1 & & 0 & 0 \\ k & & & * \end{array} \end{array} \Rightarrow \dots \Rightarrow \begin{array}{c} \begin{array}{c|ccccc} & 0 & 1 & \dots & k-1 & k \\ \hline 0 & * & * & \dots & *
& * \\ 1 & & 0 & \dots & 0 & 0 \\ \vdots & & & \ddots & \vdots & \vdots \\ k-1 & & & & 0 & 0 \\ k & & & & & * \end{array} \end{array}\]
More precisely, for \(m\le k-1\) we prove the following formula, which expresses \(\ell_k\) in terms of the entries at \((0,r)\) for \(k-m\le r\le k-1\), the entries at \((r,k-m)\) for \(1\le r\le k-m\), and the diagonal term \((k,k)\): (We still omit the
parameter \(n\).) \[\begin{align} &\ell_k(\lambda_1,\cdots,\lambda_k)\nonumber\\ =& \ell_{k,(k)}-\sum_{r=1}^{m-1}\sum_{1\le a_1<\cdots<a_r\le
k}\widetilde{R}_{a_1,\cdots,a_r}\ell_{k-r}^{\widehat{a_1,\cdots,a_r}} \nonumber\\ & -\sum_{r=0}^{m-1}\sum_{r+1<a_{r+1}<\cdots<a_{m-1}\le
k-1}\widetilde{R}_{1,\cdots,r,a_{r+1},\cdots,a_{m-1}}\frac{\ell_{k-m}^{\widehat{1,\cdots,r,a_{r+1},\cdots,a_{m-1},k}}-\ell_{k-m}^{\widehat{1,\cdots,r,r+1,a_{r+1},\cdots,a_{m-1}}}}{\lambda_k-\lambda_1} \nonumber\\ &
-\sum_{t=1}^{k-m-1}\sum_{r=0}^{m-1}\sum_{r+1<a_{r+1}<\cdots<a_m=k-t} \frac{\lambda_k-\lambda_{r+1}}{\lambda_k-\lambda_1}\widetilde{R}_{1,\cdots,r,a_{r+1},\cdots,a_m}\ell_{k-m,(t)}^{\widehat{1,\cdots,r,a_{r+1},\cdots,a_m}}\nonumber\\
&-\frac{\lambda_k-\lambda_{m+1}}{\lambda_k-\lambda_1}\widetilde{R}_{1,\cdots,m}\ell_{k-m,(k-m)}^{\widehat{1,\cdots,m}}, \label{lind}
\end{align}\tag{27}\] where for \(a_{1}<\cdots<a_{m}\) we denote \[\begin{align} \widetilde{R}_{a_1,\cdots,a_m}:=& \prod_{s=1}^{m}\frac{1}{\lambda_{a_s}-\lambda_{a_s+1}},
\\ \ell_{k-m}^{\widehat{a_1,\cdots,a_m}} :=& \ell_{k-m}(\lambda_1,\cdots,\lambda_{a_1-1},\lambda_{a_1+1},\cdots,\lambda_{a_m-1},\lambda_{a_m+1},\cdots,\lambda_k), \\ \ell_{k-m,(t)}^{\widehat{a_1,\cdots,a_m}} :=&
\ell_{k-m,(t)}(\lambda_1,\cdots,\lambda_{a_1-1},\lambda_{a_1+1},\cdots,\lambda_{a_m-1},\lambda_{a_m+1},\cdots,\lambda_k).
\end{align}\]
We prove 27 by induction on \(m\). First, for \(m=1\), \[\begin{align} \ell_{k}(\lambda_1,\cdots,\lambda_k) &=
\ell_{k,(1)}-\frac{1}{\lambda_k-\lambda_1}(\ell_{k-1}^{\widehat k}-\ell_{k-1}^{\widehat1}) \nonumber\\ &= \ell_{k,(2)}-\frac{1}{\lambda_k-\lambda_1}(\ell_{k-1}^{\widehat
k}-\ell_{k-1}^{\widehat1})-\frac{1}{\lambda_{k-1}-\lambda_k}\ell_{k-1,(1)}^{\widehat{k-1}} \nonumber\\ &= \ell_{k,(k-1)}-\frac{1}{\lambda_k-\lambda_1}(\ell_{k-1}^{\widehat
k}-\ell_{k-1}^{\widehat1})-\sum_{t=1}^{k-2}\frac{1}{\lambda_{k-t}-\lambda_{k-t+1}}\ell_{k-1,(t)}^{\widehat{k-t}} \nonumber\\ &= \ell_{k,(k)}-\frac{1}{\lambda_k-\lambda_1}(\ell_{k-1}^{\widehat
k}-\ell_{k-1}^{\widehat1})-\sum_{t=1}^{k-2}\widetilde{R}_{k-t}\ell_{k-1,(t)}^{\widehat{k-t}} -\frac{\lambda_k-\lambda_2}{\lambda_k-\lambda_1}\widetilde{R}_1\ell_{k-1,(k-1)}^{\widehat1}.
\end{align}\]
Now assume that 27 holds for \(m\). For \(m+1\), we transition from the \((k-m)\)-th column to the \((k-m-1)\)-th column. Consequently, we need to compute the terms at the entries \((t,k-m-1)\) for \(0\le t\le k-m-1\), as well as the boundary term \((0,k-m)\). The term at \((k-m-1,k-m-1)\) is derived from the term at \((k-m,k-m)\) and is given by \[\begin{align}
&-\left(\frac{1}{\lambda_{m+1}-\lambda_{m+2}}+\frac{1}{\lambda_k-\lambda_{m+1}}\right)\frac{\lambda_k-\lambda_{m+1}}{\lambda_k-\lambda_1}\widetilde{R}_{1,\cdots,m}\ell_{k-m-1,(k-m-1)}^{\widehat{1,\cdots,m,m+1}}\nonumber\\
=&-\frac{\lambda_k-\lambda_{m+2}}{\lambda_k-\lambda_1}\widetilde{R}_{1,\cdots,m+1}\ell_{k-m-1,(k-m-1)}^{\widehat{1,\cdots,m+1}}. \label{part1}
\end{align}\tag{28}\] The term at \((t,k-m-1)\) for \(1\le t\le k-m-2\) is \[\begin{align}
&-\frac{1}{\lambda_{k-t}-\lambda_{k-t+1}}\sum_{r=0}^{m}\sum_{r+1<a_{r+1}<\cdots<a_m\le
k-t-1}\frac{\lambda_k-\lambda_{r+1}}{\lambda_k-\lambda_1}\widetilde{R}_{1,\cdots,r,a_{r+1},\cdots,a_m}\ell_{k-m-1,(t)}^{\widehat{1,\cdots,r,a_{r+1},\cdots,a_m,k-t}}\nonumber\\
=&-\sum_{r=0}^{m+1}\sum_{r+1<a_{r+1}<\cdots<a_{m+1}=k-t}\frac{\lambda_k-\lambda_{r+1}}{\lambda_k-\lambda_1}\widetilde{R}_{1,\cdots,r,a_{r+1},\cdots,a_{m+1}}\ell_{k-m-1,(t)}^{\widehat{1,\cdots,r,a_{r+1},\cdots,a_{m+1}}}. \label{part2}
\end{align}\tag{29}\] Similarly, the term at \((0,k-m-1)\) reads \[\begin{align} &-\sum_{r=0}^{m}\sum_{r+1<a_{r+1}<\cdots<a_m\le
k-1}\frac{\lambda_k-\lambda_{r+1}}{\lambda_k-\lambda_1}\widetilde{R}_{1,\cdots,r,a_{r+1},\cdots,a_m}\frac{\ell_{k-m-1}^{\widehat{1,\cdots,r,a_{r+1},\cdots,a_m,k}}-\ell_{k-m-1}^{\widehat{1,\cdots,r+1,a_{r+1},\cdots,a_m}}}{\lambda_k-\lambda_{r+1}}\nonumber\\
=&-\sum_{r=0}^{m}\sum_{r+1<a_{r+1}<\cdots<a_m\le k-1}\widetilde{R}_{1,\cdots,r,a_{r+1},\cdots,a_m}\frac{\ell_{k-m-1}^{\widehat{1,\cdots,r,a_{r+1},\cdots,a_m,k}}-\ell_{k-m-1}^{\widehat{1,\cdots,r+1,a_{r+1},\cdots,a_m}}}{\lambda_k-\lambda_1}.
