A determinant formula associated with the elliptic hypergeometric integrals of type $BC_n$

TL;DR

This paper derives a determinant formula for elliptic hypergeometric integrals of type BC_n by analyzing q-difference equations and asymptotic behavior, advancing the understanding of their algebraic structure.

Contribution

It introduces a new determinant formula for elliptic hypergeometric integrals of type BC_n using q-difference equations and asymptotic analysis.

Findings

Established a determinant formula for the bilinear form of elliptic hypergeometric integrals of type BC_n

Connected elliptic interpolation functions to the study of q-difference equations

Provided asymptotic analysis along singularities to prove the formula

Abstract

We establish a determinant formula for the bilinear form associated with the elliptic hypergeometric integrals of type $BC_n$ by studying the structure of $q$-difference equations to be satisfied by them. The determinant formula is proved by combining the $q$-difference equations of the determinant and its asymptotic analysis along the singularities. The elliptic interpolation functions of type $BC_n$ are essentially used in the study of the $q$-difference equations.

Equations

$$\theta (u;p)=(u;p{)}_{\infty }(p/u;p{)}_{\infty },\Gamma (u;p,q)=\frac{(pq/u;p,q{)}_{\infty }}{(u;p,q{)}_{\infty }}(u\in {\mathbb{C}}^{\ast })$$

$$\theta (pu;p)=-u^{-1}\theta (u;p)\text{mbox}and\Gamma (qu;p,q)=\theta (u;p)\Gamma (u;p,q),\Gamma (pu;p,q)=\theta (u;q)\Gamma (u;p,q),$$

$$e(u,v;p)=u^{-1}\theta (uv;p)\theta (u{v}^{-1};p).$$

$$\begin{array}{rl}\theta (u;p{)}_{t,k}& =\theta (u;p)\theta (tu;p)\cdots \theta (t^{k-1}u;p),\\ e(u,v;p{)}_{t,k}& =e(u,v;p)e(tu,v;p)\cdots e(t^{k-1}u,v;p)\end{array}$$

$$\Phi (z)=\Phi (z;a;p,q)=\prod _{i=1}^n\frac{{\prod }_{k=1}^m\Gamma (a_k{z}_i^{\pm 1};p,q)}{\Gamma (z_i^{\pm 2};p,q)}\prod _{1\le i<j\le n}\frac{\Gamma (t{z}_i^{\pm 1}{z}_j^{\pm 1};p,q)}{\Gamma (z_i^{\pm 1}{z}_j^{\pm 1};p,q)}.$$

$$f(u^{\pm 1})=f(u)f(u^{-1}),f(u^{\pm 1}{v}^{\pm 1})=f(uv)f(u{v}^{-1})f(u^{-1}v)f(u^{-1}{v}^{-1}).$$

$$\mathbb{T}^{n}=\left\{\,z=(z_{1},\ldots,z_{n})\in(\mathbb{C}^{\ast})^{n}\ \big{|}\ |z_{i}|=1\quad(i=1,\ldots,n)\,\right\},$$

$$⟨f{⟩}_{\Phi }={\int }_{{\mathbb{T}}^n}f(z)\Phi (z){\omega }_n(z),{\omega }_n(z)=\frac{1}{(2\pi \sqrt{-1}{)}^n}\frac{d{z}_1\cdots d{z}_n}{z_1\cdots z_n}.$$

$$\begin{split}\langle\,1\rangle_{\Phi}=\int_{\mathbb{T}^{n}}\Phi(z)\omega_{n}(z)=\frac{2^{n}n!}{(p;p)_{\infty}^{n}(q;q)_{\infty}^{n}}\prod_{i=0}^{n-1}\bigg{(}\frac{\Gamma(t^{i+1};p,q)}{\Gamma(t;p,q)}\prod_{1\leq k<l\leq 6}\Gamma(t^{i}a_{k}a_{l};p,q)\bigg{)},\end{split}$$

$$I(a_1,\dots ,q{a}_k,\dots ,a_5,q^{-1}{a}_6)=I(a_1,\dots ,a_5,a_6)\prod _{i=0}^{n-1}\prod _{\begin{array}{c}1\le l\le 5\\ l\ne k\end{array}}\frac{\theta (a_l{a}_k{t}^i;p)}{\theta (q^{-1}{a}_l{a}_6{t}^i;p)}$$

$$⟨,{⟩}_{\Phi }:{\mathcal{H}}_{r-1,n}^{(p)}\times {\mathcal{H}}_{r-1,n}^{(q)}\to \mathbb{C}$$

$$\begin{array}{rl}& ⟨f,g{⟩}_{\Phi }=⟨fg{⟩}_{\Phi }={\int }_{{\mathbb{T}}^n}f(z)g(z)\Phi (z){\omega }_n(z)(f\in {\mathcal{H}}_{r-1,n}^{(p)},g\in {\mathcal{H}}_{r-1,n}^{(q)}).\end{array}$$

$$T_{p,z_i}f(z_1,\dots ,z_n)=f(z_1,\dots ,p{z}_i,\dots ,z_n)(i=1,\dots ,n).$$

$$\begin{split}\mathcal{H}_{s-1,n}^{(p)}=\left\{\,f(z)\in\mathcal{O}((\mathbb{C}^{\ast})^{n})^{W_{n}}\ \big{|}\ T_{p,z_{i}}f(z)=f(z)(pz_{i}^{2})^{-s+1}\quad(i=1,\ldots,n)\,\right\}\end{split}$$

$$Z_{s,n}=\left\{\,\mu=(\mu_{1},\ldots,\mu_{s})\in\mathbb{N}^{s}\ \big{|}\ |\mu|=\mu_{1}+\cdots+\mu_{s}=n\,\right\}$$

$$(c{)}_{t,\mu }=(c_1,t{c}_1,\dots ,t^{{\mu }_1-1}{c}_1,c_2,t{c}_2,\dots ,t^{{\mu }_2-1}{c}_2,\dots ,c_s,t{c}_s,\dots ,t^{{\mu }_s-1}{c}_s)\in ({\mathbb{C}}^{\ast }{)}^n$$

$$E_{\mu }(c;(c{)}_{t,\nu };p)={\delta }_{\mu ,\nu }(\mu ,\nu \in Z_{s,n})$$

$$f(z)=\sum _{\mu \in Z_{s,m}}f((c{)}_{t,\mu })E_{\mu }(c;z;p)$$

$$\begin{array}{r}{\mathcal{H}}_{r-1,n}^{(p)}=\underset{\mu \in Z_{r,n}}{⨁}\mathbb{C}{E}_{\mu }(x;z;p),{\mathcal{H}}_{r-1,n}^{(q)}=\underset{\mu \in Z_{r,n}}{⨁}\mathbb{C}{E}_{\mu }(y;z;q).\end{array}$$

$$\begin{array}{rl}& K_{\mu ,\nu }(a;x,y)=⟨E_{\mu }(x;z;p),E_{\nu }(y;z;q){⟩}_{\Phi }\\ & ={\int }_{{\mathbb{T}}^n}{E}_{\mu }(x;z;p)E_{\nu }(y;z;q)\Phi (z;a;p,q){\omega }_n(z)(\mu ,\nu \in Z_{r,n}),\end{array}$$

$$K(a;x,y)=\big{(}K_{\mu,\nu}(a;x,y)\big{)}_{\mu,\nu\in Z_{r,n}}$$

$$\begin{split}&\det K(a;x,y)\\ &=\bigg{(}\frac{2^{n}n!}{(p;p)_{\infty}^{n}(q;q)_{\infty}^{n}}\bigg{)}^{\binom{n+r-1}{r-1}}\prod_{i=0}^{n-1}\bigg{(}\frac{\Gamma(t^{i+1};p,q)^{r}}{\Gamma(t;p,q)^{r}}\prod_{1\leq k<l\leq m}\Gamma(t^{i}a_{k}a_{l};p,q)\bigg{)}^{\binom{n-i+r-2}{r-1}}\\ &\qquad\cdot\Bigg{(}\prod_{0\leq i+j<n}\prod_{1\leq k<l\leq r}\big{(}e(t^{i}x_{k},t^{j}x_{l};p)e(t^{i}y_{k},t^{j}y_{l};q)\big{)}^{\binom{n-i-j+r-3}{r-2}}\Bigg{)}^{\!\!-1}.\end{split}$$

$$\begin{split}\det\Big{(}\langle E_{\epsilon_{i}}(x;z;p),E_{\epsilon_{j}}(y;z;q)\rangle_{\Phi}\Big{)}_{i,j=1}^{r}=\frac{2^{r}}{(p;p)_{\infty}^{r}(q;q)_{\infty}^{r}}\frac{\prod_{1\leq k<l\leq 2r+4}\Gamma(a_{k}a_{l};p,q)}{\prod_{1\leq k<l\leq r}e(x_{k},x_{l};p)e(y_{k},y_{l};q)}\end{split}$$

$$T_{q,a_k}{T}_{q,a_l}^{-1}K(a;x,y)=M^{k,l}(a;x;p,q)K(a;x,y)(1\le k<l\le m),$$

$$M^{k,l}(a;x;p,q)=\big{(}M^{k,l}_{\mu,\nu}(a;x;p,q)\big{)}_{\mu,\nu\in Z_{r,n}}\quad(1\leq k<l\leq m)$$

$$T_{p,a_k}{T}_{p,a_l}^{-1}K(a;x,y)=K(a;x,y)M^{k,l}(a;y;q,p{)}^{\mathrm{t}}(1\le k<l\le m),$$

$$E_{{ϵ}_k}(c;u)=\prod _{\begin{array}{c}1\le l\le s\\ l\ne k\end{array}}\frac{e(u,c_l)}{e(c_k,c_l)}=\prod _{\begin{array}{c}1\le l\le s\\ l\ne k\end{array}}\frac{\theta (c_l/u)\theta (c_{l}u)}{\theta (c_l/c_k)\theta (c_l{c}_k)}.$$

$$\begin{array}{r}{E}_{\mu }(c;z)=\sum _{\begin{array}{c}1\le k\le s\\ {\mu }_k>0\end{array}}{E}_{\mu -{ϵ}_k}(c;z_1,\dots ,z_{n-1})E_{{ϵ}_k}(t^{\mu -{ϵ}_k}c;z_n)(\mu \in Z_{s,n}),\end{array}$$

$$\begin{array}{rl}{E}_{\mu }(c;z)& =\sum _{k_1,\dots ,k_n\in \left\{1,\dots ,s\right\}}{E}_{{ϵ}_{k_1}}(c;z_1)E_{{ϵ}_{k_2}}(t^{{ϵ}_{k_1}}c;z_2)\cdots E_{{ϵ}_{k_n}}(t^{{ϵ}_{k_1}+\cdots +{ϵ}_{k_{n-1}}}c;z_n)\\ & =\sum _{k_1,\dots ,k_n\in \left\{1,\dots ,s\right\}}\prod _{i=1}^n\prod _{\begin{array}{c}1\le l\le s\\ l\ne k_i\end{array}}\frac{e(z_i,t^{{\mu }^{(i)}}{c}_l)}{e(t^{{\mu }_{k_i}^{(i-1)}}{c}_{k_i},t^{{\mu }_l^{(i-1)}}{c}_l)},\end{array}$$

$$\left\{\,\mu=(\mu_{1},\ldots,\mu_{s})\in\mathbb{R}^{s}\ \big{|}\ \mu_{1}+\cdots+\mu_{s}=n;\quad\mu_{k}\geq 0\ \ (k=1,\ldots,s)\,\right\}$$

Full text

000 2010 Mathematics Subject Classification. Primary 33D67; Secondary 33D65, 39A13.000 Key words and phrases. elliptic hypergeometric integral, determinant formula, elliptic interpolation function

1. 1 Introduction

2. 1.1 Elliptic hypergeometric integrals of type $BC_{n}$

3. 1.2 Interpolation basis for a space of $W_{n}$-invariant quasi-periodic functions.

