May 04, 2026
We prove a fermionic–bosonic duality relation for the Macdonald index in Argyres–Douglas theories of type \((A_1, D_{2k+1})\), thereby yielding a conjectural fermionic formula due to Andrews et al. Our duality is built upon a new conjugate Bailey pair to be established using techniques from orthogonal polynomials and basic hypergeometric series. In addition, this fermionic formula implies another sum-like expression independently conjectured by Andrews et al. and Kim et al. for the same Macdonald index.
Originating in [1], the Argyres–Douglas theories are families of four-dimensional \(\mathcal{N} = 2\) superconformal field theories to describe the interactions of massless, mutually non-local BPS particles. Geometrically, they are realized by compactifying six-dimensional \(\mathcal{N} = (2,0)\) theories on a Riemann surface with irregular punctures [2], [3]. In particular, these theories have marginal deformations and the Coulomb branch spectrum is of fractional scaling dimension; see [4]–[6] for more background in physics and recent advances on this topic.
Due to the strong-coupling effects, no Lagrangian description exists in any duality frame for Argyres–Douglas theories, and there is a lack of direct ways for computing the superconformal indices, such as Schur index, Hall–Littlewood index, and Macdonald index. For the latter challenge, Buican and Nishinaka [7] made the very first progress on the Macdonald index in \((A_1, A_{2k+1})\) and \((A_1, D_{2k})\) Argyres–Douglas theories. In addition, they [8] proposed an Ansatz for the topological quantum description of Schur index, agreeing with the \(S\)-duality of Argyres–Douglas theories [9], [10], and this construction was subsequently shown to be consistent with \(S^1\) reductions of Argyres–Douglas theories [11]. Alternatively, Córdova and Shao [12] predicted the Schur index according to the correspondence between superconformal field theories and vertex operator algebras, and their results match those in [13]. Along these directions, Song [14] made further headway on the computation of the three types of superconformal indices.
In this work, we focus on the Macdonald index in Argyres–Douglas theories of type \((A_1, D_{2k+1})\). Briefly speaking, the Macdonald index \(\mathcal{I}_{\mathsf{Mac}}^{(2k+1)}(z,t;q)\), carrying fugacities \(z\), \(t\), and \(q\), comes from a trace formula [15] for certain BPS states associated with the \((A_1, D_{2k+1})\) theory. This Macdonald index reduces to the Schur index at the \(q = t\) case, and to the Hall–Littlewood index when \(q = 0\).
There is a bosonic1 sum-like representation for \(\mathcal{I}_{\mathsf{Mac}}^{(2k+1)}(z,t;q)\) [16]: \[\begin{align} \label{eq:Mac-bosonic} \mathcal{I}_{\mathsf{Mac}}^{(2k+1)}(z,t;q) &= \frac{1}{(t,tz^2,tz^{-2};q)_\infty}\sum_{n\ge 0} (-1)^n t^{(k+1)n} q^{kn^2+\binom{n}{2}} \frac{(q^{n+1};q)_n (t^2q^{2n};q)_\infty}{(tq^n;q)_n (tq^{2n+1};q)_\infty}\notag\\ &\quad\times \sum_{j=0}^{2n} \frac{(t;q)_j (t;q)_{2n-j}}{(q;q)_j (q;q)_{2n-j}}z^{2j-2n}. \end{align}\tag{1}\] Throughout, we adopt the conventional \(q\)-Pochhammer symbols: \[\begin{align} (a;q)_\infty := \prod_{k\ge 0} (1-aq^k),\qquad\qquad (a;q)_n := \frac{(a;q)_\infty}{(aq^n;q)_\infty}, \end{align}\] with the compact notation \[\begin{align} (a_1,\ldots,a_M;q)_\infty &:= (a_1;q)_\infty \cdots (a_M;q)_\infty,\\ (a_1,\ldots,a_M;q)_n &:= (a_1;q)_n \cdots (a_M;q)_n. \end{align}\] In addition, we need the \(q\)-binomial coefficients: \[\begin{align} {M\brack N}_q:=\begin{cases} \dfrac{(q;q)_M}{(q;q)_N(q;q)_{M-N}}, & \text{if 0\le N\le M},\\[10pt] 0, & \text{otherwise}. \end{cases} \end{align}\]
Recently, Andrews et al. [17], on the other hand, made the following conjecture2 about the fermionic expression for \(\mathcal{I}_{\mathsf{Mac}}^{(2k+1)}(1,t;q)\).
Conjecture 1 (Andrews et al. [17]). \[\begin{align} \label{eq:ABBST-conj} \mathcal{I}_{\mathsf{Mac}}^{(2k+1)}(1,t;q) \overset{?}{=} \sum_{n_k\ge \cdots\ge n_1 \ge 0} \frac{t^{n_1+\cdots+n_{k}} q^{n_1^2+\cdots + n_{k-1}^2}}{(q;q)_{n_k-n_{k-1}}\cdots (q;q)_{n_2-n_1} (q;q)_{n_1}}\sum_{j=0}^{2n_k} {2n_k\brack j}_q. \end{align}\qquad{(1)}\]
Amongst basic hypergeometric series, especially (finite) Rogers–Ramanujan type identities [18], the fermionic–bosonic duality plays a significant role. In principle, it refers to equalities of the form \[\begin{align} \text{``multiple q-summation} = \text{single q-summation.''} \end{align}\] A typical example is \[\begin{align} \sum_{n_{k}\ge \cdots\ge n_1 \ge 0} \frac{q^{n_1^2+\cdots + n_{k}^2}}{(q;q)_{n_{k}-n_{k-1}}\cdots (q;q)_{n_2-n_1} (q;q)_{n_1}} = \frac{1}{(q;q)_\infty}\sum_{n= -\infty}^\infty (-1)^n q^{(k+1)n^2 + \binom{n}{2}}, \end{align}\] which produces Andrews’ multiple generalization of the first Rogers–Ramanujan identity [19] after applying Jacobi’s triple product [20] to the bosonic sum on the right-hand side.