\label{part3}
\end{align}\tag{30}\] Finally, the term at \((0,k-m)\) changes to \[\begin{align} & -\sum_{r=0}^{m-1}\sum_{r+1<a_{r+1}<\cdots<a_{m-1}\le
k-1}\widetilde{R}_{1,\cdots,r,a_{r+1},\cdots,a_{m-1}}\frac{\ell_{k-m}^{\widehat{1,\cdots,r,a_{r+1},\cdots,a_{m-1},k}}-\ell_{k-m}^{\widehat{1,\cdots,r+1,a_{r+1},\cdots,a_{m-1}}}}{\lambda_k-\lambda_1} \nonumber\\ &\qquad
-\sum_{t=1}^{k-m}\sum_{r=0}^{m}\sum_{r+1<a_{r+1}<\cdots<a_m=k-t} \frac{\lambda_k-\lambda_{r+1}}{\lambda_k-\lambda_1}\widetilde{R}_{1,\cdots,r,a_{r+1},\cdots,a_m}\ell_{k-m}^{\widehat{1,\cdots,r,a_{r+1},\cdots,a_m}}\nonumber\\ &\qquad
-\frac{\lambda_k-\lambda_{m+1}}{\lambda_k-\lambda_1}\widetilde{R}_{1,\cdots,m}\ell_{k-m}^{\widehat{1,\cdots,m}}.\label{0k-m}
\end{align}\tag{31}\] In 31 we compute the rational function before \(\ell_{k-m}^{\widehat{1,\cdots,r,a_{r+1},\cdots,a_{m}}}\) where \(a_{r+1}>r+1\)
and \(a_{m}<k\). It is \[\begin{align}
&-\sum_{t=0}^{r-1}\frac{1}{\lambda_1-\lambda_k}\widetilde{R}_{1,\cdots,t,t+2,\cdots,r,a_{r+1},\cdots,a_{m-1}}-\frac{\lambda_k-\lambda_{r+1}}{\lambda_k-\lambda_1}\widetilde{R}_{1,\cdots,r,a_{r+1},\cdots,a_m} \nonumber\\
=&-\frac{1}{\lambda_1-\lambda_k}\left(\sum_{t=0}^{r-1}(\lambda_{t+1}-\lambda_{t+2})-(\lambda_k-\lambda_{r+1})\right)\widetilde{R}_{1,\cdots,r,a_{r+1},\cdots,a_{m}} \nonumber\\ =&-\widetilde{R}_{1,\cdots,r,a_{r+1},\cdots,a_m}.
\end{align}\] Similarly, the rational function before \(\ell_{k-m}^{\widehat{1,\cdots,m}}\) is \(-\widetilde{R}_{1,\cdots,m}\). And lastly, the rational function before \(\ell_{k-m}^{\widehat{1,\cdots,r,a_{r+1},\cdots,a_{m-1},k}}\) where \(a_{r+1}>r+1\) is \[\begin{align}
-\frac{1}{\lambda_k-\lambda_1}\widetilde{R}_{1,\cdots,r,a_{r+1},\cdots,a_{m-1}} =-\widetilde{R}_{1,\cdots,r,a_{r+1},\cdots,a_{m-1},k}.
\end{align}\] Hence 31 reads \[-\sum_{1\le a_1<\cdots<a_m\le k}\widetilde{R}_{a_1,\cdots,a_m}\ell_{k-m}^{\widehat{a_1,\cdots,a_m}}.
\label{part4}\tag{32}\] Thus by 28 , 29 , 30 and 32 , we obtain 27 for \(m+1\).
Now following 27 , for \(m=k-1\) we have \[\begin{align} &\ell_k(\lambda_1,\cdots,\lambda_k)\nonumber\\ =& \ell_{k,(k)}-\sum_{r=1}^{k-2}\sum_{1\le
a_1<\cdots<a_r\le k}\widetilde{R}_{a_1,\cdots,a_r}\ell_{k-r}^{\widehat{a_1,\cdots,a_r}}-\sum_{r=0}^{k-2}\widetilde{R}_{1,\cdots,r,r+2,\cdots,k-1}\frac{\ell_1(\lambda_{r+1})-\ell_1(\lambda_k)}{\lambda_k-\lambda_1} \nonumber\\ =&
\ell_{k,(k)}-\sum_{r=1}^{k-1}\sum_{1\le a_1<\cdots<a_r\le k}\widetilde{R}_{a_1,\cdots,a_r}\ell_{k-r}^{\widehat{a_1,\cdots,a_r}} \nonumber\\
=&(-1)^k\prod_{i=1}^k\mathcal{J}_n(\lambda_{i+1},\lambda_i)-\frac{1}{\prod_{i=1}^k(\lambda_i-\lambda_{i+1})}-\sum_{t=1}^{k-1}\sum_{1\le a_1<\cdots<a_t\le k}R_{a_1,\cdots,a_t}(\lambda_1,\cdots,\lambda_k)\ell_t(\lambda_{a_1},\cdots,\lambda_{a_t}).
\end{align}\] Thus the lemma is proved. ◻
Proof. By ?? , noting that \(\ell_{k}\) and \(\ell_{k,(k)}\) are invariant under the cyclic group \(C_{k}\), we have \[\begin{align} &\sum_{\sigma\in S_k/C_k}\ell_k(\lambda_{\sigma(1)},\cdots,\lambda_{\sigma(k)})\nonumber\\ =&(-1)^k\sum_{\sigma\in S_k/C_k}\prod_{i=1}^k\mathcal{J}_n(\lambda_{\sigma(i)},\lambda_{\sigma(i+1)})-\sum_{\sigma\in
S_k/C_k}\prod_{i=1}^k\frac{1}{\lambda_{\sigma(i)}-\lambda_{\sigma(i+1)}} \nonumber\\ &-\sum_{\sigma\in S_{k}/C_{k}}\sum_{t=1}^{k-1}\sum_{1\le a_1<\cdots<a_t\le
k}R_{\sigma(a_1),\cdots,\sigma(a_t)}(\lambda_{\sigma(1)},\cdots,\lambda_{\sigma(k)})\ell_{t}(\lambda_{\sigma(a_1)},\cdots,\lambda_{\sigma(a_t)}).
\end{align}\] For \(k\ge3\), \[\begin{align} \sum_{\sigma\in S_k/C_k}\prod_{i=1}^k\frac{1}{\lambda_{\sigma(i)}-\lambda_{\sigma(i+1)}}=\sum_{\sigma\in
S_{k-1}}\left(\prod_{i=1}^{k-1}\frac{1}{\lambda_{\sigma(i)}-\lambda_{\sigma(i+1)}}\right)\left(\frac{1}{\lambda_{\sigma(k-1)}-\lambda_k}-\frac{1}{\lambda_{\sigma(1)}-\lambda_k}\right)=0.
\end{align}\] Given \(1\le a_{1}<\cdots<a_{t}\le k\), \(\ell_{t}(\lambda_{a_{\sigma(1)}},\cdots,\lambda_{a_{\sigma(t)}})\) has the same rational function before it for any \(\sigma\in S_{t}/C_{t}\), which is \[\begin{align} \sum_{\substack{\sqcup_{s=1}^tP_s=\mathbf{Z}_k\backslash\{a_1,\cdots,a_t\}\\P_s:=\{p_1,\cdots,p_{g_s}\}}}\prod_{s=1}^{t}\sum_{\sigma\in
S_{g_s}}\left(\prod_{b=1}^{g_s-1}\frac{1}{\lambda_{p_{\sigma(b)}}-\lambda_{p_{\sigma(b+1)}}}\right)\frac{1}{\lambda_{p_{\sigma(g_s)}}-\lambda_{a_s}}.