4. 1.3 Bilinear form associated with the elliptic hypergeometric integral

5. 1.4 Plan of this paper

6. 2 Elliptic interpolation functions of type $BC_{n}$

7. 2.1 Recursion formula

8. 2.2 Explicit formula

9. 2.3 Interpolation functions on the vertices and the faces

10. 2.4 Dual Cauchy formula

11. 2.5 Partition of the variables

12. 2.6 Special value

13. 2.7 Change of parameters

14. 3 $q$-Difference de Rham theory

15. 3.1 Definition of $H_{\mathrm{sym}}^{\Psi}$

16. 3.2 Reduction of cocycles

17. 3.3 Base change in $H_{\mathrm{sym}}^{\Psi}$

18. 4 System of $q$-difference equations

19. 4.1 System of $q$-difference equations associated with a basis of $\mathcal{H}^{(p)}_{r-1,n}$

20. 4.2 Derivation of $q$-difference equations

21. 4.3 System of $q$-difference equations associated with an interpolation basis

22. 4.4 Symmetry of the difference system with respect to $(p,q)$.

23. 5 Computation of the constants $c_{r,n}$

24. 5.1 Determinant of the bilinear form

25. 5.2 Asymptotic behavior of $L_{r,n}(a)$

26. 5.3 Remark on analytic continuation

27. 5.4 Asymptotic behavior of $\det K^{(r,n)}(a)$

28. 5.5 Determination of $c_{r,n}$

29. 6 Determinant formula for $q$-hypergeometric integrals of type $BC_{n}$

1 Introduction

The purpose of this paper is to provide a foundation for the study of $q$-difference equations satisfied by the elliptic hypergeometric integrals of type $BC_{n}$, and to establish a determinant formula for the bilinear form associated with them. We summarize below the main results of this paper.

Throughout this paper we fix two bases $p,q\in\mathbb{C}^{\ast}$ with $|p|<1$, $|q|<1$, and use the notation of the multiplicative theta function $\theta(u;p)$ and the elliptic gamma function $\Gamma(u;p,q)$ specified by

[TABLE]

where $(u;p)_{\infty}=\prod_{i=0}^{\infty}(1-p^{i}u)$ and $(u;p,q)_{\infty}=\prod_{i,j=0}^{\infty}(1-p^{i}q^{j}u)$. They satisfy

[TABLE]

respectively. In the $BC_{n}$ context of this paper, we define the function $e(u,v;p)$ of two variables by

[TABLE]

We also use the notation of $t$-shifted factorials

[TABLE]

for $k=0,1,2,\ldots$.

1.1 Elliptic hypergeometric integrals of type $BC_{n}$

Let $z=(z_{1},\ldots,z_{n})$ be the canonical coordinates of the $n$-dimensional algebraic torus $(\mathbb{C}^{\ast})^{n}$. We denote by $W_{n}=\left\{\pm 1\right\}^{n}\rtimes\mathfrak{S}_{n}$ the hyperoctahedral group of degree $n$ (the Weyl group of type $BC_{n}$), acting on $(\mathbb{C}^{\ast})^{n}$ through permutations and inversions of the coordinates $z_{i}$ ($i=1,\ldots,n$). Fixing a constant $t\in\mathbb{C}^{*}$ with $|t|<1$, for each $m=1,2,\ldots$ we define a $W_{n}$-invariant meromorphic function $\Phi(z)$ on $(\mathbb{C}^{\ast})^{n}$ with parameters $a=(a_{1},\ldots,a_{m})\in(\mathbb{C}^{\ast})^{m}$ by

[TABLE]

The double sign $\pm$ indicates, here and hereafter, the product of factors with possible combinations of signs as

[TABLE]

We now suppose that the parameters satisfy the conditions $|a_{k}|<1$ $(k=1,\ldots,m)$. Noting that $\Phi(z)$ is holomorphic in a neighborhood of the real torus

[TABLE]

for each holomorphic function $f(z)$ on $(\mathbb{C}^{\ast})^{n}$ we consider the elliptic hypergeometric integral

[TABLE]

When $m=6$ and $f=1$, the integral (1.2) can be evaluated as

[TABLE]

provided that $a_{1}\cdots a_{6}t^{2n-2}=pq$. This is known as the evaluation formula of the elliptic Selberg integral of type $BC_{n}$ due to van Diejen and Spiridonov [22]. If we denote by $I(a_{1},\ldots,a_{6})$ the left -hand side of (1.3), then it satisfies the $q$-difference equations

[TABLE]

for $k=1,\ldots,5$. In [11, 12], we gave a proof of the formula (1.3) based on the $q$-difference equations (1.4) and singularity analysis of the integral.

When we study the general case where $m\geq 6$, we need to consider an appropriate class of functions $f(z)$ in the integral (1.2). In the following, we introduce two vector spaces $\mathcal{H}^{(p)}_{r-1,n}$, $\mathcal{H}^{(q)}_{r-1,n}$ of $W_{n}$-invariant quasi-periodic functions with respect to $p$ and $q$, respectively, and investigate the $\mathbb{C}$-bilinear form

[TABLE]

defined by

[TABLE]

This bilinear form is a $BC_{n}$ elliptic extension of the hypergeometric pairing of Tarasov and Varchenko [20], studied in the context of the $q$-KZ equations (of type $A_{n}$). From the viewpoint of $q$-difference de Rham theory, $\mathcal{H}_{r-1,n}^{(p)}$ plays the role of the space of $n$-cocycles, and $\mathcal{H}_{r-1,n}^{(q)}$ that of $n$-cycles, respectively. One of the main goals of this paper is to provide an explicit formula for the determinant of this bilinear form with respect to a pair of certain interpolation bases for $\mathcal{H}^{(p)}_{r-1,n}$ and $\mathcal{H}^{(q)}_{r-1,n}$.

1.2 Interpolation basis for a space of

$W_{n}$-invariant quasi-periodic functions.

In what follows we denote by $T_{p,z_{i}}$ the $p$-shift operator with respect to the variable $z_{i}$:

[TABLE]

As in [13], for each $s=1,2,\ldots$ we introduce the $\mathbb{C}$-vector space

[TABLE]

of all $W_{n}$-invariant holomorphic functions on $(\mathbb{C}^{\ast})^{n}$ with quasi-periodicity of degree $s-1$. This vector space has dimension $\binom{n+s-1}{s-1}$, which coincides with the cardinality of the set

[TABLE]

of multiindices, where $\mathbb{N}=\left\{0,1,2,\ldots\right\}$.

Fixing a set $c=(c_{1},c_{2},\ldots,c_{s})\in(\mathbb{C}^{\ast})^{s}$ of generic parameters, for each $\mu\in Z_{s,n}$ we consider the reference point

[TABLE]

in $(\mathbb{C}^{\ast})^{n}$, where the indexing set $\left\{1,\ldots,n\right\}$ is divided into $s$ blocks of size $\mu_{1},\mu_{2},\ldots,\mu_{s}$. Then, for the set $\{(c)_{t,\mu}\ |\ \mu\in Z_{s,n}\}$ of reference points, it is known by [13] that there exists a unique $\mathbb{C}$-basis $\{E_{\mu}(c;z;p)\ |\ \mu\in Z_{s,n}\}$ of $\mathcal{H}_{s-1,n}^{(p)}$ satisfying the interpolation condition

[TABLE]

where $\delta_{\mu,\nu}$ denotes the Kronecker delta; we call this $\{E_{\mu}(c;z;p)\ |\ \mu\in Z_{s,n}\}$ the interpolation basis of $\mathcal{H}_{s-1,n}^{(p)}$ with respect to $c\in(\mathbb{C}^{\ast})^{s}$. Note that each $f(z)\in\mathcal{H}^{(p)}_{s-1,n}$ is expanded as

[TABLE]

in terms of the interpolation basis. Fundamental properties of this interpolation basis are summarized below in Section 2.

1.3 Bilinear form associated with the elliptic hypergeometric integral

Returning to the meromorphic function $\Phi(z)$ of (1.1), we assume that $m=2r+4$ $(r=1,2,\ldots)$. With respect to the two bases $p$, $q$, we consider the two vector spaces $\mathcal{H}^{(p)}_{r-1,n}$, $\mathcal{H}^{(q)}_{r-1,n}$ of $W_{n}$-invariant quasi-periodic functions of degree $r-1$, respectively, and define the $\mathbb{C}$-bilinear form $\langle\,,\,\rangle_{\Phi}:\mathcal{H}^{(p)}_{r-1,n}\times\mathcal{H}^{(q)}_{r-1,n}\to\mathbb{C}$ by (1.5). We propose an explicit formula for the determinant of this bilinear form with respect to a pair of interpolation bases for $\mathcal{H}^{(p)}_{r-1,n}$ and $\mathcal{H}^{(q)}_{r-1,n}$.

Fixing generic parameters $x=(x_{1},\ldots,x_{r})$ and $y=(y_{1},\ldots,y_{r})$, we take the interpolation bases for these two vector spaces with respect to $x$ and $y$ respectively:

[TABLE]

For each pair $(\mu,\nu)\in Z_{r,n}\times Z_{r,n}$, we introduce the elliptic hypergeometric integral

[TABLE]

which is a holomorphic function on the domain $|a_{k}|<1$ $(k=1,\ldots,m)$ of $(\mathbb{C}^{*})^{m}$. We consider the $\binom{n+r-1}{r-1}\times\binom{n+r-1}{r-1}$ matrix

[TABLE]

as a matrix representation of the bilinear form (1.5) with respect to the interpolation bases.