Now the first objective of this paper is to establish the following fermionic dual for 1 : \[\begin{align} \label{eq:ABBST-z} \mathcal{I}_{\mathsf{Mac}}^{(2k+1)}(z,t;q) = \sum_{n_k\ge \cdots\ge n_1 \ge 0} \frac{t^{n_1+\cdots+n_{k}} q^{n_1^2+\cdots + n_{k-1}^2}}{(q;q)_{n_k-n_{k-1}}\cdots (q;q)_{n_2-n_1} (q;q)_{n_1}}\sum_{j=0}^{2n_k} {2n_k\brack j}_q z^{2j - 2n_k}, \end{align}\tag{2}\] which reduces to the conjectural formula of Andrews et al. [17], that is, ?? , at \(z=1\). More precisely, we show the fermionic–bosonic duality:
Theorem 2. For every \(k\ge 1\), \[\begin{align} \label{eq:fermionic-bosonic-main} &\sum_{n_k\ge \cdots\ge n_1 \ge 0} \frac{t^{n_1+\cdots+n_{k}} q^{n_1^2+\cdots + n_{k-1}^2}}{(q;q)_{n_k-n_{k-1}}\cdots (q;q)_{n_2-n_1} (q;q)_{n_1}}\sum_{j=0}^{2n_k} {2n_k\brack j}_q z^{2j - 2n_k}\notag\\ &\qquad\qquad = \frac{1}{(t,tz^2,tz^{-2};q)_\infty}\sum_{n\ge 0} (-1)^n t^{(k+1)n} q^{kn^2+\binom{n}{2}} \frac{(q^{n+1};q)_n (t^2q^{2n};q)_\infty}{(tq^n;q)_n (tq^{2n+1};q)_\infty}\notag\\ &\qquad\qquad\quad\times \sum_{j=0}^{2n} \frac{(t;q)_j (t;q)_{2n-j}}{(q;q)_j (q;q)_{2n-j}}z^{2j-2n}. \end{align}\tag{3}\]
Andrews et al. [21] and Kim et al. [16] independently conjectured another fermionic sum for \(\mathcal{I}_{\mathsf{Mac}}^{(2k+1)}(z,t;q)\). Our second purpose is to show that this formula, as stated below using an equivalent form:3 \[\begin{align} \label{eq:KKS-24626} \mathcal{I}_{\mathsf{Mac}}^{(2k+1)}(z,t;q) &= (t,q;q)_\infty^k \sum_{\substack{r_1,\ldots,r_k\ge 0\\s_1,\ldots,s_k\ge 0}} \frac{t^{\sum_{i=1}^k s_i} q^{\sum_{i=1}^k r_i(s_{i-1}+s_i+1)}}{(t,q;q)_{r_1}\cdots (t,q;q)_{r_k} (q;q)_{s_1}^2 \cdots (q;q)_{s_k}^2}\notag\\ &\quad\times \sum_{u_1,u_2\ge 0} {s_k\brack u_1}_q {s_k\brack u_2}_q z^{2u_1-2u_2}, \end{align}\tag{4}\] with \(s_0:=0\), can be transformed to that in 2 , thereby linking all three expressions for the Macdonald index in Argyres–Douglas theories of type \((A_1, D_{2k+1})\). In other words, we prove the following identity:
Theorem 3. For every \(k\ge 1\), \[\begin{align} \label{eq:KKS-24626-id} &\sum_{n_k\ge \cdots\ge n_1 \ge 0} \frac{t^{n_1+\cdots+n_{k}} q^{n_1^2+\cdots + n_{k-1}^2}}{(q;q)_{n_k-n_{k-1}}\cdots (q;q)_{n_2-n_1} (q;q)_{n_1}}\sum_{j=0}^{2n_k} {2n_k\brack j}_q z^{2j - 2n_k}\notag\\ &\qquad\qquad = (t,q;q)_\infty^k \sum_{\substack{r_1,\ldots,r_k\ge 0\\s_1,\ldots,s_k\ge 0}} \frac{t^{\sum_{i=1}^k s_i} q^{\sum_{i=1}^k r_i(s_{i-1}+s_i+1)}}{(t,q;q)_{r_1}\cdots (t,q;q)_{r_k} (q;q)_{s_1}^2 \cdots (q;q)_{s_k}^2}\notag\\ &\qquad\qquad\quad\times \sum_{u_1,u_2\ge 0} {s_k\brack u_1}_q {s_k\brack u_2}_q z^{2u_1-2u_2}, \end{align}\tag{5}\] where we put \(s_0:=0\).
This paper is structured as follows. First, in Section 2, we review Bailey’s method on summations. Then a new conjugate Bailey pair is constructed in Section 3. In this course, techniques from orthogonal polynomials and basic hypergeometric series are needed, including a basic hypergeometric evaluation to be established in Appendix 8. Next, we choose a proper seed Bailey pair in Section 4 to conclude a strengthening of Theorem 2. In Section 5, we prove Theorem 3. Finally, we provide a well-poised generalization of our new conjugate Bailey pair as closing remarks in Section 6.
In this section, we briefly recall Bailey’s theory on summations [22], which is streamlined in [23]. Let \((u_n)_{n\ge 0}\) and \((v_n)_{n\ge 0}\) be generic sequences. Subject to suitable convergence conditions, we assume that sequences \((\alpha_n)_{n\ge 0}\), \((\beta_n)_{n\ge 0}\), \((\gamma_n)_{n\ge 0}\), and \((\delta_n)_{n\ge 0}\) satisfy the following relations for all \(n\ge 0\): \[\begin{align} \beta_n = \sum_{l=0}^n u_{n-l} v_{n+l} \alpha_l, \end{align}\] and \[\begin{align} \gamma_n = \sum_{l\ge n} u_{l-n} v_{l+n} \delta_l. \end{align}\] Then Bailey’s transform [23] asserts that \[\begin{align} \label{eq:Bailey-trans} \sum_{n\ge 0} \alpha_n \gamma_n = \sum_{n\ge 0} \beta_n \delta_n. \end{align}\tag{6}\] A typical choice of the sequences \(u_n\) and \(v_n\) is \[\begin{align} u_n = \frac{1}{(q;q)_n} \qquad \text{and} \qquad v_n = \frac{1}{(tq;q)_n}, \end{align}\] where \(t\) is an indeterminate. We say sequences \(\alpha_n\) and \(\beta_n\) form a Bailey pair relative to \((t,q)\) [23] if for each \(n\ge 0\), \[\begin{align} \label{eq:Bailey-beta} \beta_n = \sum_{l=0}^n \frac{\alpha_l}{(q;q)_{n-l} (tq; q)_{n+l}}. \end{align}\tag{7}\] We may boost this Bailey pair to a Bailey chain [23]. To be precise, for each \(k\ge 1\), the sequences \[\begin{align} \label{eq:Bailey-chain-k-a} \alpha_n^{(k)} &:= \frac{(b_1^{-1},c_1^{-1},\ldots,b_k^{-1},c_k^{-1};q)_n (b_1c_1\cdots b_kc_k)^n}{(b_1tq,c_1tq,\ldots,b_ktq,c_ktq;q)_n}\cdot (tq)^{kn} \alpha_n \end{align}\tag{8}\] and \[\begin{align} \label{eq:Bailey-chain-k-b} \beta_n^{(k)} &:= \sum_{0 \le n_0\le \cdots\le n_{k-1}\le n} \frac{(tq)^{n_0+\cdots+n_{k-1}} \beta_{n_0}}{(q;q)_{n-n_{k-1}}(q;q)_{n_{k-1}-n_{k-2}}\cdots (q;q)_{n_1-n_0}}\notag\\ &\quad\;\times \frac{(b_kc_ktq;q)_{n-n_{k-1}}(b_{k-1}c_{k-1}tq;q)_{n_{k-1}-n_{k-2}}\cdots (b_1c_1tq;q)_{n_1-n_0}}{(b_1tq,c_1tq;q)_{n_1}\cdots (b_{k-1}tq,c_{k-1}tq;q)_{n_{k-1}}(b_{k}tq,c_{k}tq;q)_{n}}\notag\\ &\quad\;\times (b_1^{-1},c_1^{-1};q)_{n_0}\cdots (b_k^{-1},c_k^{-1};q)_{n_{k-1}}(b_1 c_1)^{n_0} \cdots (b_kc_k)^{n_{k-1}} \end{align}\tag{9}\] also form a Bailey pair relative to \((t,q)\). This comes from iterations of Bailey’s lemma [23]. Finally, the conjugate Bailey pair \((\gamma_n,\delta_n)\) relative to \((t,q)\) is such that \[\begin{align} \label{eq:Bailey-conj-gamma} \gamma_n = \sum_{l\ge n} \frac{\delta_l}{(q;q)_{l-n} (tq; q)_{l+n}}. \end{align}\tag{10}\] Under different choices of Bailey and conjugate Bailey pairs, we are taken to a realm of summation formulas, with Bailey’s transform recalled.