\end{align}\] Now we claim that for any set \(P=\{p_{1},\cdots,p_{s}\}\subset\mathbf{Z}_k\) and \(a\in\mathbf{Z}_k\backslash P\)\[\sum_{\sigma\in
S_s}\left(\prod_{b=1}^{s-1}\frac{1}{\lambda_{p_{\sigma(b)}}-\lambda_{p_{\sigma(b+1)}}}\right)\frac{1}{\lambda_{p_{\sigma(s)}}-\lambda_{a}}=\prod_{b=1}^s\frac{1}{\lambda_{p_{b}}-\lambda_{a}},\label{claimQ}\tag{33}\] where the RHS is just \(Q_{a,P}(\lambda_1,\cdots,\lambda_k)\). We prove 33 by induction on s. The base case \(s=1\) is straightforward. Assume that 33 holds for \(s\ge1\). Then for \(s+1\), we consider the two sides of 33 as meromorphic functions in \(\lambda_{a}\), treating the other \(\lambda_{j}\)’s as fixed parameters. Both rational functions possess only simple poles at \(\lambda_{p_{1}},\cdots,\lambda_{p_{s+1}}\). Therefore, it suffices to verify that the residues at
these poles match. Without loss of generality, we focus on the coefficient of the term \(\frac{1}{\lambda_{p_{s+1}}-\lambda_{a}}\). Specifically, the coefficient on the LHS of 33 is \[\begin{align} \sum_{\substack{\sigma\in S_{s+1}\\\sigma(s+1)=s+1}}\prod_{b=1}^{s}\frac{1}{\lambda_{p_{\sigma(b)}}-\lambda_{p_{\sigma(b+1)}}} &=\sum_{\sigma\in
S_s}\left(\prod_{b=1}^{s-1}\frac{1}{\lambda_{p_{\sigma(b)}}-\lambda_{p_{\sigma(b+1)}}}\right)\frac{1}{\lambda_{p_{\sigma(s)}}-\lambda_{p_{s+1}}}=\prod_{b=1}^s\frac{1}{\lambda_{p_b}-\lambda_{p_{s+1}}},
\end{align}\] which is the same as the coefficient on the RHS of 33 . These prove the lemma. ◻
Now we are ready to finish the proof of Theorem 1. For \(k\ge2\), \[\begin{align}
&C_{k}(n;\lambda_1,\cdots,\lambda_k)\nonumber\\ =&-\sum_{\text{partition }\Pi\text{ of }\mathbf{Z}_k}\sum_{(a_1,\cdots,a_{|\Pi|})\in\Pi_1\times\cdots\times\Pi_{|\Pi|}} \left(\prod_{i=1}^{|\Pi|}
Q_{a_i,\Pi_i\backslash\{a_i\}}(\lambda_1,\cdots,\lambda_k)\right)\frac{1}{|\Pi|}\sum_{\sigma\in S_{|\Pi|}} \ell_{|\Pi|}(n;\lambda_{a_{\sigma(1)}},\cdots,\lambda_{a_{\sigma(|\Pi|)}})\nonumber\\ =&-\sum_{l=1}^k\sum_{1\le a_1<\cdots<a_l\le k}
\sum_{\sqcup_{i=1}^l\Pi_i=\mathbf{Z}_k\setminus\{a_1,\cdots,a_l\}}\left(\prod_{i=1}^{l}Q_{a_i,\Pi_i}(\lambda_1,\cdots,\lambda_k)\right)\frac{1}{l}\sum_{\sigma\in S_l}\ell_l(n;\lambda_{a_{\sigma(1)}},\cdots,\lambda_{a_{\sigma(l)}})\nonumber\\
=&\frac{(-1)^{k-1}}{k}\sum_{\sigma\in S_k}\prod_{i=1}^k\mathcal{J}_n(\lambda_{\sigma(i)},\lambda_{\sigma(i+1)})-\frac{\delta_{k,2}}{(\lambda_1-\lambda_2)^2},
\end{align}\] where the last equation follows from ?? . Therefore we obtain the theorem. ◻
Remark 2. The matrix-resolvent method of the 1D Toda lattice hierarchy also applies to more general solutions, such as the one given by the Gromov-Witten invariants of \(\mathbb{CP}^1\)[13], [21], [41].
Remark 3. The formulas 17 and 18 can be equivalently (cf. [13], [21], [27], [29], [41], [43]–[46]) written as \[\begin{align} \mathcal{C}_k(n;\lambda_1,\cdots,\lambda_k)=&-\sum_{\sigma\in
S_k/C_k}\frac{\mathop{\mathrm{tr}}(R_n(\lambda_{\sigma(1)})\cdots R_n(\lambda_{\sigma(k)}))}{\prod_{i=1}^k(\lambda_{\sigma(i)}-\lambda_{\sigma(i+1)})}-\frac{\delta_{k,2}}{(\lambda_1-\lambda_2)^2},\qquad k\ge2,
\end{align}\] (cf. also [47], [48]), where \[\begin{align}
R_n(\lambda):=&\frac{1}{h_{n-1}}\begin{pmatrix} -\pi_n(\lambda)\widehat\pi_{n-1}(\lambda)&-\pi_n(\lambda)\widehat\pi_n(\lambda)\\ \pi_{n-1}(\lambda)\widehat\pi_{n-1}(\lambda)&\pi_{n-1}(\lambda)\widehat\pi_n(\lambda) \end{pmatrix}.
\end{align}\]
Remark 4. For the cases of Jacobi unitary ensemble and Laguerre unitary ensemble, Gisonni, Grava and Ruzza [43]–[45] also gave explicit formulas in terms of generalized hypergeometric functions using different methods (cf. [46]).
Remark 5. For the case of Gaussian unitary ensemble (GUE), the Christoffel–Darboux kernel reads \[K_n^\text{GUE}(\lambda_1,\lambda_2)=\frac{1}{h_{n-1}}\frac{\mathrm{He}_n(\lambda_1)\mathrm{He}_{n-1}(\lambda_2)-\mathrm{He}_{n-1}(\lambda_1)\mathrm{He}_n(\lambda_2)}{\lambda_1-\lambda_2}.\] It is well known [49]–[51] that the double-scaling limit \(n\to\infty\) of \(K_n^\text{GUE}(\lambda_1,\lambda_2)\) gives the Airy kernel \(K^\text{Airy}(\xi_1,\xi_2)\): \[\lim_{n\to\infty}n^{-1/6}K_n^\mathrm{GUE}(\lambda_1,\lambda_2)e^{-\frac{\lambda_1^2+\lambda_2^2}{4}}=K^\mathrm{Airy}(\xi_1,\xi_2),\] where \[\lambda_1=2\sqrt{n}+n^{-1/6}\xi_1,\qquad
\lambda_2=2\sqrt{n}+n^{-1/6}\xi_2,\] and \[K^\text{Airy}(\xi_1,\xi_2):=\frac{\mathrm{Ai}(\xi_1)\mathrm{Ai}'(\xi_2)-\mathrm{Ai}'(\xi_1)\mathrm{Ai}(\xi_2)}{\xi_1-\xi_2}.\] In addition, in the lower half plane,
the \(\mathcal{J}_n\) kernel (cf. 14 ) for GUE \[\mathcal{J}_n^\mathrm{GUE}(\lambda_1,\lambda_2)=-\frac{1}{h_{n-1}}\frac{\mathrm{He}_n(\lambda_1)\widehat{\mathrm{He}}_{n-1}(\lambda_2)-\mathrm{He}_{n-1}(\lambda_1)\widehat{\mathrm{He}}_n(\lambda_2)}{\lambda_1-\lambda_2}\]
(cf. [21], [42]) gives the Airy–Bairy kernel as \(n\to\infty\):
\[\label{dsl} \lim_{n\to\infty}n^{-1/6}e^{\frac{\lambda_1^2-\lambda_2^2}{4}}\mathcal{J}_n^\mathrm{GUE}(\lambda_1,\lambda_2)=2\pi i\omega
\mathcal{J}^\mathrm{Ai-Bi}(\xi_1,\xi_2),\tag{34}\] where \[\mathcal{J}^\mathrm{Ai-Bi}(\xi_1,\xi_2):=-\frac{\mathrm{Ai}(\xi_1)\omega\mathrm{Ai}'(\omega\xi_2)-\mathrm{Ai}'(\xi_1)\mathrm{Ai}(\omega\xi_2)}{\xi_1-\xi_2},\] and \(\omega:=e^{2\pi
i/3}\). We study application of 34 and of the determinantal formulas ?? and ?? in a subsequent publication.