Theorem 1.1

Set $m=2r+4$ $(r=1,2,\ldots)$. Under the balancing condition $a_{1}\cdots a_{m}t^{2n-2}=pq$ for the parameters, the determinant of the $\binom{n+r-1}{r-1}\times\binom{n+r-1}{r-1}$ matrix $K(a;x,y)=(K_{\mu,\nu}(a;x,y))_{\mu,\nu\in Z_{r,n}}$ is given explicitly by

[TABLE]

Remark 1.2

We comment on two special cases of Theorem 1.1. When $r=1$ (i.e., $m=6$), the matrix size of $K(a;x,y)$ reduces to 1 and Theorem 1.1 gives the formula (1.3). On the other hand, when $n=1$, the matrix size of $K(a;x,y)$ reduces to $r$, and Theorem 1.1 means that, for $x,y\in(\mathbb{C}^{*})^{r}$

[TABLE]

under the condition $a_{1}\cdots a_{2r+4}=pq$ with $|a_{k}|<1$ $(k=1,\ldots,2r+4)$, where $E_{\epsilon_{i}}(x;z;p)=\prod_{1\leq k\leq r\atop k\neq i}e(z,x_{k};p)/e(x_{i},x_{k};p)$ $(z\in\mathbb{C}^{*})$ for $i=1,\ldots,r$, as mentioned in (2.1). This is also a special case of the determinant formula of type I  found by Rains and Spiridonov [19]. **

Under the balancing condition, it turns out that the integrals $K_{\mu,\nu}(a;x,y)$ are continued to meromorphic functions on the whole hypersurface $a_{1}\cdots a_{m}t^{2n-2}=pq$ of $(\mathbb{C}^{*})^{m}$, provided that $|p|$ or $|q|$ is sufficiently small.

Theorem 1.3

Suppose that $|p|<|t|^{2n-2}$. Under the condition $a_{1}\cdots a_{m}t^{2n-2}=pq$ $(m=2r+4),$ the matrix $K(a;x,y)$ satisfies a system of $q$-difference equations with respect of $a=(a_{1},\ldots,a_{m})$ in the form

[TABLE]

where

[TABLE]

are $\binom{n+r-1}{r-1}\times\binom{n+r-1}{r-1}$ matrices whose entries are meromorphic functions in $a$.

Theorem 1.1 indicates that $K(a;x,y)$ is a fundamental matrix of solutions of the $q$-difference system (1.7).

Remark 1.4

If $|q|<|t|^{2n-2}$, the matrix $K(a;x,y)$ also satisfies the system of $p$-difference equations

[TABLE]

by symmetry with respect to $p$ and $q$. Although we imposed the condition $|p|<|t|^{2n-2}$ in Theorem 1.3, it may be possible to relax this restriction on $|p|$. **

1.4 Plan of this paper

This paper is organized as follows. In Section 2, we give a review of the elliptic interpolation functions of type $BC_{n}$ based on our previous work [13]. We also propose explicit formulas for the special values and for the transition coefficients between interpolation bases with different parameters. After this preparation, in Section 3 we formulate the method of $q$-difference de Rham theory in terms of the $q$-difference coboundary operator $\nabla_{\rm sym}^{\Psi}:\mathcal{H}_{s-1,n-1}^{(p)}\to\mathcal{H}_{s-1,n}^{(p)}$ $(m=2s+2)$. The cokernel of this operator, denoted by $H^{\Psi}_{\mathrm{sym}}$, plays the role of the “symmetrized $n$th $q$-difference de Rham cohomology group”. This method enables us to describe linear relations among the hypergeometric integrals on the algebraic level (Theorem 3.5). We prove in particular that $\dim_{\mathbb{C}}H_{\mathrm{sym}}^{\Psi}=\binom{n+s-2}{s-2}$ giving a $\mathbb{C}$-basis consisting of interpolation functions (Theorem 3.7). In Section 4, we apply the results of Section 3 to derive the system of $q$-difference equations for the elliptic hypergeometric integrals with respect to an interpolation basis of $H_{\mathrm{sym}}^{\Psi}$ (Theorems 4.1). Taking the determinant of the coefficient matrices, we obtain the system of $q$-difference equations for the determinant of the bilinear form $\langle f,g\rangle_{\Phi}$. In particular, we see that the determinant is expressed as a product of elliptic gamma functions, up to an unknown constant $c_{r,n}$ (Theorem 4.4). We determine in Section 5 the explicit value of the unknown constant through the recursive structure of asymptotic behavior of the integrals along the singularities. In the final section, we investigate the limiting cases of our main theorems as $p\to 0$, and derive determinant formulas for two types of $q$-hypergeometric integrals of type $BC_{n}$.

We expect that the results of this paper will be used as a foundation for further analysis of elliptic hypergeometric integrals.

2 Elliptic interpolation functions of type $BC_{n}$

As in Section 1.1, we consider the $\mathbb{C}$-vector space $\mathcal{H}_{s-1,n}^{(p)}$ ($s=1,2,\ldots$) of all quasi-periodic $W_{n}$-invariant holomorphic functions on $(\mathbb{C}^{\ast})^{n}$ of degree $s-1$ with respect to $p$. In this section we recall from [13] basic properties of the interpolation functions $E_{\mu}(c;z;p)$ $(\mu\in Z_{s,n})$. We remark that our elliptic interpolation functions for $s=2$ are essentially the special cases of interpolation theta functions of Coskun–Gustafson [5] and Rains [18] attached to single columns of partitions. In fact, $E_{\mu}(c;z;p)$ $(\mu\in Z_{s,n})$ for $s=2$ are compared explicitly with the functions of [5] and [18], respectively, in [11, Introduction]. Since our functions $E_{\mu}(c;z;p)$ for general $s$ are defined by an interpolation property of Lagrange type, they are different in nature from those of Coskun–Gustafson and Rains which are based on the triangularity with respect to partitions. It would be an interesting problem to clarify how these two approaches are related to each other.

We also propose explicit formulas for the special values $E_{\mu}(c;(u)_{t,n};p)$ and for the transition coefficients between interpolation bases with different $c$ parameters. Throughout this section we use the base $p$ only, and simply set $E_{\mu}(c;z)=E_{\mu}(c;z;p)$, $\theta(u)=\theta(u;p)$ and $e(u,v)=e(u,v;p)$.

2.1 Recursion formula

When $n=1$, the interpolation functions are parametrized by the canonical basis $\epsilon_{1},\ldots,\epsilon_{s}$ of $\mathbb{N}^{s}$, and given explicitly as

[TABLE]

When $n\geq 2$, they satisfy the recursion formula

[TABLE]

where $t^{\nu}c=(t^{\nu_{1}}c_{1},\ldots,t^{\nu_{s}}c_{s})$. This formula apparently depends on the ordering of the variables $z_{1},\ldots,z_{n}$, while $E_{\mu}(c;z)$ are $W_{n}$-invariant.

2.2 Explicit formula

[TABLE]

where $\mu^{(i)}=\epsilon_{k_{1}}+\cdots+\epsilon_{k_{i}}$ ($i=1,\ldots,n$).

2.3 Interpolation functions on the vertices and the faces

The indexing set $Z_{s,n}$ for the interpolation functions are the lattice points on the $(s-1)$-simplex

[TABLE]

in $\mathbb{R}^{s}$. On the vertices $\mu=n\epsilon_{k}$ ($k=1,\ldots,s$), the interpolation functions are factorized as

[TABLE]

where $e(u,v)_{k}=e(u,v)e(tu,v)\cdots e(t^{k-1}u,v)$ denotes the $t$-shifted factorial with respect to the first argument ($k=0,1,2,\ldots$). When $\mu\in Z_{s,n}$ is on the $s$th face ($\mu_{s}=0$), they are expressed as

[TABLE]

in terms of the interpolation functions with $s-1$ parameters $(c_{1},\ldots,c_{s-1})$.

2.4 Dual Cauchy formula

For each $\mu\in\mathbb{N}^{s}$ we define a holomorphic function $F_{\mu}(c;w)$ in $s-1$ variables $w=(w_{1},\ldots,w_{s-1})$ by

[TABLE]

Then we have the dual Cauchy formula

[TABLE]

2.5 Partition of the variables

For the variables $z=(z^{\prime},z^{\prime\prime})\in(\mathbb{C}^{*})^{n}$ divided into two parts $z^{\prime}=(z_{1},\ldots,z_{l})$ and $z^{\prime\prime}=(z_{l+1},\ldots,z_{n})$, the interpolation function $E_{\lambda}(c;z)$ is expressed as

[TABLE]

2.6 Special value

Theorem 2.1

For each $\mu\in Z_{s,n}$ the special value of $E_{\mu}(c;z)$ at $z=(u)_{t,n}=(u,tu,\ldots,t^{n-1}u)$ is given explicitly by

[TABLE]

where $[u]=u^{-\frac{1}{2}}\theta(u)$ and $[u]_{k}=[u][tu]\cdots[t^{k-1}u]$.

Remark 2.2

The above special value $E_{\mu}(c;(u)_{t,n})$ is expressed as

[TABLE]

in terms of the $t$-shifted factorials $\theta(u)_{k}=\theta(u)\theta(tu)\cdots\theta(t^{k-1}u)$ $(k=0,1,2,\ldots)$ of the theta function $\theta(u)$.

Proof of Theorem 2.1 $:$ 

By (2.2) the special value $E_{\mu}(c;(u)_{t,n})$ satisfies the recurrence formula

[TABLE]

Denoting by $C_{\mu}$ $(n=|\mu|)$ the right-hand side of (2.6), we verify that $C_{\mu}$ satisfies the same recurrence formula. Since we have

[TABLE]

for $\mu_{k}>0$, the above recurrence formula is equivalent to

[TABLE]

namely,

[TABLE]

This formula follows from the partial fraction decomposition

[TABLE]

as the special case where $z=t^{n-1}u$. $\square$

2.7 Change of parameters

We consider to expand the interpolation function $E_{\mu}(c;z)=E_{\mu}(c_{1},\ldots,c_{s};z)$ in terms of another interpolation basis $E_{\nu}(c_{1},\ldots,c_{s-1},d_{s};z)$ $(\nu\in Z_{s,n})$ by replacing $c_{s}$ with $d_{s}$:

[TABLE]

where $c^{\prime}=(c_{1},\ldots,c_{s-1})$. The transition coefficients $C_{\mu,\nu}(c^{\prime};c_{s},d_{s})$ in this case are computed as follows:

[TABLE]

where $\nu^{\prime}=(\nu_{1},\ldots,\nu_{s-1})$. Since

[TABLE]

we have

[TABLE]

which can be computed by Theorem 2.1.