Conjugate Bailey pairs are not as well explored as Bailey pairs; see the work of Schilling and Warnaar [24] as an instance. Now our task is to establish the following new conjugate Bailey pair.
Theorem 4. The sequences \[\begin{align} \gamma_n = \frac{t^n (q;q)_{2n} (t^2;q)_\infty}{(t^2;q)_{2n} (t,tq,tz,tz^{-1};q)_\infty} \sum_{j=0}^{2n} \frac{(t;q)_j (t;q)_{2n-j}}{(q;q)_j (q;q)_{2n-j}}z^{j-n} \end{align}\] and \[\begin{align} \delta_n = t^n \sum_{j=0}^{2n} {2n\brack j}_q z^{j-n} \end{align}\] form a conjugate Bailey pair relative to \((t,q)\).
Proof. Let \(z:=\operatorname{e}^{\operatorname{i}\theta}\). Recall the continuous \(q\)-Hermite polynomials [25] (or essentially the Rogers–Szegő polynomials; see [26]): \[\begin{align} \label{eq:q-Hermite-def} H_n(z;q) := \sum_{j=0}^n {n\brack j}_q z^{n-2j}, \end{align}\tag{11}\] and the continuous \(q\)-ultraspherical polynomials [25]: \[\begin{align} \label{eq:q-ultraspherical-def} C_n(z,t;q) := \sum_{j=0}^n \frac{(t;q)_j (t;q)_{n-j}}{(q;q)_j (q;q)_{n-j}} z^{n-2j}. \end{align}\tag{12}\]
It is a standard result that these continuous \(q\)-Hermite polynomials form an orthogonal basis of symmetric Laurent polynomials in \(z\). As such, we may assume that \(c_{n,l}\) are coefficients in the infinite expansion of the symmetric bilateral even series \[\begin{align} \label{eq:C-expansion} \frac{C_{2n}(z,t;q)}{(t z^2, t z^{-2};q)_\infty} = \sum_{l\ge 0} c_{n,l}\, H_{2l}(z;q). \end{align}\tag{13}\] In particular, we do not have odd-indexed \(q\)-Hermite polynomials in this expansion because they are odd functions in \(z\). Recall the orthogonality of continuous \(q\)-Hermite polynomials [25]: \[\begin{align} \int_0^\pi H_n(z;q) H_{n'}(z;q) (z^2, z^{-2};q)_\infty \operatorname{d}\theta = \frac{2\pi (q; q)_n}{(q; q)_\infty} \delta_{n,n'}, \end{align}\] where \(\delta_{n,n'}\) is the Kronecker delta. Therefore, \[\begin{align} c_{n,l} = \frac{(q;q)_\infty}{2 (q;q)_{2l}} \int_0^\pi C_{2n}(z,t;q) H_{2l}(z;q) \frac{(z^2, z^{-2};q)_\infty}{(t z^2, t z^{-2};q)_\infty} \frac{\operatorname{d}\theta}{\pi}. \end{align}\] Using 11 for \(H_{2l}(z;q)\), we further write \(c_{n,l}\) as \[\begin{align} \label{eq:c-expression-int} c_{n,l} = \frac{(q;q)_\infty}{2 (q;q)_{2l}} \sum_{j=0}^{2l} {2l\brack j}_q \int_0^\pi z^{2l-2j} C_{2n}(z,t;q) \frac{(z^2, z^{-2};q)_\infty}{(t z^2, t z^{-2};q)_\infty} \frac{\operatorname{d}\theta}{\pi}. \end{align}\tag{14}\]
For the integral \[\begin{align} \label{eq:Imn-def} I_{m,n} := \int_0^\pi z^{-2 (m+n)} C_{2n}(z,t;q) \frac{(z^2, z^{-2};q)_\infty}{(t z^2, t z^{-2};q)_\infty} \frac{\operatorname{d}\theta}{\pi}, \end{align}\tag{15}\] we apply 12 for \(C_{2n}(z,t;q)\) and then invoke Ramanujan’s \({}_{1} \psi_1\) sum [20]: \[\begin{align} \sum_{k=-\infty}^\infty \frac{(a;q)_k z^k}{(b;q)_k} = \frac{(q,b/a,az,q/(az);q)_\infty}{(b,q/a,z,b/(az);q)_\infty} \end{align}\] with \((a,b,z)\mapsto (t^{-1},t,tz^2)\) so that \[\begin{align} \frac{(z^2, z^{-2};q)_\infty}{(t z^2, t z^{-2};q)_\infty} = \frac{(t,tq;q)_\infty (1-z^{-2})}{(q,t^2;q)_\infty} \sum_{k=-\infty}^{\infty} \frac{(t^{-1};q)_k t^k z^{2k}}{(t;q)_k}. \end{align}\] Hence, \(I_{m,n}\) equals \[\begin{align} \frac{(t,tq;q)_\infty}{(q,t^2;q)_\infty} \sum_{j=0}^{2n} \frac{(t;q)_j (t;q)_{2n-j}}{(q;q)_j (q;q)_{2n-j}} \sum_{k=-\infty}^{\infty} \frac{(t^{-1};q)_k t^k}{(t;q)_k} \int_0^\pi z^{-2 (m+j-k)} (1-z^{-2}) \frac{\operatorname{d}\theta}{\pi}, \end{align}\] yielding the relation \[\begin{align} \label{eq:I-expression} I_{m,n} = \frac{(t,tq;q)_\infty}{(q,t^2;q)_\infty} \big(S_{m,n} - S_{m+1,n}\big), \end{align}\tag{16}\] where \[\begin{align} S_{d,n} := \sum_{j=0}^{2n} \frac{(t;q)_j (t;q)_{2n-j} (t^{-1};q)_{j+d} t^{j+d}}{(q;q)_j (q;q)_{2n-j} (t;q)_{j+d}}. \end{align}\]
Now we write \(S_{d,n}\) in terms of a \({}_3 \phi_{2}\) series \[\begin{align} S_{d,n} = \frac{(t;q)_{2n} (t^{-1};q)_d t^d}{(q;q)_{2n} (t;q)_d}\, {}_3 \phi_{2}\left(\begin{matrix} t,t^{-1}q^d,q^{-2n}\\ tq^d,t^{-1}q^{1-2n} \end{matrix};q,q\right), \end{align}\] and further evaluate it as \[\begin{align} \label{eq:S-expression} S_{d,n} = \frac{(t^2;q)_{2n} (q^{d};q)_{2n} (t^{-1};q)_d t^d}{(q;q)_{2n} (t;q)_{2n+d}}, \end{align}\tag{17}\] where we have applied the \(q\)-Pfaff–Saalschütz sum [20]: \[\begin{align} {}_3 \phi_{2}\left(\begin{matrix} a,b,q^{-n}\\ c,abq^{1-n}/c \end{matrix};q,q\right) = \frac{(c/a,c/b;q)_n}{(c,c/(ab);q)_n}. \end{align}\]