Proof of Lemma 1. The case \(n=0\) is direct. For the case \(n\ge1\), along the lines of the proof
of [32], we denote for \(n\in\mathbb{Z}_{\ge0}\), \[\begin{align}
&\phi_A(\lambda,n):=\lambda^{-n}\psi_A^\star(\lambda,n),\qquad\phi_B(\lambda,n):=h_n^{-1}\lambda^n\psi_B^\star(\lambda,n).
\end{align}\] Then for \(n\in\mathbb{Z}_{\ge1}\), \[\label{an}
\frac{\lambda_2^n}{\lambda_1^n}B_n(\lambda_1,\lambda_2)=\frac{\phi_A(\lambda_1,n)\phi_B(\lambda_2,n-1)-\gamma_n\frac{\phi_A(\lambda_1,n-1)}{\lambda_1}\frac{\phi_B(\lambda_2,n)}{\lambda_2}}{\lambda_1-\lambda_2},\tag{35}\] where we recall that
\(\gamma_n=\frac{h_n}{h_{n-1}}\) for \(n\ge1\). Similar to [32], [48], [52], we have \[\begin{align}
\phi_B(\lambda_2)=&\phi_B(\lambda_1)+\phi'_B(\lambda_1)(\lambda_2-\lambda_1)+(\lambda_2-\lambda_1)^2\frac{\mathop{}\!\mathrm{d}}{\mathop{}\!\mathrm{d}\lambda_1}\left(\frac{\phi_B(\lambda_1)-\phi_B(\lambda_2)}{\lambda_1-\lambda_2}\right),\tag{36}\\
\frac{\phi_B(\lambda_2)}{\lambda_2}=&\frac{\phi_B(\lambda_1)}{\lambda_1}+\frac{\mathop{}\!\mathrm{d}}{\mathop{}\!\mathrm{d}\lambda_1}\left(\frac{\phi_B(\lambda_1)}{\lambda_1}\right)(\lambda_2-\lambda_1)+(\lambda_2-\lambda_1)^2\frac{\mathop{}\!\mathrm{d}}{\mathop{}\!\mathrm{d}\lambda_1}\left(\frac{\phi_B(\lambda_1)/\lambda_1-\phi_B(\lambda_2)/\lambda_2}{\lambda_1-\lambda_2}\right),\tag{37}
\end{align}\] where we omit the parameter \(n\). Since \(\phi_A(\lambda,n)\sim1+O(\lambda^{-1})\) and \(\phi_B(\lambda,n)\sim1+O(\lambda^{-1})\), substituting 36 , 37 into 35 , and
using 16 , we obtain the lemma. ◻
Proof of Corollary 1. Using Theorem 1, Lemma 1 and the converse theorem of Zhou’s theorem, we obtain the corollary. ◻
Denote \[\begin{align}
\label{pi} \pi_l(\lambda)=\sum_{k=0}^la_k(l)\lambda^{l-k},\qquad\lambda^l=\sum_{k=0}^lc_k(l)\pi_{l-k}(\lambda),
\end{align}\tag{38}\] and let \[T_{k,l}(n):=A_{k-1,l}(n)-A_{k,l-1}(n).\] We may prove a useful lemma.
Lemma 8. We have \[\begin{align} &A_{k,0}(n)=c_{k+1}(n+k)~~(n\ge1),\qquad A_{0,l}(n)=-a_{l+1}(n)~~(n>l), \label{0-case}\\
&T_{k,l}(n)=a_l(n)c_k(n-1+k)-a_{l-1}(n-1)c_{k-1}(n+k-1)\gamma_n~~(n>l). \label{klcase}
\end{align}\] {#eq: sublabel=eq:0-case,eq:klcase}
Proof. By definition, \[c_k(l+k)=\frac{1}{h_l}\int_{\mathbb{R}}y^{l+k}\pi_l(y)\mathop{}\!\mathrm{d}\mu(y).\] By comparing the coefficients in the two hand sides of ?? , we obtain the lemma. ◻
Now using Theorem 1, Corollary 1 and Lemma 8, we
prove Theorem 2. For a matrix \(M\), denote \(M_{i_1,\cdots,i_m}^{j_1,\cdots,j_n}\) as the matrix obtained from the
matrix \(M\) deleting the \(i_1\)th, …, \(i_m\)th rows and the \(j_1\)th, …, \(j_n\)th
columns. We denote \[\mathcal{H}:=H_{(0,1,\cdots,n;0,1,\cdots,n)}\] We first state a lemma.
Lemma 9. For any \(2\le i<j\le n-1\), \[\det(\mathcal{H}_{i,j}^{n,n+1})+\det(\mathcal{H}_{j,n+1}^{i-1,n+1})-\det(\mathcal{H}_{j-1,n+1}^{i,n+1})=0.\]
Proof. Denote by \(\mathcal{K}\) the adjoint matrix of \(\mathcal{H}\). Then from [53], we only need to prove that \[\label{adj} \det\begin{pmatrix} \mathcal{K}_{n,i}&\mathcal{K}_{n,j}\\\mathcal{K}_{n+1,i}&\mathcal{K}_{n+1,j}
\end{pmatrix}+\det\begin{pmatrix} \mathcal{K}_{i-1,j}&\mathcal{K}_{i-1,n+1}\\\mathcal{K}_{n+1,j}&\mathcal{K}_{n+1,n+1} \end{pmatrix}-\det\begin{pmatrix} \mathcal{K}_{i,j-1}&\mathcal{K}_{i,n+1}\\\mathcal{K}_{n+1,j-1}&\mathcal{K}_{n+1,n+1}
\end{pmatrix}=0.\tag{39}\] It is known [54] (see also [55]) that \(\mathcal{K}\) is a Bezoutian matrix with respect to the orthogonal polynomials associated to moments \(\{m_k\}_{k\ge0}\): \[K_{n+1}(\lambda_1,\lambda_2)=\sum_{l=0}^{n}\frac{\pi_l(\lambda_1)\pi_l(\lambda_2)}{h_l}=\frac{1}{h_{n}}\frac{\pi_{n+1}(\lambda_1)\pi_n(\lambda_2)-\pi_n(\lambda_1)\pi_{n+1}(\lambda_2)}{\lambda_1-\lambda_2}=\frac{1}{\Delta_n}\sum_{i,j=0}^{n}\mathcal{K}_{i+1,j+1}\lambda_1^i\lambda_2^j.\]
Thus \[\mathcal{K}_{i,j}=\Delta_n\sum_{k=\max\{i-1,j-1\}}^n\frac{a_{k-i+1}(k)a_{k-j+1}(k)}{h_k}.\] Then 39 is equivalent to \[\begin{align}
&-\frac{a_{j-i-1}(j-2)}{h_{j-2}}-\frac{a_{n-i+1}(n)a_{n-j}(n-1)-a_{n-i}(n-1)a_{n-j+1}(n)}{h_{n-1}}\nonumber\\ &\qquad+\sum_{k=j-1}^{n-1}\frac{a_{k-i+2}(k)a_{k-j+1}(k)-a_{k-i+1}(k)a_{k-j+2}(k)}{h_k}=0. \label{adj1}
\end{align}\tag{40}\]
We write \[\alpha_k:=\frac{1}{h_k}(a_{k-i+2}(k+1)a_{k-j+1}(k)-a_{k-i+1}(k)a_{k-j+2}(k+1)).\] By 2 , we have \[a_{k-i}(k)=a_{k-i}(k+1)+\beta_ka_{k-i-1}(k)+\frac{h_k}{h_{k-1}}a_{k-i-2}(k-1).\] Thus \[\begin{align} &\frac{1}{h_k}(a_{k-i+2}(k)a_{k-j+1}(k)-a_{k-i+1}(k)a_{k-j+2}(k))\nonumber\\
=&\frac{1}{h_k}\biggl(\left(a_{k-i+2}(k+1)+\beta_ka_{k-i+1}(k)+\frac{h_k}{h_{k-1}}a_{k-i}(k-1)\right)a_{k-j+1}(k)\nonumber\\ &\qquad-a_{k-i+1}(k)\left(a_{k-j+2}(k+1)+\beta_ka_{k-j+1}(k)+\frac{h_k}{h_{k-1}}a_{k-j}(k-1)\right)\biggr)\nonumber\\
=&\alpha_k-\alpha_{k-1}.