Theorem 2.3

When we change the parameters $c=(c^{\prime},c_{s})$ to $(c^{\prime},d_{s})$, we have

[TABLE]

where $\mu^{\prime}\geq\nu^{\prime}$ means that $\mu_{k}\geq\nu_{k}$ $(k=1,\ldots,s-1)$. The coefficients $C_{\mu,\nu}(c^{\prime};c_{s},d_{s})$ are given explicitly by

[TABLE]

$\square$**

Remark 2.4

In terms of the ordinary $\theta(u)$ notation, we have

[TABLE]

In the succeeding section, Theorem 2.3 will be applied to interpolation functions with respect to subsets of the parameters $a=(a_{1},\ldots,a_{m})$ of $\Phi(z)$ where $m=2s+2$. For each subset $K\subseteq\left\{1,\ldots,m\right\}$ of the indexing set with $|K|=s$, we consider the interpolation basis

[TABLE]

of $\mathcal{H}_{s-1,n}^{(p)}$ with respect to the parameters $a_{K}=(a_{k})_{k\in K}$, where

[TABLE]

Let $I\subseteq\left\{1,\ldots,m\right\}$ be a subset with $|I|=s-1$, and choose two indices $k,l\in\left\{1,\ldots,m\right\}\backslash I$. Then Theorem 2.3 is reformulated in terms of the transition between the two bases $E_{\mu}(a_{I\cup\left\{k\right\}};z)$ and $E_{\nu}(a_{I\cup\left\{l\right\}};z)$:

[TABLE]

The transition coefficients $C^{I;k,l}_{\mu,\nu}$ are nonzero only if $\mu_{\widehat{k}}\geq\nu_{\widehat{l}}$, namely $\mu_{i}\geq\nu_{i}$ ($i\in I$); they are given explicitly by

[TABLE]

We denote by C^{I;k,l}=\big{(}C^{I;k,l}_{\mu,\nu}\big{)}_{\mu,\nu} the square matrix with row indices $\mu\in Z_{I\cup\left\{k\right\},n}$ and column indices $\nu\in Z_{I\cup\left\{l\right\},n}$ arranged by the partial ordering of $\mu_{\widehat{k}},\nu_{\widehat{l}}\in\mathbb{N}^{I}$. Then the diagonal entries of $C^{I;k,l}$ are given by

[TABLE]

for each pair $(\mu,\nu)$ such that $\mu_{i}=\nu_{i}$ $(i\in I)$ and $\mu_{k}=\nu_{l}$. Hence the determinant of the transition matrix C^{I;k,l}=\big{(}C_{\mu,\nu}^{I;k,l}\big{)}_{\mu,\nu} is computed as

[TABLE]

3 $q$-Difference de Rham theory

3.1 Definition of $H_{\mathrm{sym}}^{\Psi}$

In this section we suppose that $m=2r+4$ ($r=1,2,\ldots$) and set $s=r+1$. Fixing a nonzero $W_{n}$-invariant holomorphic function $g(z)\in\mathcal{H}^{(q)}_{s-1,n}$ with quasi-periodicity of degree $s-1$, we set

[TABLE]

For each $i=1,\ldots,n$, the function

[TABLE]

is computed as follows:

[TABLE]

Hence, $b_{i}^{\Psi}(z)$ is expressed as

[TABLE]

where

[TABLE]

This implies that

[TABLE]

Note that $f_{i}^{+}(z)$ and $f_{i}^{-}(z)$ have the following quasi-periodicity with respect to the $p$-shifts:

[TABLE]

If the parameters satisfy the balancing condition $t^{2n-2}a_{1}\cdots a_{m}=1$, then we have

[TABLE]

so that $f_{i}^{+}(z)$ and $f_{i}^{-}(z)$ have the same quasi-periodicity.

Lemma 3.1

Under the balancing condition $t^{2n-2}a_{1}\cdots a_{m}=1$ $(m=2s+2)$, for each $\varphi\in\mathcal{H}^{(p)}_{s-1,n-1}$,

[TABLE]

belongs to $\mathcal{H}_{s-1,n}^{(p)}$.

From the definition (3.1), it directly follows that $\psi(z)$ is a $W_{n}$-invariant meromorphic function with quasi-periodicity of degree $s-1$; $\psi(z)$ is in fact holomorphic on $(\mathbb{C}^{\ast})^{n}$, since $\psi(z)$ is $W_{n}$-invariant and $\Delta_{C}(z;p)\psi(z)$ is holomorphic, where

[TABLE]

denotes the elliptic version of the Weyl denominator of type $C_{n}$.

In view of this lemma, we define the $\mathbb{C}$-linear mapping $\nabla_{\mathrm{sym}}^{\Psi}:\ \mathcal{H}^{(p)}_{s-1,n-1}\to\mathcal{H}^{(p)}_{s-1,n}$ by

[TABLE]

Since $\nabla^{\Psi}_{\mathrm{sym}}\varphi$ is rewritten as

[TABLE]

it can be regarded as symmetrization of the coboudary operator for the $q$-difference de Rham cohomology [1, 2]. The cokernel

[TABLE]

plays the role of the “symmetrized $n$th $q$-difference de Rham cohomology group” associated with $\Psi(z)$. In this context the elements of $\mathcal{H}_{s-1,n}^{(p)}$ will be called $q$-cocyles, or simply cocycles. When $\psi_{1},\psi_{2}\in\mathcal{H}^{(p)}_{s-1,n}$, we denote by

[TABLE]

the congruence modulo $\nabla_{\mathrm{sym}}^{\Psi}\mathcal{H}^{(p)}_{s-1,n-1}$, i.e. $\psi_{1}(z)-\psi_{2}(z)\in\nabla_{\mathrm{sym}}^{\Psi}\mathcal{H}^{(p)}_{s-1,n-1}.$ If this is the case, it turns out that $\langle\psi_{1}\rangle_{\Psi}=\langle\psi_{2}\rangle_{\Psi}$, namely,

[TABLE]

by the Cauchy theorem, provided that $|a_{k}|<1$ ($k=1,\ldots,,m$) and $|t|<1$. In fact, we have

Lemma 3.2

Suppose that $|a_{k}|<1$ $(k=1,\ldots,,m)$ and $|t|<1$. If $\psi(z)\equiv_{\Psi}0$ for $\psi\in\mathcal{H}^{(p)}_{s-1,n}$, then we have $\langle\psi\rangle_{\Psi}=0$, namely,

[TABLE]

Proof $:$ 

For $\psi\in\mathcal{H}^{(p)}_{s-1,n}$, $\psi(z)\equiv_{\Psi}0$ is equivalent to

[TABLE]

If $|a_{k}|<1$ $(k=1,\ldots,,m)$ and $|t|<1$, one can verify that $\Psi(z)f_{i}^{-}(z)\varphi(z_{\,\widehat{i}})$ is holomorphic in a neighborhood of the compact set

[TABLE]

In fact, since $f_{i}^{-}(z)$ is explicitly written as

[TABLE]

in the product $\Psi(z)f_{i}^{-}(z)$, all possible poles of each of the two functions $\Psi(z)$ and $f_{i}^{-}(z)$

[TABLE]

relevant to the region (3.2) are eliminated by zeros of the other. Hence, by the Cauchy theorem, we have

[TABLE]

so that

[TABLE]

This implies that

[TABLE]

This completes the proof. $\square$

3.2 Reduction of cocycles

Choosing $s$ parameters $a_{1},\ldots,a_{s}$ from $a=(a_{1},\ldots,a_{m})$, we consider the interpolation basis

[TABLE]

for $\mathcal{H}_{s-1,n}^{(p)}$ with respect to the parameters $a_{\left\{1,\ldots,s\right\}}=(a_{1},\ldots,a_{s})$. In this subsection we use the notations $\theta(z)=\theta(z;p)$ and $E_{\mu}(a_{\left\{1,\ldots,s\right\}};z)=E_{\mu}(a_{\left\{1,\ldots,s\right\}};z;p)$, omitting the base $p$.

For each $\lambda\in Z_{s,n-1}$, we take the interpolation function $\varphi_{\lambda}=E_{\lambda}(a_{\left\{1,\ldots,s\right\}};\cdot)\in\mathcal{H}_{s-1,n-1}^{(p)}$ and set $\psi_{\lambda}=\nabla_{\mathrm{sym}}^{\Psi}\varphi_{\lambda}\in\mathcal{H}^{(p)}_{s-1,n}$:

[TABLE]

Lemma 3.3

For each $\lambda\in Z_{s,n-1}$, $\psi_{\lambda}(z)$ defined by (3.3) is expressed in terms of the interpolation basis of $\mathcal{H}_{s-1,n}^{(p)}$ as

[TABLE]

where

[TABLE]

Proof $:$ 

Noting that

[TABLE]

we determine the values of $\psi_{\lambda}(z)$ at $z=(a_{\left\{1,\ldots,s\right\}})_{t,\mu}$ by means of (3.3).

Fixing $\mu\in Z_{s,n}$, suppose that $z_{i}=t^{\nu}a_{k}$ ($k=1,\ldots,s$; $0\leq\nu<\mu_{k}$) under the substitution $z=(a_{\left\{1,\ldots,s\right\}})_{t,\mu}$. If $\nu>0$, we have $f_{i}^{-}((a_{\left\{1,\ldots,s\right\}})_{t,\lambda})=0$ since $i\geq 2$ and $f_{i}^{-}(z)$ has the factor $\theta(tz_{i-1}/z_{i})$. If $\nu=0$, then $z_{i}=a_{k}$, and $f_{i}^{-}((a_{\left\{1,\ldots,s\right\}})_{t,\mu})=0$ since $f_{i}^{-}(z)$ has the factor $\theta(a_{k}/z_{i})$. This means that $f^{-}_{i}((a_{\left\{1,\ldots,s\right\}})_{t,\mu})=0$ for any $\mu$.