Recall from 14 and 16 that \[\begin{align} c_{n,l} &= \frac{(q;q)_\infty}{2 (q;q)_{2l}} \sum_{j=0}^{2l} {2l\brack j}_q I_{j-l-n,n}\\ &= \frac{(t,tq;q)_\infty}{2(t^2;q)_\infty (q;q)_{2l}} \sum_{j=0}^{2l} {2l\brack j}_q \big(S_{j-l-n,n} - S_{j-l-n+1,n}\big). \end{align}\] For every \(j\) with \(0\le j\le 2l\), write \(j' := 2l-j\). Our key observation is that for \(j\) in this range, it is always true that \[\begin{align} {2l\brack j}_q S_{j-l-n,n} = - {2l\brack j'}_q S_{j'-l-n+1,n}. \end{align}\] Sum the above for \(j\) with \(0\le j\le 2l\) so that \(j'\) also runs from \(0\) to \(2l\). Thus, \[\begin{align} \label{eq:S-symmetry} \sum_{j=0}^{2l} {2l\brack j}_q S_{j-l-n,n} = - \sum_{j=0}^{2l} {2l\brack j}_q S_{j-l-n+1,n}. \end{align}\tag{18}\] Consequently, \[\begin{align} c_{n,l} &= \frac{(t,tq;q)_\infty}{(t^2;q)_\infty (q;q)_{2l}} \sum_{j=0}^{2l} {2l\brack j}_q S_{j-l-n,n}\\ &= \frac{(t,tq;q)_\infty (t^2;q)_{2n}}{(t^2;q)_\infty (q;q)_{2n}} \sum_{j=0}^{2l} \frac{(q^{j-l-n};q)_{2n} (t^{-1};q)_{j-l-n} t^{j-l-n}}{(q;q)_j (q;q)_{2l-j} (t;q)_{j-l+n}}, \end{align}\] where 17 has been utilized. It then follows from 27 that \[\begin{align} c_{n,l} = \frac{(t,tq;q)_\infty (t^2;q)_{2n} t^{l-n}}{(t^2;q)_\infty (q;q)_{2n} (q;q)_{l-n} (tq;q)_{l+n}}. \end{align}\] In particular, \[\begin{align} c_{n,l} = 0, \qquad (0\le l\le n-1). \end{align}\]
Finally, substituting these \(c_{n,l}\) into 13 , we have \[\begin{align} \frac{t^n (q;q)_{2n} (t^2;q)_\infty}{(t^2;q)_{2n} (t,tq,tz^2,tz^{-2};q)_\infty} C_{2n}(z,t;q) = \sum_{l\ge n} \frac{t^l}{(q;q)_{l-n} (tq;q)_{l+n}} H_{2l}(z;q). \end{align}\] Recalling 10 , this exactly confirms the claimed conjugate Bailey pair under the change of variables \(z \mapsto z^{-1/2}\). ◻
Corollary 1. For any Bailey pair \((\alpha_n,\beta_n)\) relative to \((t,q)\), we have \[\begin{align} \label{eq:Bailey-special} &\sum_{n\ge 0} \beta_n\cdot t^n \sum_{j=0}^{2n} {2n\brack j}_q z^{j-n}\notag\\ &\qquad = \frac{(t^2;q)_\infty}{(t,tq,tz,tz^{-1};q)_\infty} \sum_{n\ge 0} \alpha_n \cdot \frac{t^n (q;q)_{2n}}{(t^2;q)_{2n}} \sum_{j=0}^{2n} \frac{(t;q)_j (t;q)_{2n-j}}{(q;q)_j (q;q)_{2n-j}}z^{j-n}. \end{align}\tag{19}\]
For the moment, we are ready to close our proof of the fermionic–bosonic duality in Theorem 2. Starting from the most elementary Bailey pair relative to \((t,q)\) [23]: \[\begin{align} \alpha_n^{(0)} = \frac{(-1)^n q^{\binom{n}{2}} (1-t q^{2n}) (t;q)_n}{(1-t) (q;q)_n}, \qquad\qquad\qquad \beta_n^{(0)} = \delta_{n,0}, \end{align}\] we produce a chain of Bailey pairs in light of 8 and 9 : \[\begin{align} \alpha_n^{(1)} &= \frac{(-1)^n t^n q^{n+\binom{n}{2}} (1-t q^{2n}) (t;q)_n}{(1-t) (q;q)_n}\cdot \frac{(b_1^{-1},c_1^{-1};q)_n (b_1 c_1)^n}{(b_1tq,c_1tq;q)_n},\\ \beta_n^{(1)} &= \frac{1}{(q;q)_n}\cdot \frac{(b_1c_1tq;q)_n}{(b_1tq,c_1tq;q)_n}, \end{align}\] and for \(k\ge 2\), \[\begin{align} \alpha_n^{(k)} &= \frac{(-1)^n t^{kn} q^{kn+\binom{n}{2}} (1-t q^{2n}) (t;q)_n}{(1-t) (q;q)_n}\\ &\quad \times \frac{(b_1^{-1},c_1^{-1},\ldots,b_k^{-1},c_k^{-1};q)_n (b_1c_1\cdots b_kc_k)^n}{(b_1tq,c_1tq,\ldots,b_ktq,c_ktq;q)_n},\\ \beta_n^{(k)} &= \sum_{0 \le n_1\le \cdots\le n_{k-1}\le n} \frac{(tq)^{n_1+\cdots+n_{k-1}}}{(q;q)_{n-n_{k-1}}(q;q)_{n_{k-1}-n_{k-2}}\cdots (q;q)_{n_2-n_1} (q;q)_{n_1}}\\ &\quad\times \frac{(b_kc_ktq;q)_{n-n_{k-1}}(b_{k-1}c_{k-1}tq;q)_{n_{k-1}-n_{k-2}}\cdots (b_2c_2tq;q)_{n_2-n_1} (b_1c_1tq;q)_{n_1}}{(b_1tq,c_1tq;q)_{n_1}\cdots (b_{k-1}tq,c_{k-1}tq;q)_{n_{k-1}}(b_{k}tq,c_{k}tq;q)_{n}}\notag\\ &\quad\times (b_2^{-1},c_2^{-1};q)_{n_1}\cdots (b_k^{-1},c_k^{-1};q)_{n_{k-1}} (b_2 c_2)^{n_1} \cdots (b_kc_k)^{n_{k-1}}. \end{align}\] Apply them to 19 and make the substitution \(z\mapsto z^{2}\). We immediately arrive at the following strengthening of Theorem 2, which reduces to 3 at the limiting case where all \(b\)’s and \(c\)’s go to zero.