\end{align}\] Then the LHS of 40 writes \[\alpha_{j-2}-\alpha_{n-1}+\sum_{k=j-1}^{n-1}(\alpha_k-\alpha_{k-1})=0.\] Thus the lemma is proved. ◻
Proof of Theorem 2. The orthogonal polynomials \(\{\pi_k\}_{k\ge0}\) with respect to \(\mathop{}\!\mathrm{d}\mu(\lambda)\) read \[\pi_k(\lambda)=\frac{1}{\Delta_{k-1}}\det\begin{pmatrix} m_0&m_1&\cdots&m_k\\ m_1&m_2&\cdots&m_{k+1}\\
\cdots&\cdots&\cdots&\cdots\\ m_{k-1}&m_k&\cdots&m_{2k-1}\\ 1&\lambda&\cdots&\lambda^k \end{pmatrix}.\] Thus we have \[\begin{align} a_{n-k}(n)&=\frac{(-1)^{n-k}}{\Delta_{n-1}}\det
H_{(0,1,\cdots,n-1;0,1,\cdots,\widehat{k},\cdots,n)},\\ c_{n-l}(n)&=\frac{1}{\Delta_l}\det H_{(0,1,\cdots,l-1,n;0,1,\cdots,l)},\\ \gamma_n&=\frac{\Delta_n\Delta_{n-2}}{\Delta_{n-1}^2}.
\end{align}\] Therefore, \(A_{0,j}(n)=-a_{j+1}(n)\) and \(A_{i,0}(n)=c_{i+1}(n+i)\) satisfy ?? . Then by ?? , \[\begin{align}
T_{i,j}(n)=&\frac{(-1)^j}{\Delta_{n-1}^2}\biggl(\det H_{(0,1,\cdots,\widehat{n-j},\cdots,n;0,1,\cdots,n-1)}\det H_{(0,1,\cdots,n-2,n+i-1;0,1,\cdots,n-1)}\nonumber\\ &\qquad\quad+\det H_{(0,1,\cdots,\widehat{n-j},\cdots,n-1;0,1,\cdots,n-2)}\det
H_{(0,1,\cdots,n-1,n+i-1;0,1,\cdots,n)}\biggr)\nonumber\\ =&\frac{(-1)^j}{\Delta_{n-1}}\det H_{(0,1,\cdots,\widehat{n-j},\cdots,n;0,1,\cdots,n-2,n+i-1)},
\end{align}\] where the last equation follows from the Sylvester’s determinant identity [53]: \[\det(M)\det(M_{u,v}^{s,t})=\det(M_u^s)\det(M_v^t)-\det(M_u^t)\det(M_v^s),\] for any \(u<v\) and \(s<t\).
Now we assume that \(A_{i,j}(n),j<n-1\) satisfies ?? , we want to show that \(A_{i-1,j+1}(n)\) also satisfies ?? . Then the proof is complete following the fact that \(A_{i,0}(n)\) and \(A_{0,j}(n)\) satisfy ?? . Since \[A_{i-1,j+1}(n)=A_{i,j}(n)+T_{i,j+1}(n),\] we need to prove that \[\begin{align}
0=&\det H_{(0,1,\cdots,\widehat{n-j-1},\cdots,n;0,1,\cdots,n-2,n+i-1)}-\det H_{(0,1,\cdots,\widehat{n-j-1},\cdots,n-1,n+i;0,1,\cdots,n-1)}\nonumber\\ &-\det H_{(0,1,\cdots,\widehat{n-j-2},\cdots,n-1,n+i-1;0,1,\cdots,n-1)}\nonumber\\
=&\sum_{\substack{2\le k\le n\\k\neq n-j}}(-1)^{k+n}m_{n+i+k-2}(\det(\mathcal{H}_{k,n-j}^{n,n+1})+\det(\mathcal{H}_{n-j,n+1}^{k-1,n+1})-\det(\mathcal{H}_{n-j-1,n+1}^{k,n+1})),
\end{align}\] where we use the notation that \(\det(\mathcal{H}_{j,i}^{n,n+1})=-\det(\mathcal{H}_{i,j}^{n,n+1})\) for \(i<j\). This follows from Lemma 9. Thus we obtain the theorem. ◻
Theorem 4. For any integers \(i_1,\cdots,i_k\ge1\), \(\langle\mathop{\mathrm{tr}}M^{i_1}\cdots\mathop{\mathrm{tr}}M^{i_k}\rangle_c(n)\) is a polynomial of \(a_j(m)\), \(1\le j\le i_1+\cdots+i_k\), \(n-1\le m\le n+i_1+\cdots+i_k\). Here \(a_j(m)\) are defined in 38
.
Proof. First, note that \(A_{0,j}(n)=-a_{j+1}(n)\). Also by comparing the coefficients of \(x^n\) and \(x^{n-1}\) in the three-term recurrence
relation 2 , we have that \[\begin{align} \beta_n=&a_1(n)-a_1(n+1), \\ \gamma_n=&a_2(n)-a_2(n+1)-\beta_na_1(n),
\end{align}\] are polynomials in \(a_j(m)\), \(j=1,2\), \(m=n,n+1\).
Next, \(A_{i,0}(n)=c_{i+1}(n+i)\), where \(c_j(n)\) are defined in 38 . We prove that for any \(j\ge0\), \(c_{j}(n)\) is a polynomial in the coefficients of the orthogonal polynomials by induction on \(j\). For \(j=0\), \(c_0(n)=1\).
Assume that \(c_k(n)\) is a polynomial in the coefficients of the orthogonal polynomials for any \(k<j\). Then by definition, \[\label{cnj} c_{j}(n)=-\sum_{k=0}^{j-1}c_{k}(n)a_{j-k}(n-k)\tag{41}\] is a polynomial in \(a_j(m)\). Hence \(A_{i,0}(n)\) is also a polynomial in \(a_j(m)\), \(1\le j\le i+1\), \(n\le m\le n+i\).
Finally, letting \(A_{0,-1}(n)=-a_0(n)=-1\) and \(A_{-1,0}(n)=c_0(n)=1\), for \(k,l\ge1\), \[T_{k,l}(n)=-A_{0,l-1}(n)A_{k-1,0}(n)+A_{0,l-2}(n-1)A_{k-2,0}(n+1)\gamma_n \label{tij}\tag{42}\] is a polynomial in \(a_j(m)\), \(1\le
j\le\max\{2,k,l\}\), \(n-1\le m\le\max\{n+1,n+k-1\}\). Thus by ?? , \(A_{k,l}(n)\) are polynomials in \(a_j(m)\), \(1\le
j\le k+l+1\), \(n-1\le m\le n+k+l\). Now by comparing the coefficients in ?? , we obtain the theorem. ◻
As a direct corollary, we obtain:
Corollary 2. If the orthogonal polynomials \(\{\pi_l\}_{l\ge0}\) satisfy that for any \(k\ge0\), \(a_k(n)\) is rational (resp.
polynomial, holomorphic, meromorphic) in \(n\), then for any \(i_1,\cdots,i_k\ge1\), \(\langle\mathop{\mathrm{tr}}M^{i_1}\cdots\mathop{\mathrm{tr}}M^{i_k}\rangle_c(n)\) is rational (resp. polynomial, holomorphic, meromorphic) in \(n\).