As to $f_{i}^{+}(z)$, if $\nu<\mu_{k}-1$, then $f_{i}^{+}((a_{\left\{1,\ldots,s\right\}})_{t,\mu})=0$ since $i<n$ and $f_{i}^{+}(z)$ have the factor $\theta(tz_{i}/z_{i+1})$. Assume that $\nu=\mu_{k}-1$. Since $z_{\,\widehat{i}}=(a_{\left\{1,\ldots,s\right\}})_{t,\mu-\epsilon_{k}}$ in this case, we have

[TABLE]

Hence, $\psi_{\lambda}((a_{\left\{1,\ldots,s\right\}}))_{t,\mu})$ is nontrivial only when $\mu=\lambda+\epsilon_{k}$ for some $k=1,\ldots,m$, and

[TABLE]

which gives the explicit expression of $C_{\lambda,k}$. $\square$

Remark 3.4

In the expansion of $\psi_{\lambda}(z)$ $(\lambda\in Z_{s,n-1})$ in terms of the interpolation basis of $\mathcal{H}^{(p)}_{s,n}$, its leading term is given by $E_{\lambda+\epsilon_{1}}(a_{\left\{1,\ldots,s\right\}};z)$ with respect to the lexicographic ordering of $Z_{s,n}$. Hence, for generic values of $a_{1},\ldots,a_{s}$, the functions $\psi_{\lambda}(z)$ $(\lambda\in Z_{s,n-1})$ are linearly independent over $\mathbb{C}$. This means that the $\mathbb{C}$-linear mapping $\nabla_{\mathrm{sym}}^{\Psi}:\ \mathcal{H}^{(p)}_{s-1,n-1}\to\mathcal{H}^{(p)}_{s-1,n}$ is injective. Hence

[TABLE]

In particular, we have $\dim_{\mathbb{C}}H_{\mathrm{sym}}^{\Psi}=1$ if $s=2$ ($r=1$), and $\dim_{\mathbb{C}}H_{\mathrm{sym}}^{\Psi}=n+1$ if $s=3$ ($r=2$). $\square$

From Lemma 3.3, for each $\lambda\in Z_{s,n-1}$ we have the congruence

[TABLE]

modulo $\nabla_{\mathrm{sym}}^{\Psi}\mathcal{H}_{s-1,n-1}^{\Psi}$, or equivalently,

[TABLE]

in terms of the odd theta function $[u]=u^{-\frac{1}{2}}\theta(u)$. For each $\mu\in\mathbb{N}^{s}$ we set

[TABLE]

so that

[TABLE]

Then, formula (3.4) implies that the renormalized interpolation functions

[TABLE]

satisfy

[TABLE]

Therefore we obtain

[TABLE]

and by putting $\lambda+\epsilon_{s}=\mu$,

[TABLE]

Using this formula, we can lower the last component of $\mu$ step by step.

Theorem 3.5

Suppose that $t^{2n-2}a_{1}\cdots a_{m}=1$ $(m=2s+2)$. For each $\mu\in Z_{s,n}$ and for each integer $l$ satisfying $0\leq l\leq\mu_{s}$, we have

[TABLE]

where

[TABLE]

Remark 3.6

The coefficient $R_{\mu,\nu}$ is also written as

[TABLE]

Proof $:$  We use induction on $l$ to prove that the coefficients $R_{\mu,\nu}$ defined by (3.7) satisfy (3.6). When $l=1$,

[TABLE]

recover the coefficients in (3.5). Under (3.6) for $l-1$ as the assumption of induction, we have

[TABLE]

Hence it suffices to show

[TABLE]

for each $\nu\in Z_{s,n}$ with $\nu_{s}=\mu_{s}-l$, $\mu^{\prime}\leq\nu^{\prime}$. Under the condition $\mu^{\prime}\leq\lambda^{\prime}\leq\nu^{\prime}$, by (3.7) we have

[TABLE]

In the right-hand side of (3.8), each $\lambda$ is expressed as $\lambda=\nu-\epsilon_{k}+\epsilon_{s}$ for some $k\in\left\{1,\ldots,s-1\right\}$ such that $\mu_{k}<\nu_{k}$. Then (3.9) is rewritten as

[TABLE]

Note that the partial fraction decomposition

[TABLE]

implies

[TABLE]

as the special case $z=t^{\nu_{s}}a_{s}$. By (3.10) and (3.11) the right-hand side of (3.8) is computed as

[TABLE]

which completes the proof of Theorem 3.5. $\square$

Theorem 3.5 implies that each $E_{\mu}(a_{\left\{1,\ldots,s\right\}};z)$ can be reduced to a linear combination of interpolation functions on the $s$th face under the congruence in $H_{\mathrm{sym}}^{\Psi}=\mathcal{H}^{(p)}_{s-1,n}/\nabla_{\mathrm{sym}}^{\Psi}\mathcal{H}^{(p)}_{s-1,n-1}$. Namely, for each $\mu\in Z_{s,n}$ we have

[TABLE]

where the coefficients are given by

[TABLE]

Recall that

[TABLE]

by (2.4), and that $E_{\nu^{\prime}}(a_{\left\{1,\ldots,s-1\right\}};z)$ $(\nu^{\prime}\in Z_{s-1,n})$ form a basis of $\mathcal{H}^{(p)}_{s-2,n}$. The above argument implies that the composition

[TABLE]

is surjective. Since $\dim_{\mathbb{C}}\mathcal{H}^{(p)}_{s-2,n}=\dim_{\mathbb{C}}H_{\mathrm{sym}}^{\Psi}=\binom{n+s-2}{s-2}$, we obtain a natural $\mathbb{C}$-isomorphism

[TABLE]

Hence we have

Theorem 3.7

Under the assumption of Theorem 3.5, the classes

[TABLE]

modulo $\nabla_{\mathrm{sym}}^{\Psi}\mathcal{H}^{(p)}_{s-1,n-1}$ form a $\mathbb{C}$-basis of $H_{\mathrm{sym}}^{\Psi}=\mathcal{H}^{(p)}_{s-1,n}/\nabla_{\mathrm{sym}}^{\Psi}\mathcal{H}^{(p)}_{s-1,n-1}$. $\square$

3.3 Base change in $H_{\mathrm{sym}}^{\Psi}$

In what follows we set $s=r+1$, so that $m=2r+4$ and $H_{\mathrm{sym}}^{\Psi}=\mathcal{H}^{(p)}_{r,n}/\nabla_{\mathrm{sym}}^{\Psi}\mathcal{H}^{(p)}_{r,n-1}$. Let $K$ be a subset of the indexing set $\left\{1,\ldots,m\right\}$ with $|K|=r+1$, and consider the the interpolation basis

[TABLE]

with respect to the parameters $a_{K}=(a_{k})_{k\in K}\in(\mathbb{C}^{\ast})^{K}$, where

[TABLE]

Then, for each subset $J\subset K$ with $|J|=r$, $K=J\cup\left\{k\right\}$, the classes

[TABLE]

modulo $\nabla_{\mathrm{sym}}^{\Psi}\mathcal{H}^{(p)}_{r,n-1}$ form a $\mathbb{C}$-basis of $H_{\mathrm{sym}}^{\Psi}$.

We now fix a subset $I$ of $\left\{1,\ldots,m\right\}$ with $|I|=r-1$, and choose two indices $k,l\in\left\{1,\ldots,m\right\}\backslash I$. In this setting, we consider the transition between the two bases of the form (3.14) of $H_{\mathrm{sym}}^{\Psi}$ specified by $I\cup\left\{k\right\}$ and $I\cup\left\{l\right\}$. We define the transition coefficients $B_{\mu,\nu}^{I;k,l}$ through the relation

[TABLE]

where $\mu_{\widehat{k}}=(\mu_{i})_{i\in I}$ and $\nu_{\widehat{l}}=(\nu_{i})_{i\in I}$. These coefficients are directly computed by (3.12) and (3.13) as follows:

[TABLE]

We remark that the matrix B^{I;k,l}=\big{(}B_{\mu,\nu}^{I;k,l}\big{)}_{\mu,\nu} is upper triangular with respect to the partial ordering of $\mu_{\widehat{k}},\nu_{\widehat{l}}\in\mathbb{N}^{I}$. The diagonal components (with $\mu_{\widehat{k}}=\nu_{\widehat{l}}$ , $\mu_{k}=\nu_{l}$) are given by

[TABLE]

Hence the determinant of $B^{I;k,l}$ is computed as

[TABLE]

4 System of $q$-difference equations

4.1 System of $q$-difference equations

associated with a basis of $\mathcal{H}^{(p)}_{r-1,n}$

In this section, in view of the parameter dependence of $\Phi(z)=\Phi(z;a)$ we investigate $q$-difference equations to be satisfied by the integrals

[TABLE]

with respect to $a$. Here we assume that $m=2r+4$ and that the parameters $a=(a_{1},\ldots,a_{m})\in(\mathbb{C}^{\ast})^{m}$ satisfy the conditions $|a_{k}|<1$ ($k=1,\ldots,m$) and $|t|<1$ as in Lemma 3.2. Fixing a $\mathbb{C}$-basis $\{f_{\mu}(z)\ |\ \mu\in Z_{r,n}\}$ of $\mathcal{H}_{r-1,n}^{(p)}$ and a holomorphic function $g(z)$ in $\mathcal{H}^{(q)}_{r-1,n}$, we consider the integrals

[TABLE]

As we will see below, under the balancing condition $t^{2n-2}a_{1}\cdots a_{m}=pq$, the column vector $\mathcal{I}=(\mathcal{I}_{\mu})_{\mu\in Z_{r,n}}$ satisfies a system of $q$-difference equations of the form

[TABLE]

of rank $\binom{n+r-1}{n}$, where the coefficient matrices $\mathcal{A}^{k,l}(a)=(\mathcal{A}_{\mu\nu}^{k,l}(a))_{\mu,\nu\in Z_{r,n}}$ are determined independently of the choice of $g(z)$. In this setting, $f_{\mu}(z)$ and $g(z)$ may depend meromorphically on $a\in(\mathbb{C}^{*})^{m}$, while we assume that $g(z)$ satisfies the condition

[TABLE]

We say that a meromorphic function $g(z)=g(z;a)$ on $(\mathbb{C}^{*})^{n}\times(\mathbb{C}^{*})^{m}$ depends meromorphically on $a$, if there exists a holomorphic function $h(a)$ on $(\mathbb{C}^{*})^{m}$ such that $h(a)g(z;a)$ is holomorphic on $(\mathbb{C}^{*})^{n}\times(\mathbb{C}^{*})^{m}$.

4.2 Derivation of $q$-difference equations

We explain how one can derive the $q$-difference equations (4.2) in the case $l=m$. Shifting $a_{m}$ by $pq$, we have

[TABLE]

and hence

[TABLE]

where $\widetilde{f}_{\mu}(z)=T_{pq,a_{m}}f_{\mu}(z)$ and $\widetilde{g}(z)=T_{pq,a_{m}}g(z)$. Setting

[TABLE]

we rewrite the formula above as

[TABLE]

Note that $\varphi_{\mu}(z)\in\mathcal{H}_{r,n}^{(p)}$ $(\mu\in Z_{r,n})$ and $\psi(z)\in\mathcal{H}_{r,n}^{(q)}$, since $f_{\mu}(z)\in\mathcal{H}_{r-1,n}^{(p)}$ and $g(z)\in\mathcal{H}_{r-1,n}^{(q)}$. Then we have

[TABLE]

where $\Psi(z)=\psi(z)\Phi(z)$. We first prove that the integrals

[TABLE]

satisfy a system of $q$-difference equations of the form

[TABLE]

for $k=1,\ldots,m-1$, under the balancing condition $t^{2n-2}a_{1}\cdots a_{m}=1$. Then, applying $T_{pq,a_{m}}^{-1}$ we see that $\mathcal{I}_{\mu}$ $(\mu\in Z_{r,n})$ satisfy the system of $q$-difference equations

[TABLE]

under the balancing condition $t^{2n-2}a_{1}\cdots a_{m}=pq$, where $\mathcal{A}_{\mu,\nu}^{k,m}(a)=T_{pq,a_{m}}^{-1}\widetilde{\mathcal{A}}_{\mu,\nu}^{k,m}(a)$.