Theorem 5. For every \(k\ge 1\), \[\begin{align} &\sum_{n_k\ge \cdots\ge n_1 \ge 0} \frac{t^{n_1+\cdots+n_{k}} q^{n_1+\cdots+n_{k-1}}}{(q;q)_{n_k-n_{k-1}}\cdots (q;q)_{n_2-n_1} (q;q)_{n_1}} \sum_{j=0}^{2n_k} {2n_k\brack j}_q z^{2j-2n_k}\notag\\ &\times \frac{(b_kc_ktq;q)_{n_k-n_{k-1}}\cdots (b_2c_2tq;q)_{n_2-n_1} (b_1c_1tq;q)_{n_1}}{(b_1tq,c_1tq;q)_{n_1}\cdots (b_{k}tq,c_{k}tq;q)_{n_k}}\notag\\ &\times (b_2^{-1},c_2^{-1};q)_{n_1}\cdots (b_k^{-1},c_k^{-1};q)_{n_{k-1}} (b_2 c_2)^{n_1} \cdots (b_kc_k)^{n_{k-1}}\notag\\ &\qquad = \frac{1}{(t,tz^2,tz^{-2};q)_\infty}\sum_{n\ge 0} (-1)^n t^{(k+1)n} q^{kn+\binom{n}{2}} \frac{(q^{n+1};q)_n (t^2q^{2n};q)_\infty}{(tq^n;q)_n (tq^{2n+1};q)_\infty}\notag\\ &\qquad\quad\times \sum_{j=0}^{2n} \frac{(t;q)_j (t;q)_{2n-j} z^{2j-2n}}{(q;q)_j (q;q)_{2n-j}} \cdot \frac{(b_1^{-1},c_1^{-1},\ldots,b_k^{-1},c_k^{-1};q)_n (b_1c_1\cdots b_kc_k)^n}{(b_1tq,c_1tq,\ldots,b_ktq,c_ktq;q)_n}. \end{align}\]
The goal of this section is to show the equivalence between 2 and 4 as stated in Theorem 3. To begin with, we single out summations over \(r\)’s on the right-hand side of 5 . In particular, for each \(i\) with \(1\le i\le k\), \[\begin{align} \sum_{r_i\ge 0} \frac{q^{r_i(s_{i-1}+s_i+1)}}{(t,q;q)_{r_i}} &= \lim_{\tau\to 0}{}_{2} \phi_{1} \left(\begin{matrix} 0,\tau\\ t \end{matrix};q,q^{s_{i-1}+s_i+1}\right)\\ &= \lim_{\tau\to 0} \frac{(\tau,0;q)_\infty}{(t,q^{s_{i-1}+s_i+1};q)_\infty} {}_{2} \phi_{1} \left(\begin{matrix} t/\tau,q^{s_{i-1}+s_i+1}\\ 0 \end{matrix};q,\tau\right), \end{align}\] where we have applied Heine’s first transformation [20]: \[\begin{align} {}_{2} \phi_{1} \left(\begin{matrix} a,b\\ c \end{matrix};q,z\right) = \frac{(b,az;q)_\infty}{(c,z;q)_\infty} {}_{2} \phi_{1} \left(\begin{matrix} c/b,z\\ az \end{matrix};q,b\right). \end{align}\] Thus, \[\begin{align} (t,q;q)_\infty \sum_{r_i\ge 0} \frac{q^{r_i(s_{i-1}+s_i+1)}}{(t,q;q)_{r_i}} = \sum_{r_i\ge 0} \frac{(-1)^{r_i} t^{r_i} q^{\binom{r_i}{2}} (q;q)_{s_{i-1}+s_i+r_i}}{(q;q)_{r_i}}. \end{align}\] Before substituting the above relations into the right-hand side of 5 , we change the indices for each \(i\) with \(1\le i\le k\), \[\begin{align} n_i := r_i + s_i. \end{align}\] It follows that \[\begin{align} \label{eq:KKS-24626-middle-step} \operatorname{RHS}\eqref{eq:KKS-24626-id} = \sum_{n_1,\ldots,n_k\ge 0} t^{\sum_{i=1}^k n_i} (q;q)_{n_1} B_{n_k}(z;q) \prod_{i=1}^{k-1} \Phi_{n_i,n_{i+1}}(q), \end{align}\tag{20}\] where \[\begin{align} B_{n}(z;q) := \sum_{s = 0}^n \frac{(-1)^{n-s} q^{\binom{n-s}{2}}}{(q;q)_{s}^2 (q;q)_{n-s}}\sum_{u_1,u_2\ge 0} {s\brack u_1}_q {s\brack u_2}_q z^{2u_1-2u_2}, \end{align}\] and \[\begin{align} \Phi_{n,n'}(q) := \sum_{s = 0}^n \frac{(-1)^{n-s} q^{\binom{n-s}{2}}(q;q)_{s+n'}}{(q;q)_{s}^2(q;q)_{n-s}}. \end{align}\]
For \(B_{n}(z;q)\), we write it in terms of the continuous \(q\)-Hermite polynomial 11 : \[\begin{align} B_n(z;q) = \sum_{s = 0}^n \frac{(-1)^{n-s} q^{\binom{n-s}{2}}}{(q;q)_{s}^2 (q;q)_{n-s}} H_{s}(z;q)^2, \end{align}\] where we have noted the symmetry \(H_n(z;q) = H_n(z^{-1};q)\). Now we require the linearization of products of \(H_n(z;q)\) [25]: \[\begin{align} H_m(z;q) H_n(z;q) = \sum_{l = 0}^{\min(m,n)} \frac{(q;q)_m (q;q)_n}{(q;q)_l (q;q)_{m-l} (q;q)_{n-l}} H_{m+n-2l}(z;q). \end{align}\] It follows that \[\begin{align} B_n(z;q) = \sum_{s = 0}^n \frac{(-1)^{n-s} q^{\binom{n-s}{2}}}{(q;q)_{s}^2 (q;q)_{n-s}} \sum_{l = 0}^s \frac{(q;q)_s^2}{(q;q)_l (q;q)_{s-l}^2} H_{2s-2l}(z;q). \end{align}\] Substituting \(l\mapsto s-j\) and interchanging the summations, we have \[\begin{align} B_n(z;q) = \sum_{j = 0}^n \frac{H_{2j}(z;q)}{(q;q)_j^2} \sum_{s = j}^n \frac{(-1)^{n-s} q^{\binom{n-s}{2}}}{(q;q)_{n-s} (q;q)_{s-j}}. \end{align}\] For the inner sum, we further make the change of indices \(s\mapsto s+j\) and write it as a \({}_{1} \phi_{0}\) series so that \[\begin{align} B_n(z;q) = \sum_{j = 0}^n \frac{(-1)^{n-j} q^{\binom{n}{2}+\binom{j}{2}+j(1-n)} H_{2j}(z;q)}{(q;q)_j^2} {}_{1} \phi_{0} \left(\begin{matrix} q^{-(n-j)}\\ - \end{matrix};q,q\right), \end{align}\] while this \({}_{1} \phi_{0}\) series becomes the Kronecker delta \(\delta_{j,n}\) according to the \(q\)-binomial theorem [20]: \[\begin{align} {}_{1} \phi_{0} \left(\begin{matrix} q^{-n}\\ - \end{matrix};q,z\right) = (zq^{-n};q)_n. \end{align}\] Therefore, by further using the symmetry \(H_n(z;q) = H_n(z^{-1};q)\), \[\begin{align} \label{eq:B-eva} B_n(z;q) = \frac{H_{2n}(z;q)}{(q;q)_n^2} = \frac{1}{(q;q)_n^2} \sum_{j=0}^{2n} {2n\brack j}_q z^{2j-2n}. \end{align}\tag{21}\]
For \(\Phi_{n,n'}(q)\), we have \[\begin{align} \Phi_{n,n'}(q) &= \frac{(-1)^n q^{\binom{n}{2}} (q;q)_{n'}}{(q;q)_n} {}_{2} \phi_{1} \left(\begin{matrix} q^{n'+1},q^{-n}\\ q \end{matrix};q,q\right)\\ &= \frac{(-1)^n q^{\binom{n}{2}} (q;q)_{n'}}{(q;q)_n} \cdot \frac{q^{n(n'+1)} (q^{-n'};q)_n}{(q;q)_n}, \end{align}\] where for the evaluation of the \({}_{2} \phi_{1}\) series, we have used the second \(q\)-Chu–Vandermonde sum [20]: \[\begin{align} {}_{2} \phi_{1} \left(\begin{matrix} a,q^{-n}\\ c \end{matrix};q,q\right) = \frac{a^n (c/a;q)_n}{(c;q)_n}. \end{align}\] Hence, \[\begin{align} \label{eq:Phi-eva} \Phi_{n,n'}(q) = \frac{q^{n^2}(q;q)_{n'}}{(q;q)_{n}} {n'\brack n}_q. \end{align}\tag{22}\]
Finally, applying 21 and 22 to 20 , we find that \[\begin{align} \operatorname{RHS}\eqref{eq:KKS-24626-id} = \sum_{n_k\ge \cdots\ge n_1 \ge 0} \frac{t^{n_1+\cdots+n_{k}} q^{n_1^2+\cdots + n_{k-1}^2}}{(q;q)_{n_k-n_{k-1}}\cdots (q;q)_{n_2-n_1} (q;q)_{n_1}}\sum_{j=0}^{2n_k} {2n_k\brack j}_q z^{2j - 2n_k}, \end{align}\] as claimed.