Example 1 (GUE: Gaussian unitary ensemble). GUE is defined on the space of hermitian matrices \(\mathcal{H}_n\) with measure \[\mathop{}\!\mathrm{d}\mu(\lambda)=e^{-\frac{1}{2}\lambda^2}\mathop{}\!\mathrm{d}\lambda.\] The corresponding orthogonal polynomials are known [50] to be the Hermite polynomials \[\label{orth-herm}
\mathrm{He}_n(\lambda)=\sum_{m=0}^{[\frac{n}{2}]}\frac{(-1)^m}{m!2^m}(n-2m)_{2m+1}\lambda^{n-2m}.\tag{43}\] In this case, the affine coordinates are given in 7 , which are obviously polynomials in \(n\). By 43 and the proof of Theorem 4, we can also see that the affine coordinates are polynomials in \(n\). In addition, by Corollary 2, the connected correlators are also polynomials in \(n\).
Example 2 (LUE: Laguerre unitary ensemble [10], [43], [46], [50], [56], [57]). LUE is defined on the space of \(n\times n\) positive definite hermitian matrices \(\mathcal{H}_n^+\) with measure \[\mathop{}\!\mathrm{d}\mu(\lambda)=\lambda^\alpha
e^{-\lambda}\mathbf{1}_{(0,+\infty)}(\lambda)\mathop{}\!\mathrm{d}\lambda,\] where \(\Re(\alpha)>-1\). The corresponding orthogonal polynomials are known to be the generalized Laguerre polynomials
\[\label{orth-lue} L_n(\alpha;\lambda)=\sum_{k=0}^n\frac{(-1)^k}{k!}(n-k+1)_k(n+\alpha-k+1)_k\lambda^{n-k}.\tag{44}\] In this case, the explicit expressions of affine coordinates
are derived in [46], [58]: \[A_{i,j}^{\mathrm{LUE}}(n)=\frac{(-1)^j}{i!j!(i+j+1)}(n-j)_{i+j+1}(n+\alpha-j)_{i+j+1}.\] Obviously, they are polynomials in \(n\). By 44 and the proof of Theorem 4, we can also see that the affine coordinates are polynomials in \(n\). In addition, by Corollary 2, the connected correlators are also polynomials in \(n\).
Example 3 (JUE: Jacobi unitary ensemble [10], [44], [45], [50], [59]). JUE is defined on the space \(\mathcal{H}_n(0,1)\) with measure \[\mathop{}\!\mathrm{d}\mu(\lambda)=\lambda^\alpha(1-\lambda)^\beta\mathbf{1}_{(0,1)}(\lambda)\mathop{}\!\mathrm{d}\lambda,\] where \(\Re(\alpha),\Re(\beta)>-1\). The corresponding orthogonal
polynomials are known to be the Jacobi polynomials \[\label{orth-jue}
J_n(\alpha,\beta;\lambda)=\frac{n!}{(\alpha+\beta+n+1)_n}\sum_{k=0}^n\binom{n+\alpha}{k}\binom{n+\beta}{n-k}(\lambda-1)^k\lambda^{n-k}.\tag{45}\] In this case, the explicit expressions of affine coordinates are derived in [44], [45] (see [15]–[17]): \[A_{i,j}^\mathrm{JUE}(n)=\frac{(-1)^j}{(i+j+1)i!j!}\frac{(n-j)_{i+j+1}(n+\alpha-j)_{i+j+1}}{(2n+\alpha+\beta-j)_{i+j+1}}.\] Obviously,
they are rational functions in \(n\). By 45 and the proof of Theorem 4, we can also see that the affine coordinates are
rational functions in \(n\). In addition, by Corollary 2, the connected correlators are also rational functions in \(n\).
Example 4 (Atkin polynomials [60]). Let \(j=j(\tau)=q^{-1}+744+196884q+\cdots\) be the modular invariant, where
\(q=e^{2\pi i\tau}\). Let \(\theta:[0,1728]\to[\pi/3,\pi/2]\) be the inverse of \(\theta\mapsto j(e^{i\theta})\). Following [60], consider the measure \[\mathop{}\!\mathrm{d}\mu(j)=\frac{6}{\pi}\theta'(j)\mathbf{1}_{[0,1728]}(j)\mathop{}\!\mathrm{d}j.\] This gives the
so-called Atkin polynomials \(A_n(j)\) with three-term recurrence relation \[jA_n(j)=A_{n+1}(j)+(\iota_{2n}+\iota_{2n+1})A_n(j)+\iota_{2n-1}\iota_{2n}A_{n-1}(j),\qquad n\ge0\] where \[\iota_n=\begin{cases} 0,&n=0,\\ 720,&n=1,\\ 12\left(6+\frac{(-1)^n}{n-1}\right)\left(6+\frac{(-1)^n}{n}\right),&n\ge2. \end{cases}\] For Atkin polynomials, \(a_k(n)\) is rational in
\(n\) for each \(k\ge0\) (see [60]). By the proof of Theorem 4, we can see that the affine coordinates are rational functions in \(n\). In addition, by Corollary 2, the connected correlators are also rational functions in \(n\).
In introduction, we mentioned the KP dual for hermitian matrix models which is characterized by \[\widetilde{\langle s_\rho\rangle}(n)=(-1)^{|\rho|}\langle s_{\rho'}\rangle(-n),\qquad n\ge\max\{2,|\rho|\}.\] Now we
prove Theorem 3.
Proof of Theorem 3. The three-term recurrence coefficients associated to \(\mathop{}\!\mathrm{d}\mu\) read \[\beta_n=\beta,n\ge0,\qquad\gamma_n=\begin{cases} \gamma,&n\ge1,\\ 0,&n=0. \end{cases}\] The associated orthogonal polynomials \(\pi_n\) read \[\pi_n(\lambda)=\sum_{k=0}^na_k(n)\lambda^{n-k},\qquad\lambda^n=\sum_{k=0}^nc_k(n)\pi_{n-k}(\lambda),\] where \(a_0(n)\equiv1,c_0(n)\equiv1\), and for \(k\ge1\),
\[\begin{align} a_k(n)=&\sum_{j=0}^{[k/2]}(-1)^{k-j}\frac{\beta^{k-2j}\gamma^j}{j!(k-2j)!}(n-k+1)_{k-j}, \label{ak}\\
c_k(n)=&(n-k+1)\sum_{j=0}^{[k/2]}\frac{\beta^{k-2j}\gamma^j}{j!(k-2j)!}(n-k+j+2)_{k-j-1}.
\end{align}\tag{46}\] Let \[\widetilde{a}_k(n):=c_k(-n-1+k).\] Then the polynomials \[\widetilde{\pi}_n(\lambda):=\sum_{k=0}^n\widetilde{a}_k(n)\lambda^{n-k},\qquad n\ge0,\]
satisfy the three-term recurrence relation \[\lambda\widetilde{\pi}_n(\lambda)=\widetilde{\pi}_{n+1}(\lambda)+\widetilde{\beta}_n\widetilde{\pi}_n(\lambda)+\widetilde{\gamma}_n\widetilde{\pi}_{n-1}(\lambda),\qquad n\ge0,\]
where for \(n\ge0\), \[\widetilde{\beta}_n=\beta,\qquad\widetilde{\gamma}_n=\begin{cases} \gamma+\delta_{n,1}\gamma,&n\ge1,\\ 0,&n=0. \end{cases}\] The corresponding measure is
\(\mathop{}\!\mathrm{d}\widetilde{\mu}\). Let \[\widetilde{c}_k(n):=a_k(-n-1+k).\] Then \[\lambda^n=\sum_{k=0}^n\widetilde{c}_k(n)\widetilde{\pi}_{n-k}(\lambda).\]
For the hermitian matrix models \(M,\widetilde{M}\) associated to \(\mathop{}\!\mathrm{d}\mu,\mathop{}\!\mathrm{d}\widetilde{\mu}\), the affine coordinates satisfy \[\widetilde{A}_{0,i}(n)=-A_{i,0}(-n),\qquad\widetilde{A}_{i,0}(n)=-A_{0,i}(-n),\qquad |n|\ge i.\] Thus for \(|n|\ge\max\{2,i,j\}\), \[\widetilde{T}_{i,j}(n)=T_{j,i}(-n).\] Therefore, for \(|n|\ge\max\{2,i,j\}\), \[\widetilde{A}_{i,j}(n)=-A_{j,i}(-n).\] Then using Theorem 1, for \(|n|\ge\max\{2,i_1+\cdots+i_k\}\), \[\begin{align}
\widetilde{\langle\mathop{\mathrm{tr}}M^{i_1}\cdots\mathop{\mathrm{tr}}M^{i_k}\rangle}_c(n)&=(-1)^k\langle\mathop{\mathrm{tr}}M^{i_1}\cdots\mathop{\mathrm{tr}}M^{i_k}\rangle_c(-n),\\
\widetilde{\langle\mathop{\mathrm{tr}}M^{i_1}\cdots\mathop{\mathrm{tr}}M^{i_k}\rangle}(n)&=(-1)^k\langle\mathop{\mathrm{tr}}M^{i_1}\cdots\mathop{\mathrm{tr}}M^{i_k}\rangle(-n).