We now assume that $t^{2n-2}a_{1}\cdots a_{m}=1$. Note that

[TABLE]

Also, by (4.3) we have

[TABLE]

Hence obtain

[TABLE]

Since $f_{\nu}(z)$ ($\nu\in Z_{r,n}$) form a $\mathbb{C}$-basis of $\mathcal{H}_{r-1,n}^{(p)}$, by Theorem 3.7 the congruence classes of

[TABLE]

form a $\mathbb{C}$-basis of $H_{\mathrm{sym}}^{\Psi}=\mathcal{H}_{r,n}^{(p)}/\nabla^{\Psi}_{\mathrm{sym}}\mathcal{H}_{r,n-1}^{(p)}$. This implies that

[TABLE]

for some $\widetilde{\mathcal{A}}_{\mu,\nu}^{k,m}(a)$. Hence we have

[TABLE]

4.3 System of $q$-difference equations

associated with an interpolation basis

We now consider the case of the interpolation basis

[TABLE]

and investigate the system of $q$-difference equations (4.2) to be satisfied by the integrals

[TABLE]

for a fixed $g(z)\in\mathcal{H}^{(q)}_{r-1,n}$. We assume that $g(z)$ depends meromorphically on $a\in(\mathbb{C}^{*})^{m}$ and satisfies the $q$-shift invariance $T_{q,a_{k}}T_{q,a_{l}}^{-1}g(z)=g(z)$ ($1\leq k<l\leq m$).

We suppose below that $a_{1}\cdots a_{m}t^{2n-2}=pq$, and regard $a_{m}=pq/a_{1}\cdots a_{m-1}t^{2n-2}$ as a function of $(a_{1},\ldots,a_{m-1})$. Then the integral $\mathcal{I}_{\mu}(a)$, regarded as a function of $(a_{1},\ldots,a_{m-1})$, is meromorphic on the open subset

[TABLE]

of $(\mathbb{C}^{\ast})^{m-1}$; we need to assume $|p||q|<|t|^{2n-2}$ in order to ensure that $U_{0}$ is not empty. If we assume further that $|p|<|t|^{2n-2}$, the integrals $T_{q,a_{k}}T_{q,a_{m}}^{-1}\mathcal{I}_{\mu}(a)$ $(k=1,\ldots,m-1)$ as well as $\mathcal{I}_{\mu}(a)$ are meromorphic on the nonempty open subset

[TABLE]

of $U_{0}$.

Theorem 4.1

Suppose that $|p|<|t|^{2n-2}$. Under the balancing condition $t^{2n-2}a_{1}\cdots a_{m}=pq$, the integrals $\mathcal{I}_{\mu}$ of (4.4) satisfy a system of $q$-difference equations of the form

[TABLE]

for each $k\in\left\{1,\ldots,m-1\right\}$, on the nonempty open set $V_{0}\subset U_{0}$. Here the coefficients $\mathcal{A}^{k,m}_{\mu,\nu}(a)$ are meromorphic functions in $a$, and do not depend on the choice of $g(z)$. Furthermore, the determinant of the coefficient matrices $\mathcal{A}^{k,m}(a)$ are given as follows $:$ For $k\in\left\{1,\ldots,r\right\}$

[TABLE]

and for $k\in\left\{r+1,\ldots,m-1\right\}$

[TABLE]

Proof $:$ 

Note that $\widetilde{f}_{\mu}(z)=f_{\mu}(z)$ in this case, and that

[TABLE]

for $k=1,\ldots,m-1$. We investigate the two cases $k\in\left\{1,\ldots,r\right\}$ and $k\in\left\{r+1,\ldots,m-1\right\}$ separately.

When $k\in\left\{1,\ldots,r\right\}$, by the change of parameters $a_{k}\to a_{m}$, we have

[TABLE]

where $I_{k}=\left\{1,\ldots,r\right\}\backslash\left\{k\right\}$ and the coefficients $C_{\mu,\alpha}^{I_{k};k,m}$ are specified by (2.8). Hence,

[TABLE]

We now apply (3.15) for the reduction in $H^{\Psi}_{\mathrm{sym}}$ from the $k$th face to the $m$th face:

[TABLE]

Hence we have

[TABLE]

When $k\in\left\{r+1,\ldots,m-1\right\}$, choosing an index $l\in\left\{1,\ldots,r\right\}$ arbitrarily, we apply the change of parameters $a_{l}\to a_{m}$ in advance, and then perform the reduction from the $k$th face to the $m$th face:

[TABLE]

Hence we have

[TABLE]

As we explained before, the coefficient matrices $\mathcal{A}^{k,m}(a)$ are determined from $\widetilde{\mathcal{A}}^{k,m}(a)$ by $\mathcal{A}^{k,m}(a)=T_{pq,a_{m}}^{-1}(\widetilde{\mathcal{A}}^{k,m}(a))$.

Note that the matrices C^{l;k,l}=\big{(}C_{\mu,\nu}^{I;k,l}\big{)}_{\mu,\nu} and B^{I;k,l}=\big{(}B_{\mu,\nu}^{I;k,l}\big{)}_{\mu,\nu} are lower triangular and upper triangular respectively, with respect to the partial ordering of $\mathbb{N}^{I}$. Hence, the determinant of $\widetilde{\mathcal{A}}^{k,m}(a)=(\widetilde{\mathcal{A}}_{\mu,\nu}^{k,m}(a))_{\mu,\nu\in Z_{r,n}}$ is computed by (2.9) and (3.16) as follows: For $k\in\left\{1,\ldots,r\right\}$,

[TABLE]

and for $k\in\left\{r+1,\ldots,m-1\right\}$

[TABLE]

The determinants $\det\mathcal{A}^{k,m}(a)$ are obtained from these by applying $T_{pq,a_{m}}^{-1}$. $\square$

Remark 4.2

Under the assumption of Theorem 4.1, by the $q$-difference equations (4.5) the integrals $\mathcal{I}_{\mu}(a)$ $(\mu\in Z_{r,n})$, regarded as functions on $(a_{1},\ldots,a_{m-1})$, are continued meromorphically to the whole algebraic torus $(\mathbb{C}^{*})^{m-1}$, and hence define meromorphic functions on the hypersurface $a_{1}\cdots a_{m}t^{2n-2}=pq$ in $(\mathbb{C}^{*})^{m}$. **

4.4 Symmetry of the difference system

with respect to $(p,q)$.

In Theorem 4.1, under the condition $|p|<|t|^{2n-2}$ we derived the system of $q$-difference equations for the integrals

[TABLE]

defined by the interpolation basis of $\mathcal{H}^{(p)}_{r-1,n}$ and a holomorphic function $g(z)\in\mathcal{H}^{(q)}_{r-1,n}$. In this formulation, we imposed on $g(z)$ the $q$-shift invariance with respect to the $a$ parameters so that the coefficient matrices should not depend on the choice of $g(z)$. We now modify the interpolation basis appropriately in order to make the $q$-difference system consistent with the $(p,q)$ symmetry of the bilinear form

[TABLE]

Recall the dual Cauchy formula (2.5) for the interpolation functions $E_{\mu}(a_{\left\{1,\ldots,r\right\}};z;p)$: For two sets of variables $z=(z_{1},\ldots,z_{n})$ and $w=(w_{1},\ldots,w_{r-1})$, we have

[TABLE]

where

[TABLE]

In view of this formula, we set

[TABLE]

Lemma 4.3

The functions $f_{\mu}(z;w;p)$ defined by (4.12) are invariant with respect to the $p$-shifts in the $a$ parameters, namely,

[TABLE]

Proof $:$ 

Since $f_{\mu}(z;w;p)$ do not depend on $a_{k}$ ($k=r+1,\ldots,m$), we show the invariance of $f_{\mu}(z;w;p)$ with respect to $T_{p,a_{k}}$ ($k=1,\ldots,r$). Applying $T_{p,a_{k}}$ to (4.10) we have

[TABLE]

By (4.11) it is directly checked that

[TABLE]

Since $F_{\mu}(a_{\left\{1,\ldots,r\right\}};w;p)$ are linearly independent as functions in $w$, we see that

[TABLE]

It also implies that the functions $f_{\mu}(z;w;p)=E_{\mu}(a_{\left\{1,\ldots,r\right\}};z;p)F_{\mu}(a_{\left\{1,\ldots,r\right\}};w;p)$ are invariant with respect to $T_{p,a_{k}}$ ($k=1,\ldots,r$). $\square$

Introducing a new set of parameters $u=(u_{1},\ldots,u_{r-1})$, we consider the integrals

[TABLE]

Then the system of $q$-difference equations to be satisfied by $I_{\mu}=I_{\mu}(a)$ are given by

[TABLE]

where

[TABLE]

To be more precise, we have

[TABLE]

Hence the determinant of the matrix $A^{k,m}(a)=(A^{k,m}_{\mu,\nu}(a))_{\mu,\nu\in Z_{r,n}}$ is computed by (4.6), (4.7) as

[TABLE]

for $k\in\left\{1,\ldots,r\right\}$, and

[TABLE]

for $k\in\left\{r+1,\ldots,m-1\right\}$. When we need to make the bases $p,q$ and the parameters $u$ explicit we use the notation $A^{k,m}(a;u;p,q)$ for $A^{k,m}(a)$.

In order to deal with the two bases $(p,q)$ on an equal footing, we introduce two sets of parameters $u=(u_{1},\ldots,u_{r-1})$, $v=(v_{1},\ldots,v_{r-1})$, and define

[TABLE]

for $\mu,\nu\in Z_{r,n}$. We suppose that $|p|<|t|^{2n-2}$ and $|q|<|t|^{2n-2}$. Then, by the symmetry with respect to $(p,q)$ the square matrix

[TABLE]

satisfies the following system of $q$- and $p$-difference equations with respect to the $a$ parameters: For each $k=1,\ldots,m-1$,

[TABLE]

or equivalently

[TABLE]

in the matrix notation. Hence the determinant $J=\det I(a;u,v)$ satisfies the $q$- and $p$-difference equations

[TABLE]

of rank one. By inspecting the explicit formulas (4.13), (4.14) for the determinants of the coefficient matrices, it is directly verified that the function

[TABLE]

provides with a particular solution of the system of $q$- and $p$- difference equations. Since the two meromorphic functions $\det I(a;u,v)$ and $J_{0}(a;u,v)$ both satisfy the difference equations (4.19), the ratio $\det I(a;u,v)/J_{0}(a;u,v)$ is invariant with respect to the $q$- and $p$-shifts in the $a$ parameters simultaneously. This implies that this ratio is a constant which does not depend on the $a$ parameters.