In [27], Andrews extended the scheme of Bailey pairs to a well-poised version. To be specific, sequences \(\alpha'_n\) and \(\beta'_n\) form a WP-Bailey pair relative to \((s,t,q)\) [27] if for each \(n\ge 0\), \[\begin{align} \label{eq:WP-Bailey-beta} \beta'_n = \sum_{l=0}^n \frac{(st^{-1};q)_{n-l} (s; q)_{n+l}}{(q;q)_{n-l} (tq; q)_{n+l}}\,\alpha'_l; \end{align}\tag{23}\] likewise, sequences \(\gamma'_n\) and \(\delta'_n\) form a conjugate WP-Bailey pair relative to \((s,t,q)\) if for each \(n\ge 0\), \[\begin{align} \label{eq:WP-Bailey-conj-gamma} \gamma'_n = \sum_{l\ge n} \frac{(st^{-1};q)_{l-n} (s;q)_{l+n}}{(q;q)_{l-n} (tq; q)_{l+n}}\,\delta'_l. \end{align}\tag{24}\] It is notable that such a well-poised extension reduces to the original form by taking \(s=0\).
Toward this direction, we may lift the conjugate Bailey pair in Theorem 4 to the following WP-extension. This conjugate WP-Bailey pair further leads us to a generalization of the relation in Theorem 5 by applying the WP-Bailey pairs acquired from iterations of [27] starting with the seed case [27]; we will not record this generalized identity due to its oversized expression. It remains unknown to understand the physical meaning conveyed by such a generalization, especially the new parameter \(s\).
Theorem 6. The sequences \[\begin{align} \gamma'_n = \frac{t^n (q;q)_{2n} (t^2, sz, sz^{-1};q)_\infty}{(t^2;q)_{2n} (t,tq,tz,tz^{-1};q)_\infty} \sum_{j=0}^{2n} \frac{(t;q)_j (t;q)_{2n-j}}{(q;q)_j (q;q)_{2n-j}}z^{j-n} \end{align}\] and \[\begin{align} \delta'_n = \frac{t^n (1-s q^{2n}) (q;q)_{2n} (s^2;q)_\infty}{(1-s) (s^2;q)_{2n} (s,sq;q)_\infty} \sum_{j=0}^{2n} \frac{(s;q)_j (s;q)_{2n-j}}{(q;q)_j (q;q)_{2n-j}}z^{j-n} \end{align}\] form a conjugate WP-Bailey pair relative to \((s,t,q)\).
Proof. See Appendix 9. ◻
Shane Chern was supported by the Austrian Science Fund (no. 10.55776/F1002). We are grateful to Ranveer Kumar Singh for informing us of their work [21], and to George Andrews, Matthew Buican, and Ole Warnaar for useful feedback. We also acknowledge conversations with GPT-5.5 Pro that suggested the bosonic formula for the Macdonald index.
Here we examine the equivalence between the fermionic sum in 4 and that in Andrews et al. [21] or Kim et al. [16]. Note that [16], with \(z\) rescaled to \(z^2\) and \(T\) substituted by \(tq^{-1}\), reads: \[\begin{align} \label{eq:fer-2-original} &(t,q;q)_\infty^{k} \sum_{\substack{l_1,\ldots,l_{2k+1}\ge 0\\m_1,\ldots,m_{2k+1}\ge 0}} \delta_{l_{2k}+l_{2k+1},m_{2k}+m_{2k+1}} \prod_{i=1}^{2k-1} \delta_{l_i,m_i}\notag\\ &\times \frac{q^{\sum_{i,j=1}^{2k+1} \frac{a_{i,j} l_i m_j}{2} + \sum_{i=1}^{k} \frac{l_{2i-1}+m_{2i-1}}{2}} t^{\sum_{i=1}^{k}\frac{l_{2i}+m_{2i}}{2}+\frac{l_{2k+1}+m_{2k+1}}{2}} z^{2m_{2k+1}-2l_{2k+1}}}{(q;q)_{l_{2k+1}}(q;q)_{m_{2k+1}} \prod_{i=1}^k (q;q)_{l_{2i}}(q;q)_{m_{2i}}(t;q)_{l_{2i-1}}(q;q)_{m_{2i-1}}}, \end{align}\tag{25}\] where \(A := (a_{i,j})_{1\le i,j\le 2k+1}\) is given by \(A = 2I_{2k+1} - M_{\mathsf{Car}}(D_{2k+1})\) with \(I_{2k+1}\) the identity matrix of dimension \(2k+1\) and \(M_{\mathsf{Car}}(D_{2k+1})\) the Cartan matrix of the \(D_{2k+1}\) Lie algebra. In particular, in this case \(A\) is the adjacency matrix for the \(D_{2k+1}\) Dynkin diagram: \[\begin{align} \vcenter{\begin{figure}\includegraphics[width=0.8\textwidth]{_pdflatex/ldhwqitb.png}\label{drshjzvp}\end{figure}}\quad , \end{align}\tag{26}\] so that \(a_{i,i+1} = a_{i+1,i} = 1\) for \(i\) with \(1\le i\le 2k-2\), \(a_{2k-1,2k} = a_{2k,2k-1} = a_{2k-1,2k+1} = a_{2k+1,2k-1} = 1\), and \(a_{i,j} = 0\) for all other cases.