\end{align}\] Now for partition \(\rho=(\rho_1,\cdots,\rho_k)\), \(l(\rho):=k,|\rho|:=\rho_1+\cdots+\rho_k\). Let \(p_\rho(M):=\prod_{j=1}^{l(\rho)}\mathop{\mathrm{tr}}M^{\rho_j}\). Then \[\widetilde{\langle p_\rho\rangle}(n) = (-1)^{l(\rho)}\langle p_\rho\rangle(-n),\qquad |n|\ge\max\{2,|\rho|\}.\] By
Frobenius character formula [61], the Schur polynomial of \(\rho\) reads \[s_\rho=\sum_{|\lambda|=|\rho|}\frac{\chi^\rho_\lambda}{z_\lambda} p_\lambda,\] where \(\chi^\rho_\lambda\) is the character of the irreducible representation of the symmetric group \(S_{|\rho|}\) on \(\lambda\). By [61], \[\chi_\lambda^\rho=\chi_\lambda^{\rho'}(-1)^{|\lambda|-l(\lambda)},\] where \(\rho'\) is the conjugate partition of \(\rho\). Then for \(|n|\ge\max\{2,|\rho|\}\), \[\begin{align} \widetilde{\langle s_\rho\rangle}(n)=&\sum_{|\lambda|=|\rho|}\frac{\chi^\rho_\lambda}{z_\lambda} (-1)^{l(\lambda)}\langle p_\lambda\rangle(-n)\nonumber\\
=&\sum_{|\lambda|=|\rho|}\frac{\chi^{\rho'}_\lambda (-1)^{|\lambda|-l(\lambda)}}{z_\lambda}(-1)^{l(\lambda)}\langle p_\lambda\rangle(-n)\nonumber\\ =&(-1)^{|\rho|}\sum_{|\lambda|=|\rho'|}\frac{\chi^{\rho'}_\lambda}{z_\lambda}\langle
p_\lambda\rangle(-n)\nonumber\\ =&(-1)^{|\rho|}\langle s_{\rho'}\rangle(-n).
\end{align}\] Thus \(\widetilde{M}\) is the conjugate dual of \(M\). ◻
Remark 6 (Relations with Kerov map). The Jacobi fraction of \(\mathop{}\!\mathrm{d}\mu\) writes \[J(t)=\dfrac{1}{1-\beta t-\dfrac{\gamma t^2}{1-\beta t-\dfrac{\gamma
t^2}{\cdots}}}.\] Then \[J(t)=\frac{1}{1-\beta t}E\left(\frac{t}{1-\beta t}\right),\] where \[E(z):=\dfrac{1}{1-\dfrac{\gamma z^2}{1-\dfrac{\gamma z^2}{\cdots}}}.\] Consider the
Kerov map [37], [62]\[\mathcal{K}:J(t)\mapsto
1+t\frac{\mathop{}\!\mathrm{d}}{\mathop{}\!\mathrm{d}t}\log J(t).\] Then let \(z:=\frac{t}{1-\beta t}\), we have \[\begin{align} \mathcal{K}(J)(t)=&1+\beta z+(1+\beta
z)z\frac{\mathop{}\!\mathrm{d}}{\mathop{}\!\mathrm{d}z}\log E(z)\nonumber\\ =&1+\beta z+(1+\beta z)(\mathcal{K}(E)(z)-1)\nonumber\\ =&(1+\beta z)\mathcal{K}(E)(z)\nonumber\\ =&\frac{1}{1-\beta t}\mathcal{K}(E)\left(\frac{t}{1-\beta t}\right).
\end{align}\] Now \(E(z)\) satisfies the relation \[E(z)=\frac{1}{1-\gamma z^2E(z)},\qquad E(0)=1,\] which implies that \[E(z)=\frac{1-\sqrt{1-4\gamma
z^2}}{2\gamma z^2}.\] Then \[\begin{align} \mathcal{K}(E)(z)=&1+z\frac{\mathop{}\!\mathrm{d}}{\mathop{}\!\mathrm{d}z}\log E(z)\nonumber\\ =&\frac{1}{\sqrt{1-4\gamma z^2}}\nonumber\\ =&\frac{1}{1-2\gamma
z^2E(z)}\nonumber\\ =&\dfrac{1}{1-\dfrac{2\gamma z^2}{1-\dfrac{\gamma z^2}{1-\dfrac{\gamma z^2}{\cdots}}}}.
\end{align}\] Thus \[\begin{align} \mathcal{K}(J)(t)=\dfrac{1}{1-\beta t-\dfrac{2\gamma t^2}{1-\beta t-\dfrac{\gamma t^2}{1-\beta t-\dfrac{\gamma t^2}{\cdots}}}}.
\end{align}\] This is exactly the Jacobi fraction of \(\mathop{}\!\mathrm{d}\widetilde{\mu}\).
When \(\beta=0,\gamma=1\), the corresponding Jacobi fractions are the generating functions of Dyck paths and of grand Dyck paths. When \(\beta=1,\gamma=1\), the corresponding Jacobi
fractions are the generating functions of Motzkin paths and of grand Motzkin paths. When \(\beta=2,\gamma=1\), the corresponding Jacobi fractions are related to the generating functions of Catalan numbers of type A and of
type B. When \(\beta=3,\gamma=2\), the corresponding Jacobi fractions are related to the generating functions of big Schröder paths and of central Delannoy paths. These examples are conjectured in [37], [38], which are now proved.
Christoffel E.B. (1858). Über die Gaußische Quadratur und eine Verallgemeinerung derselben. J. Reine Angew. Math. 55, 61–82.
[2]
Darboux G. (1878). Mémoire sur l’approximation des fonctions de très-grands nombres, et sur une classe étendue de développements en série, J. Math. Pures Appl. 4,
5–56.
[3]
Dyson F.J. (1962). Statistical theory of energy levels of complex systems III. J. Math. Phys. 3, 166–175.
[4]
Dyson F.J. (1970). Correlations between the eigenvalues of a random matrix. Comm. Math. Phys. 19, 235–250.
[5]
’t Hooft G. (1974). A planar diagram theory for strong interactions. Nucl. Phys. B. 72, 461–473.
[6]
’t Hooft G. (1974). A two-dimensional model for mesons. Nucl. Phys. B. 75, 461–470.
Bessis D., Itzykson C., Zuber J.B. (1980). Quantum field theory techniques in graphical enumeration. Adv. Appl. Math. 1, 109–157.
[9]
Harer J., Zagier D. (1986). The Euler characteristic of the moduli space of curves. Invent Math 85, 457–485.
[10]
Mehta M.L. (2004). Random Matrices. 3rd Edition. Academic Press, London.
[11]
Deift P. (2000). Orthogonal Polynomials and Random Matrices: A Riemann-Hilbert Approach(Vol. 3). American Mathematical Soc..
[12]
Morozov A., Shakirov S. (2009). Exact 2-point function in Hermitian matrix model. J. High Energy Phys. 2009, 003.
[13]
Dubrovin B., Yang D. (2017). Generating series for GUE correlators. Lett. Math. Phys. 107, 1971–2012.