Theorem 4.4

Suppose that $|p|<|t|^{2n-2}$ and $|q|<|t|^{2n-2}$. Under the condition $t^{2n-2}a_{1}\cdots a_{m}=pq$ with $m=2r+4$, let $I(a;u,v)=(I_{\mu,\nu}(a;u,v))_{\mu,\nu\in Z_{r,n}}$ be the square matrix defined by the integrals (4.15). The determinant of $I(a;u,v)$ is expressed as

[TABLE]

where $c_{r,n}$ is a constant which does not depend on the parameters $a=(a_{1},\ldots,a_{m})$.

In the next section, we will give an explicit formula for $c_{r,n}$ as a function of $(p,q,t)$ as in Theorem 1.1. Since $c_{r,n}\neq 0$, $I(a;u,v)$ is in fact a fundamental matrix solution of the system of $q$-difference equations

[TABLE]

As in the previous subsection, we consider the integrals

[TABLE]

defined by the interpolation bases for $\mathcal{H}_{r-1,n}^{(p)}$, $\mathcal{H}_{r-1,n}^{(q)}$, and set K(a)=\big{(}K_{\mu,\nu}(a)\big{)}_{\mu,\nu\in Z_{r,n}}. Then we have

[TABLE]

which implies

[TABLE]

By Theorem 4.4 the determinant of the matrix $K(a)$ is expressed as follows.

Corollary 4.5

Under the condition $t^{2n-2}a_{1}\cdots a_{m}=pq$ with $m=2r+4$, we have

[TABLE]

where $c_{r,n}$ is a constant independent of the parameters $a=(a_{1},\ldots,a_{m})$.

As in Theorem 1.1 we now consider the integrals

[TABLE]

defined by the interpolation bases for $\mathcal{H}_{r-1,n}^{(p)}$, $\mathcal{H}_{r-1,n}^{(q)}$ with respect to generic parameters $x=(x_{1},\ldots,x_{r})$, $y=(y_{1},\ldots,y_{r})$. Note that

[TABLE]

by the property of interpolation functions. Also, by [13, Theorem 4.1] the determinants of these transition matrices are given by

[TABLE]

Hence the determinant of the matrix K(a;x,y)=\big{(}K_{\mu,\nu}(a;x,y)\big{)}_{\mu,\nu\in Z_{r,n}} is computed as

[TABLE]

Then, by Corollary 4.5 we obtain the following expression for $\det K(a;x,y)$.

Corollary 4.6

Under the condition $t^{2n-2}a_{1}\cdots a_{m}=pq$ with $m=2r+4$, we have

[TABLE]

where $c_{r,n}$ is a constant independent of the parameters $a=(a_{1},\ldots,a_{m})$.

Remark 4.7

We compute the constant $c_{r,n}$ later in Section 5, and eventually see that

[TABLE]

As a consequence, Corollary 4.6 with the explicit formula (4.23) of $c_{r,n}$ implies Theorem 1.1. Once the constant $c_{r,n}$ has been determined, we see that Theorem 4.4 and its corollaries are valid for $|p|<1$ and $|q|<1$ without any particular restriction. **

The system of $q$- and $p$-difference equations for the matrix $K(a;x,y)$ as stated in Theorem 1.3 can be derived from the system (4.18) for $I(a;u,v)$. From (4.12) and (4.22) the transition between $E_{\mu}(x;z;p)$ and $f_{\alpha}(z;u;p)$ is given by

[TABLE]

where

[TABLE]

From (4.15) and (4.21) we have

[TABLE]

where

[TABLE]

Since $G(a;y,v;q)$ is invariant under the $q$-shifts in $a$ parameters, by (4.18), for $k=1,\ldots,m-1$, we have

[TABLE]

and hence,

[TABLE]

where

[TABLE]

Note that these matrices are actually independent of $u$ as can be seen from (4.24), provided $\det K(a;x,y)\neq 0$. Since $K(a;x,y)$ is invariant under the permutation of $a_{1},\ldots,a_{m}$, the $q$-difference equations (1.7) are obtained from (4.25) by symmetry. Also, the $p$-difference equations (1.8) follow from the symmetry of $K(a;x,y)$ with respect to $p$ and $q$. This completes the proof of Theorem 1.3 under the assumption $c_{r,n}\neq 0$.

5 Computation of the constants $c_{r,n}$

5.1 Determinant of the bilinear form

In this section, we use the notation $K^{(r,n)}_{\mu,\nu}(a)$ and $\Phi_{n}(z;a)$ for $K_{\mu,\nu}(a)$ and $\Phi(z)=\Phi(z;a;p,q)$ respectively, in order to make explicit the dependence on $(r,n)$ and $a=(a_{1},\ldots,a_{m})$. Namely,

[TABLE]

Under the conditions $|p|<|t|^{2n-2}$, $|q|<|t|^{2n-2}$ and $t^{2n-2}a_{1}\cdots a_{m}=pq$ $(m=2r+4)$, by Corollary 4.5 the determinant of the matrix K^{(r,n)}(a)=\big{(}K^{(r,n)}_{\mu,\nu}(a)\big{)}_{\mu,\nu\in Z_{r,n}} is expressed as

[TABLE]

with a constant $c_{r,n}$ which does not depend on $a$, where

[TABLE]

In the following, we determine the unknown constant $c_{r,n}$ by comparing the asymptotic behavior of the two meromorphic functions $\det K^{(r,n)}(a)$ and $L_{r,n}(a)$ around their poles.

5.2 Asymptotic behavior of $L_{r,n}(a)$

Among the parameters $a_{1},\ldots,a_{m}$ ($m=2r+4$), we choose two parameters $a_{1}$ and $a_{r+1}$ and analyze the singularity of $L_{r,n}(a)$ along the pole $1-a_{1}a_{r+1}=0$. Since $L_{r,n}(a)$ has the factor

[TABLE]

it has a pole of multiplicity $\binom{n+r-2}{r-1}$ along the hypersurface $1-a_{1}a_{r+1}=0$. We compute the limit

[TABLE]

as $a_{r+1}\to a_{1}^{-1}$. In this procedure, we regard $(a_{1},\ldots,a_{m-1})$ as independent variables and $a_{m}=pq/t^{2n-2}a_{1}\cdots a_{m-1}$ as a function of $(a_{1},\ldots,a_{m-1})$. Note that as $a_{r+1}\to a_{1}^{-1}$, $a_{m}$ has the limit

[TABLE]

Also, in the notation $\widetilde{L}_{r,n}(\widetilde{a})$, $\widetilde{a}$ stands for

[TABLE]

The limit $\widetilde{L}_{r,n}(\widetilde{a})$ is computed explicitly as follows. We first rewrite $L_{r,n}(a)$ as

[TABLE]

Since

[TABLE]

we obtain

[TABLE]

Here $a_{m}$ in the right-hand side should be understood as its limit $\widetilde{a}_{m}=pq/t^{2n-2}a_{2}\cdots a_{r}a_{r+2}\cdots a_{m-1}$.

5.3 Remark on analytic continuation

Before proceeding to asymptotic analysis of $\det K^{(r,n)}(a)$, we give a general remark on analytic continuation of the integral

[TABLE]

for a holomorphic function $f(z;a)$ on $(\mathbb{C}^{\ast})^{n}\times(\mathbb{C}^{\ast})^{m}$, which defines a holomorphic function on the domain

[TABLE]

This function can be continued to a holomorphic function on a larger domain by replacing $\mathbb{T}^{n}$ with an appropriate $n$-cycle depending on the parameters $(a_{1},\ldots,a_{m})$.

Notice that $\Phi_{n}(z;a)$ has poles possibly along the divisors

[TABLE]

Also, regarded as a function of $z_{i}$ ($i=1,\ldots,n$), it has poles possibly at

[TABLE]

where $1\leq k\leq m$, $1\leq j\leq n$, $j\neq i$ and $\mu,\nu\in\mathbb{N}$. In view of this fact, for each $a=(a_{1},\ldots,a_{m})\in(\mathbb{C}^{\ast})^{m}$, we define two subsets $S_{0}$, $S_{\infty}$ of $\mathbb{C}^{\ast}$ by

[TABLE]

and suppose that $S_{0}\cap S_{\infty}=\phi$, namely $a_{k}a_{l}\notin p^{-\mathbb{N}}q^{-\mathbb{N}}$ $(1\leq k,l\leq m)$. Assuming that $|t|<\rho^{2}$ for some $\rho\in(0,1]$, we choose a circle

[TABLE]

which does not intersect with $S_{0}\cup S_{\infty}$. Then we define a cycle $C$ in $\mathbb{C}^{\ast}$ by

[TABLE]

where $C_{\varepsilon}(c)$ denotes a sufficiently small circle around $c$. Note that, if $|a_{k}|<1$ ($k=1,\ldots,m$), then $C$ is homologous to the unit circle.

We now assume that $|a_{k}|<\rho^{-1}$ ($k=1,\ldots,m$). Then such a cycle $C$ can be taken inside the annulus $\{u\in\mathbb{C}^{\ast}\ |\ \rho\leq|u|\leq\rho^{-1}\}$. Since $|t|<\rho^{2}$, the meromorphic function $\Phi_{n}(z;a)$ is holomorphic in an neighborhood of the $n$-cycle $C^{n}=C\times\cdots\times C$. Hence, the integral

[TABLE]

is well defined, and does not depend on the choice of $\delta\in[\rho,\rho^{-1}]$. This implies the following lemma on analytic continuation.

Lemma 5.1

Suppose that $|t|<\rho^{2}$ for some $\rho\in(0,1]$. Then the holomorphic function (5.2) on the domain $U$ can be continued by the integral (5.3) to a holomorphic function on

[TABLE]

$\square$**

We remark that, when $f(z;a)$ depends meromorphically on $a$, the integral (5.2) is continued similarly to a meromorphic function on the domain (5.4).

5.4 Asymptotic behavior of $\det K^{(r,n)}(a)$

Applying the same procedure as in Subsection 5.2 to $\det K^{(r,n)}(a)$, we compute the limit

[TABLE]

Note here that the power $\binom{n+r-2}{r-1}$ is the cardinality of $Z_{r,n-1}$. The indexing set $Z_{r,n}$ for the matrix $K^{(r,n)}(a)$ is divided into two parts as

[TABLE]

according as $\mu_{1}=0$ or $\mu_{1}>0$. Since $Z_{r-1,n}\stackrel{{\scriptstyle\sim}}{{\to}}Z_{r,n}^{0}$ and $Z_{r,n-1}\stackrel{{\scriptstyle\sim}}{{\to}}Z_{r,n}^{+}=Z_{r,n-1}+\epsilon_{1}$,

[TABLE]

and the above decomposition of $Z_{r,n}$ corresponds the identity $\binom{n+r-1}{r-1}=\binom{n+r-2}{r-2}+\binom{n+r-2}{r-1}$ of binomial coefficients.