In light of the Kronecker delta factors in 25 , we rename the summation indices as in 4 : \[\begin{align} r_i &:= l_{2i-1} = m_{2i-1}, &&\quad (1\le i\le k),\\ s_i &:= l_{2i} = m_{2i}, &&\quad (1\le i\le k-1), \end{align}\] and \[\begin{align} s_k := l_{2k}+l_{2k+1} = m_{2k}+m_{2k+1}. \end{align}\] Thus, the exponent of \(q\) in the numerator becomes \[\begin{align} \sum_{i,j=1}^{2k+1} \frac{a_{i,j} l_i m_j}{2} + \sum_{i=1}^{k} \frac{l_{2i-1}+m_{2i-1}}{2} = \sum_{i=1}^k r_i (s_{i-1} + s_i + 1), \end{align}\] while the exponent of \(t\) is \[\begin{align} \sum_{i=1}^{k}\frac{l_{2i}+m_{2i}}{2}+\frac{l_{2k+1}+m_{2k+1}}{2} = \sum_{i=1}^k s_i. \end{align}\] We further put \[\begin{align} u_1 := l_{2k}, \qquad u_2 := m_{2k}, \end{align}\] so that \[\begin{align} l_{2k+1} = s_k - u_1, \qquad m_{2k+1} = s_k - u_2. \end{align}\] In this way, the exponent of \(z\) becomes \[\begin{align} 2m_{2k+1}-2l_{2k+1} = 2u_1 - 2u_2. \end{align}\] Finally, for the denominator, we have \[\begin{align} \frac{1}{(t,q;q)_{r_1}\cdots (t,q;q)_{r_k} (q;q)_{s_1}^2 \cdots (q;q)_{s_{k-1}}^2}\cdot \frac{1}{(q;q)_{l_{2k}}(q;q)_{m_{2k}}(q;q)_{l_{2k+1}}(q;q)_{m_{2k+1}}}, \end{align}\] while the latter factor can be further rewritten as \[\begin{align} \frac{1}{(q;q)_{l_{2k}+l_{2k+1}} (q;q)_{m_{2k}+m_{2k+1}}} {l_{2k}+l_{2k+1}\brack l_{2k}}_q {m_{2k}+m_{2k+1}\brack m_{2k}}_q = \frac{1}{(q;q)_{s_k}^2} {s_k\brack u_1}_q {s_k\brack u_2}_q. \end{align}\] Now 25 becomes exactly the same as the fermionic sum in 4 .
Here we evaluate the following basic hypergeometric series, which appears as a critical step in our proof of Theorem 4.
Lemma 1. For \(l,n\ge 0\), \[\begin{align} \label{eq:key-id-1} \sum_{j=0}^{2l} \frac{(q^{j-l-n};q)_{2n} (t^{-1};q)_{j-l-n} t^{j}}{(q;q)_j (q;q)_{2l-j} (t;q)_{j-l+n}} = \frac{t^{2l}}{(q;q)_{l-n} (tq;q)_{l+n}}. \end{align}\tag{27}\]
Proof. If \(l< n\), we have the vanishing of \((q^{j-l-n};q)_{2n}\) for all \(j\) ranging from \(0\) to \(2l\), and hence both sides of 27 trivially go to zero in this case. Now we assume \(l\ge n\). For convenience, let us write \[\begin{align} s_j := \frac{(q^{j-l-n};q)_{2n} (t^{-1};q)_{j-l-n} t^{j}}{(q;q)_j (q;q)_{2l-j} (t;q)_{j-l+n}}, \qquad (0\le j\le 2l). \end{align}\] It is clear that \(s_j\) is supported on the ranges \(0\le j\le l-n\) and \(l+n+1\le j\le 2l\). Putting in addition \[\begin{align} s_{2l+1} := 0, \end{align}\] we observe that for each \(j\) with \(0\le j\le l-n\), \[\begin{align} s_j + s_{1+2l-j} = \frac{q^j (1-q^{1+2l-2j})}{1-q^{1+2l-j}}\cdot s_j. \end{align}\] Thus, \[\begin{align} \operatorname{LHS}\eqref{eq:key-id-1} = \sum_{j=0}^{l-n} \frac{(q^{j-l-n};q)_{2n} (t^{-1};q)_{j-l-n} t^{j} q^j (1-q^{1+2l-2j})}{(q;q)_j (q;q)_{2l-j} (t;q)_{j-l+n} (1-q^{1+2l-j})}. \end{align}\] Note that \[\begin{align} (q^{j-l-n};q)_{2n} = \frac{(q;q)_{l+n-j}q^{n(2j-2l-1)}}{(q;q)_{l-n-j}}. \end{align}\] We further make the change of indices \(j\mapsto l-n-j\) so that \[\begin{align} \operatorname{LHS}\eqref{eq:key-id-1} &= \sum_{j=0}^{l-n} \frac{(q;q)_{2n+j}(t^{-1};q)_{-2n-j} t^{l-n-j} q^{l-2n(n+1)-(2n+1)j} (1-q^{1+2n+2j})}{(q;q)_j (q;q)_{l-n-j} (q;q)_{l+n+j} (t;q)_{-j} (1-q^{1+l+n+j})}\\ & = \sum_{j=0}^{l-n} \frac{(q;q)_{2n+j}(t^{-1};q)_{-2n-j} t^{l-n-j} q^{l-2n(n+1)-(2n+1)j} (1-q^{1+2n+2j})}{(q;q)_j (q;q)_{l-n-j} (q;q)_{1+l+n+j} (t;q)_{-j}}. \end{align}\] Using the relations \[\begin{align} (a;q)_{m+j} &= (a;q)_m (aq^m;q)_j,\\ (a;q)_{m-j} &= \frac{(a;q)_m}{(q^{1-m}/a;q)_j} \left(-\frac{q}{a}\right)^j q^{\binom{j}{2}-nj}, \end{align}\] we have \[\begin{align} \operatorname{LHS}\eqref{eq:key-id-1} &= \frac{(q;q)_{2n} t^{l+n} q^{l-n}}{(q;q)_{l-n}(q;q)_{l+n+1}(tq;q)_{2n}}\\ &\quad\times \sum_{j=0}^{l-n} \frac{(q^{2n+1},t^{-1}q,q^{-(l-n)};q)_jt^j q^{(l-n-1)j}(1-q^{1+2n+2j})}{(q,q^{l+n+2},tq^{2n+1};q)_j (-1)^j q^{\binom{j}{2}}}\\ &= \frac{(q;q)_{2n} t^{l+n} q^{l-n} (1-q^{2n+1})}{(q;q)_{l-n}(q;q)_{l+n+1}(tq;q)_{2n}}\\ &\quad\times \lim_{\tau\to 0} {}_6 \phi_{5}\left(\begin{matrix} q^{2n+1},q^{n+\frac{3}{2}},-q^{n+\frac{3}{2}},t^{-1}q,\tau q^{2n+2},q^{-(l-n)}\\ q^{n+\frac{1}{2}},-q^{n+\frac{1}{2}},tq^{2n+1},\tau^{-1},q^{l+n+2} \end{matrix};q,\frac{tq^{l-n-1}}{\tau}\right). \end{align}\] For this terminating \({}_6 \phi_{5}\) series, we require the following evaluation [20]: \[\begin{align} \label{eq:6phi5} {}_6 \phi_{5}\left(\begin{matrix} a, a^{\frac{1}{2}}q, -a^{\frac{1}{2}}q, b, c, q^{-n}\\ a^{\frac{1}{2}}, -a^{\frac{1}{2}}, aq/b, aq/c, aq^{n+1} \end{matrix};q,\frac{aq^{n+1}}{bc}\right) = \frac{(aq,aq/(bc);q)_n}{(aq/b,aq/c;q)_n}. \end{align}\tag{28}\] Therefore, \[\begin{align} \operatorname{LHS}\eqref{eq:key-id-1} &= \frac{(q;q)_{2n} t^{l+n} q^{l-n} (1-q^{2n+1})}{(q;q)_{l-n}(q;q)_{l+n+1}(tq;q)_{2n}}\lim_{\tau\to 0} \frac{(q^{2n+2},tq^{-1}\tau^{-1};q)_{l-n}}{(tq^{2n+1},\tau^{-1};q)_{l-n}}\\ &= \frac{t^{l+n}q^{l-n}}{(q;q)_{l-n} (tq;q)_{l+n}} \lim_{\tau\to 0} \frac{(tq^{-1}\tau^{-1};q)_{l-n}}{(\tau^{-1};q)_{l-n}}. \end{align}\] Finally, using the relation \[\begin{align} \frac{(a;q)_n}{(b;q)_n} = \frac{(q^{1-n}/a;q)_n}{(q^{1-n}/b;q)_n} \left(\frac{a}{b}\right)^n, \end{align}\] the above limit becomes \((tq^{-1})^{l-n}\), producing the right-hand side of 27 . ◻