[14]
Shaw J.C., Tu M.H., Yen H.C. (1992). Matrix Models at Finite N and the KP Hierarchy. Chinese Journ. Phys. 30, 497-507.
[15]
Zhou J. Emergent Geometry and Mirror Symmetry of A Point. arXiv:1507.01679v1.
[16]
Balogh F., Yang D. (2017). Geometric interpretation of Zhou’s explicit formula for the Witten–Kontsevich tau function. Lett. Math. Phys. 107, 1837–1857.
[17]
Harnad J., Ènolskii V.Z. (2011). Schur function expansions of KP \(\tau\)-functions associated to algebraic curves. Russ. Math. Surv.
66, 767–807.
[18]
Sato M. (1983). Soliton equations as dynamical systems on infinite dimensional Grassmann manifold. North-Holland Math. Stud. 81, 259–271.
[19]
Zhou J. Explicit Formula for Witten-Kontsevich Tau-Function. arXiv:1306.5429.
[20]
Zhou J. Hermitian One-Matrix Model and KP Hierarchy. arXiv:1809.07951v1.
[21]
Yang D. (2020). On tau-functions for the Toda lattice hierarchy. Lett. Math. Phys. 110, 555–583.
[22]
Gerasimov A., Marshakov A., Mironov A., Morozov A., Orlov A. (1991). Matrix models of two-dimensional gravity and Toda theory. Nucl. Phys. B. 357, 565–618.
[23]
Martinec E.J. (1991). On the origin of integrability in matrix models. Commun. Math. Phys. 138, 437–449.
[24]
Adler M., van Moerbeke P. (1995). Matrix integrals, Toda symmetries, Virasoro constraints, and orthogonal polynomials. Duke Math. J. 80, 863–911.
[25]
Yin S. (2022). The Generalized Hermitian Matrix Model and Examples from Statistical Physics. Doctoral dissertation, Tsinghua University.
[26]
Yang D. (2024). GUE via Frobenius manifolds. I. From matrix gravity to topological gravity and back. Acta Math. Sin. Engl. Ser. 40, 383–405.
[27]
Ruzza G. (2019). Tau functions: theory and applications to matrix models and enumerative geometry. Doctoral dissertation, SISSA.
[28]
Bergère M. (2006). Correlation functions of complex matrix models. J. Phys. A: Math. Gen. 39, 8749–8773.
[29]
Bergère M., Eynard B. Determinantal formulae and loop equations. arXiv:0901.3273v2.
[30]
König W. (2005). Orthogonal polynomial ensembles in probability theory. Probab. Surv. 2, 385–447.
[31]
Soshnikov A. (2000). Determinantal random point fields. Russian Math. Surveys. 55, 923–975.
[32]
Yang D., Zhou J. (2025). From Toda hierarchy to KP hierarchy. SIGMA. 21, 068.
[33]
Alexandrov A., Bychkov B., Dunin-Barkowski P., Kazarian M., Shadrin S. (2025). KP integrability through the \(x-y\) swap relation. Sel. Math. New Ser.
31, 42.
[34]
Haine L., Horozov E. (1993). Toda Orbits of Laguerre-polynomials and Representations of the Virasoro Algebra. Bull. Sci. Math. 117, 485–518.
[35]
Haine L., Semengue J. (1999). The Jacobi polynomial ensemble and the Painlevé VI equation. J. Math. Phys. 40, 2117–2134.
[36]
Zhou J. Integrable Hierarchies and Mirror Symmetry in Probability Theory and Combinatorics. I. Tau-Functions from Probability Theory and Lattice Path Counting.
Preprint.
[37]
Zhou J. Integrable Hierarchies and Mirror Symmetry in Probability Theory and Combinatorics. II. Mirror Symmetry for Dyck Paths and Motzkin Paths. Preprint.
[38]
Zhou J. Integrable Hierarchies and Mirror Symmetry in Probability Theory and Combinatorics. III. Mirror Symmetry for Schröder Paths and Delannoy Paths.
Preprint.
[39]
Eynard B., Kimura T., Ribault S. Random matrices. arXiv:1510.04430v2.
[40]
Rota G.C. (1964). On the foundations of combinatorial theory I. Theory of Möbius Functions. Z. Wahrscheinlichkeitstheorie verw Gebiete 2. 340–368.
[41]
Dubrovin B., Yang D., Zagier D. (2020). Gromov–Witten invariants of the Riemann sphere. Pure Appl. Math. Q. 16, 153–190.
[42]
Fu A., Yang D. (2022). The matrix-resolvent method to tau-functions for the nonlinear Schrödinger hierarchy. J. Geom. Phys. 179, 104592.
[43]
Gisonni M., Grava T., Ruzza G. (2020). Laguerre Ensemble: Correlators, Hurwitz Numbers and Hodge Integrals. Ann. Henri Poincaré 21, 3285–3339.
[44]
Gisonni M., Grava T., Ruzza G. (2021). Jacobi ensemble, Hurwitz numbers and Wilson polynomials. Lett. Math. Phys. 111, 67.
[45]
Gisonni M., Grava T., Ruzza G. (2023). Correction to: Jacobi ensemble, Hurwitz numbers and Wilson polynomials. Lett. Math. Phys. 113, 86.
[46]
Yang D., Zhou J. (2023). Grothendieck’s dessins d’enfants in a web of dualities. III. J. Phys. A: Math. Theor. 56, 055–201.
[47]
Bertola M., Dubrovin B., Yang D. (2016). Correlation functions of the KdV hierarchy and applications to intersection numbers over \(\overline{\mathcal{M}}_{g,n}\). Phys. D. 327, 30–57.
[48]
Bertola M., Dubrovin B., Yang D. (2021). Simple Lie algebras, Drinfeld–Sokolov hierarchies, and multi-point correlation functions. Mosc. Math. J. 21, 233–270.
[49]
Plancherel, M., Rotach, W. (1929). Sur les valeurs asymptotiques des polynomes d’Hermite \(H_n(x)=(-I)^ne^{\frac{{x^2}}{2}}\frac{{d^n}}{{dx^n}}\left({e^{-\frac{{x^2 }}{2}}}\right)\). Comment. Math. Helv. 1, 227–254.
[50]
Szegö G. (1975). Orthogonal Polynomials. 4th Edition. AMS.
[51]
Forrester P. (1993). The spectrum edge of random matrix ensembles. Nucl. Phys. B. 402, 709–728.
[52]
Dubrovin B., Yang D., Zagier D. (2021). On tau-functions for the KdV hierarchy. Selecta Math. 27, 47 pp.
[53]
Gantmacher F.R. (1980). The Theory of Matrices. Volume 1. Chelsea Publishing Company, New York.
[54]
Berg C. Fibonacci numbers and orthogonal polynomials. arXiv:math/0609283.
[55]
Heinig G., Rost K. (1984). Algebraic Methods for Toeplitz-like Matrices and Operators. Birkhäuser Basel, Basel.
[56]
Laguerre E. (1879). Sur l’intégrale \(\int_x^{\infty}\frac{e^{-x}dx}{x}\). Bull. Soc. Math. France. 7, 72–81.
[57]
Sonine N. (1880). Recherches sur les fonctions cylindriques et le développement des fonctions continues en séries. Math. Ann. 16, 1–80.
[58]
Zhou J. Grothendieck’s dessins d’enfants in a web of dualities. arXiv:1905.10773.
[59]
Jacobi C. (1859). Untersuchungen über die Differentialgleichung der hypergeometrischen Reihe. J. Reine Angew. Math. 56, 149–165.
[60]
Kaneko M., Zagier D. (1998). Supersingular j-invariants, hypergeometric series, and Atkin’s orthogonal polynomials. AMS/IP Stud. in Adv. Math. 7, 97–126.
[61]
Macdonald I.G. (1998). Symmetric Functions and Hall Polynomials. 2nd Edition. Clarendon Press.
[62]
Kerov S.V. (1993). Transition probabilities for continual young diagrams and the Markov moment problem. Funct. Anal. Its Appl. 27, 104–117.