Here we show the conjugate WP-Bailey pair claimed in Theorem 6; the proof is similar to that for Theorem 4. In particular, this time we use the fact that the continuous \(q\)-ultraspherical polynomials \(C_n(z,s;q)\) also form an orthogonal basis of symmetric Laurent polynomials in \(z\). Then it is possible to write \[\begin{align} \frac{(sz^2,sz^{-2};q)_\infty}{(t z^2, t z^{-2};q)_\infty} C_{2n}(z,t;q) = \sum_{l\ge 0} c'_{n,l}\, C_{2l}(z,s;q). \end{align}\] Applying the orthogonality of continuous \(q\)-ultraspherical polynomials [25]: \[\begin{align} \int_0^\pi C_n(z,s;q) C_{n'}(z,s;q) \frac{(z^2, z^{-2};q)_\infty}{(s z^2, s z^{-2};q)_\infty} \operatorname{d}\theta = \frac{2\pi (1-s) (s^2; q)_n (s, sq; q)_\infty}{(1-sq^n) (q; q)_n (q, s^2; q)_\infty} \delta_{n,n'}, \end{align}\] the coefficients \(c'_{n,l}\) can be identified by \[\begin{align} c'_{n,l} &= \frac{(1-sq^{2l}) (q; q)_{2l} (q, s^2; q)_\infty}{2 (1-s) (s^2; q)_{2l} (s, sq; q)_\infty} \int_0^\pi C_{2n}(z,t;q) C_{2l}(z,s;q) \frac{(z^2, z^{-2};q)_\infty}{(t z^2, t z^{-2};q)_\infty} \frac{\operatorname{d}\theta}{\pi}\\ &= \frac{(1-sq^{2l}) (q; q)_{2l} (q, s^2; q)_\infty}{2 (1-s) (s^2; q)_{2l} (s, sq; q)_\infty} \sum_{j=0}^{2l} \frac{(s;q)_j (s;q)_{2l-j}}{(q;q)_j (q;q)_{2l-j}}\, I_{j-l-n,n}, \end{align}\] where \(I_{j-l-n,n}\) is as in 15 . It follows from 16 and 17 , with a similar symmetry to that in 18 used, that \[\begin{align} c'_{n,l} &= \frac{t^{-l-n} (1-sq^{2l}) (q; q)_{2l} (t^2;q)_{2n} (t, tq , s^2; q)_\infty}{(1-s) (s^2; q)_{2l} (q;q)_{2n} (t^2, s, sq; q)_\infty}\\ &\quad\times \sum_{j=0}^{2l} \frac{(s;q)_j (s;q)_{2l-j} (q^{j-l-n};q)_{2n} (t^{-1};q)_{j-l-n} t^{j}}{(q;q)_j (q;q)_{2l-j} (t;q)_{j-l+n}}. \end{align}\] For the summation on the right-hand side, we proceed in the same way as that for Lemma 1 by pairing the summands, and eventually reformulate it in terms of a \({}_6 \phi_5\) series: \[\begin{align} &\sum_{j=0}^{2l} \frac{(s;q)_j (s;q)_{2l-j} (q^{j-l-n};q)_{2n} (t^{-1};q)_{j-l-n} t^{j}}{(q;q)_j (q;q)_{2l-j} (t;q)_{j-l+n}}\\ &\qquad = \frac{(s;q)_{l-n} (s;q)_{l+n} (q;q)_{2n} t^{l+n} (1-s^{-1}q) (1-q^{2n+1})}{(q;q)_{l-n}(q;q)_{l+n+1}(tq;q)_{2n} (1-s^{-1}q^{1-(l-n)})}\\ &\qquad\quad\times {}_6 \phi_{5}\left(\begin{matrix} q^{2n+1},q^{n+\frac{3}{2}},-q^{n+\frac{3}{2}},t^{-1}q,s q^{l+n},q^{-(l-n)}\\ q^{n+\frac{1}{2}},-q^{n+\frac{1}{2}},tq^{2n+1},s^{-1}q^{2-(l-n)},q^{l+n+2} \end{matrix};q,\frac{tq}{s}\right). \end{align}\] Invoking 28 , we arrive at \[\begin{align} \label{eq:c39-sum-evaluation} \sum_{j=0}^{2l} \frac{(s;q)_j (s;q)_{2l-j} (q^{j-l-n};q)_{2n} (t^{-1};q)_{j-l-n} t^{j}}{(q;q)_j (q;q)_{2l-j} (t;q)_{j-l+n}} = \frac{(st^{-1};q)_{l-n} (s;q)_{l+n} t^{2l}}{(q;q)_{l-n} (tq;q)_{l+n}}. \end{align}\tag{29}\] Substituting 29 into the expression for \(c'_{n,l}\), we conclude that \[\begin{align} c'_{n,l} = \frac{(st^{-1};q)_{l-n} (s;q)_{l+n}}{(q;q)_{l-n} (tq;q)_{l+n}} \cdot \frac{t^{l-n} (1-sq^{2l}) (q;q)_{2l} (t^2;q)_{2n} (t,tq,s^2;q)_\infty}{(1-s)(s^2;q)_{2l}(q;q)_{2n}(t^2,s,sq;q)_\infty}, \end{align}\] which, in particular, vanishes when \(l < n\). Therefore, \[\begin{align} &\frac{t^n (q;q)_{2n} (t^2,sz^2,sz^{-2};q)_\infty}{(t^2;q)_{2n} (t, tq, t z^2, t z^{-2};q)_\infty} C_{2n}(z,t;q)\\ &\qquad = \sum_{l\ge n} \frac{(st^{-1};q)_{l-n} (s;q)_{l+n}}{(q;q)_{l-n} (tq;q)_{l+n}} \cdot \frac{t^{l} (1-sq^{2l}) (q;q)_{2l} (s^2;q)_\infty}{(1-s)(s^2;q)_{2l}(s,sq;q)_\infty} C_{2l}(z,s;q). \end{align}\] Finally, we replace \(z\) with \(z^{-1/2}\). The above identity becomes exactly the required relation 24 for conjugate WP-Bailey pairs.
Bosonic and fermionic representations are two ways to describe the same physical system: the former treats the system as a collection of independent particles, usually leading to single sum-like expressions mathematically, while the latter looks at the exclusion principle of the particles, resulting in multisums.↩︎
The original formulation in [17] goes to ?? after a simple substitution of indices.↩︎
The equivalence can be seen by renaming the summation indices in [21] or [16] and invoking the Cartan matrix of the \(D_{2k+1}\) Lie algebra. For details, see Appendix 7.↩︎