In order to compute the limit (5.5), we analyze the asymptotic behavior of each matrix element $K^{(r,n)}_{\mu,\nu}(a)$ along the hypersurface $1-a_{1}a_{r+1}=0$, by the same method of pinching as we used in [12]. As we remarked in Lemma 5.1, in the region (5.4) the integral $K^{(r,n)}_{\mu,\nu}(a)$ is expressed as

[TABLE]

over a certain $n$-cycle $C^{n}$, provided that $|t|<\rho^{2}$. Assuming that $\rho\in(0,1)$ satisfies

[TABLE]

we consider the situation where

[TABLE]

In this case we can choose the cycle $C$ as

[TABLE]

with sufficiently small $\varepsilon>0$, and analyze the effect of pinching about the cycles $C_{\varepsilon}(a_{1})$, $C_{\varepsilon}(a_{1}^{-1})$ as $a_{r+1}\to a_{1}^{-1}$ in the region (5.6).

We first consider the integral with respect to the variable $z_{1}$. When $a_{r+1}$ approaches to $a_{1}^{-1}$, the contour $C$ is pinched by the two pairs of poles $(a_{r+1},a_{1}^{-1})$ and $(a_{r+1}^{-1},a_{1})$. Taking this into account we decompose $\Phi_{n}(z;a)$ as

[TABLE]

where $z_{\widehat{1}}=(z_{2},\ldots,z_{n})$, and compute the residues at the poles $z_{1}=a_{1}^{\pm 1}$. Then we obtain

[TABLE]

for $\epsilon=\pm 1$, where

[TABLE]

Setting

[TABLE]

we compute

[TABLE]

By the same argument as in [12], we repeat this computation for $z_{2},\ldots,z_{n}$. As a result we obtain

[TABLE]

We remark that the first term of the right-hand side is regular along $1-a_{1}a_{r+1}=0$, and has a finite limit in the limit as $a_{r+1}\to a_{1}^{-1}$.

If $\mu_{1}=0$ or $\nu_{1}=0$, then the first term of the right-hand side of (5.7) is 0. In fact, when $\mu_{1}=0$, we have

[TABLE]

and hence $E_{\mu}(a_{\left\{1,\ldots,r\right\}};a_{1},z_{\widehat{1}};p)=0$. Similarly, when $\nu_{1}=0$, we have $E_{\nu}(a_{\left\{1,\ldots,r\right\}};a_{1},z_{\widehat{1}};q)=0$. Therefore, when $\mu_{1}=0$ or $\nu_{1}=0$, we obtain

[TABLE]

Since the integral over $C_{0}^{n}$ is regular along $1-a_{1}a_{r+1}=0$, we obtain

[TABLE]

We now decompose the matrix $K_{r,n}(a)$ into four blocks as

[TABLE]

according to the partition $Z_{r,n}=Z_{r,n}^{0}\sqcup Z_{r,n}^{+}$ of the indexing set. Note that $\#Z_{r,n}^{+}=\binom{n+r-2}{r-1}$ and that $\lim_{a_{r+1}\to a_{1}^{-1}}(1-a_{1}a_{r+1})K^{(r,n)}_{\mu,\nu}(a)=0$ when $\nu_{1}=0$. Hence we compute

[TABLE]

When $\mu_{1}=0$, $\nu_{1}=0$, with the notation $\mu=(0,\mu^{\prime})$, $\nu=(0,\nu^{\prime})$ we have

[TABLE]

Since

[TABLE]

we obtain

[TABLE]

and hence

[TABLE]

(In the right-hand side, $a_{m}$ should be understood as $a_{m}=pq/t^{2n-2}a_{2}\cdots a_{r}{a}_{r+2}\cdots a_{m-1}$.)

We next consider the case where $\mu_{1}>0$ and $\nu_{1}>0$. From (5.7) we compute

[TABLE]

Here, by the property of interpolation functions, we have

[TABLE]

Also, noting that

[TABLE]

and that $a_{m}=pq/t^{2n-2}a_{2}\cdots a_{r}a_{r+2}\cdots a_{m-1}=pq/t^{2n-4}(ta_{1})a_{2}\cdots a_{r}(ta_{1}^{-1})a_{r+2}\cdots a_{m-1}$, we compute

[TABLE]

Passing to the determinant, we obtain

[TABLE]

Summarizing the arguments above, we obtain

[TABLE]

For $n=1$ we understand $\det K^{(r,0)}(a)=1$. This computation is valid also for $r=1$:

[TABLE]

Here we understand $\det K^{(1,0)}(a)=1$.

5.5 Determination of $c_{r,n}$

In order to compare $\widetilde{K}_{r,n}(\widetilde{a})$ with $\widetilde{L}_{r,n}(\widetilde{a})$, we compute

[TABLE]

The three factors in this expression are given as follows:

[TABLE]

[TABLE]

[TABLE]

Combining these formulas, for $r\geq 2$ we have

[TABLE]

where $L_{r,0}(a)=1$, and for $r=1$ we have

[TABLE]

where $L_{1,0}(a)=1$.

Remark 5.2

When $r=1$ $(m=6)$, by the balancing condition $t^{2n-2}a_{1}\cdots a_{6}=pq$, we have $t^{2n-2}a_{3}a_{4}a_{5}a_{6}=pq$ in the limit $a_{2}\to a_{1}^{-1}$. Since $(t^{n-1}a_{i}a_{j})(t^{n-1}a_{k}a_{l})=pq$ for $\left\{i,j,k,l\right\}=\left\{3,4,5,6\right\}$, $\Gamma(t^{n-1}a_{i}a_{j};p,q)\Gamma(t^{n-1}a_{k}a_{l};p,q)=1$. Hence we have $\prod_{3\leq k<l\leq 6}\Gamma(t^{n-1}a_{k}a_{l};p,q)=1$. **

Recall that $\det K^{(r,n)}(a)$ and $L_{r,n}(a)$ are related through the formulas

[TABLE]

Hence, by combining (5.10) and (5.11), we obtain the following recurrence formulas for $c_{r,n}$:

[TABLE]

for $r=1$, and

[TABLE]

for $r\geq 2$. Solving these recurrence formulas, we obtain

[TABLE]

for $r\geq 1$ and $n\geq 1$. This completes the proof of Theorem 1.1, as is pointed out in the remark of Corollary 4.6.

6 Determinant formula for $q$-hypergeometric integrals of type $BC_{n}$

In this section we derive a determinant formula for $q$-hypergeometric integrals of type $BC_{n}$ from Theorem 1.1. In view of the balancing condition $a_{1}\cdots a_{m}t^{2n-2}=pq$, we first replace $a_{m}$ with $pa_{m}$, and then take the limit $p\to 0$. By this procedure, from $\Phi(z;a;p,q)$ of (1.1) we obtain

[TABLE]

In place of $\mathcal{H}_{s-1,n}^{(p)}$ we use the $\mathbb{C}$-vector space of $W_{n}$-invariant Laurent polynomials in $z=(z_{1},\ldots,z_{n})$ of degree $\leq s-1$ in each variable:

[TABLE]

For generic $c=(c_{1},\ldots,c_{s})\in(\mathbb{C}^{*})^{s}$, there exists a unique basis $\{\mathsf{E}_{\mu}(c;z)\,|\,\mu\in Z_{s,n}\}$ of $\mathsf{H}_{s-1,n}$ satisfying the condition

[TABLE]

This interpolation basis is obtained from $\{E_{\mu}(c;z;p)\ |\ \mu\in Z_{s,n}\}$ simply by taking the limit $p\to 0$:

[TABLE]

Otherwise this basis can be constructed by the method of Section 2 by replacing $e(u,v;p)$ with its limit

[TABLE]

These polynomials $\mathsf{E}_{\mu}(c;z)$ $(\mu\in Z_{s,n})$ for $s=2$ are written as

[TABLE]

for $r=0,1,\ldots,n$, where the summation is taken over all pairs of sequences $1\leq i_{1}<\cdots<i_{r}\leq n$ and $1\leq j_{1}<\cdots<j_{n-r}\leq n$ such that $\{i_{1},\ldots,i_{r}\}\cup\{j_{1},\ldots,j_{n-r}\}=\{1,2,\ldots,n\}$. The polynomials (6.1) are used in the study of Jackson integrals of type $BC_{n}$ [10]. The polynomials $\mathsf{E}_{\mu}(c;z)$ $(\mu\in Z_{s,n})$ for general $s$ were defined for the first time in the present paper.

We assume $m=2r+4$ $(r=1,2,\ldots)$. Fixing generic parameters $x=(x_{1},\ldots,x_{r})$ and $y=(y_{1},\ldots,y_{r})$, we take the interpolation bases for the two vector spaces $\mathsf{H}_{s-1,n}$ and $\mathcal{H}^{(q)}_{r-1,n}$ with respect to $x$ and $y$ respectively:

[TABLE]

For each pair $(\mu,\nu)\in Z_{r,n}\times Z_{r,n}$, we consider the $q$-hypergeometric integral

[TABLE]

assuming that $|a_{k}|<1$ $(k=1,\ldots,m-1)$. By the limiting procedure as explained above, Theorem 1.1 implies the following evaluation formula.

Theorem 6.1

Set $m=2r+4$ $(r=1,2,\ldots)$. Under the balancing condition $a_{1}\cdots a_{m-1}a_{m}t^{2n-2}=q$ for the parameters, the determinant of the $\binom{n+r-1}{r-1}\times\binom{n+r-1}{r-1}$ matrix $\mathsf{K}(a;x,y)=(\mathsf{K}_{\mu,\nu}(a;x,y))_{\mu,\nu\in Z_{r,n}}$ is given explicitly by

[TABLE]

Note that the above formula with $r=1$ (i.e., the integral (1.3) with $p\to 0$) is known as Gustafson’s multivariate Nassrallah–Rahman integral [6]

[TABLE]

which recovers the Nassrallah–Rahman integral [15] when $n=1$. In [10], a proof of (6.3) is given by means of the polynomials (6.1).

Furthermore, we can take the limit $a_{2r+3}\to 0$ in Theorem 6.1. Then, without balancing condition for the parameters $a_{1},\ldots,a_{2r+2}$, we have the following.

Corollary 6.2

Let $\widetilde{\mathsf{K}}_{\mu,\nu}(a;x,y)$ be the $q$-hypergeometric integrals defined by (6.2) using

[TABLE]

with $2r+2$ parameters, instead of $\mathsf{\Phi}(z;a;q)$. The determinant of the $\binom{n+r-1}{r-1}\times\binom{n+r-1}{r-1}$ matrix $\widetilde{\mathsf{K}}(a;x,y)=(\widetilde{\mathsf{K}}_{\mu,\nu}(a;x,y))_{\mu,\nu\in Z_{r,n}}$ is given by

[TABLE]

The above formula with $r=1$ is also known as Gustafson’s multivariate Askey–Wilson integral [7]

[TABLE]

which is the Askey–Wilson integral [4] when $n=1$. In [21] and [9], proofs of (6.4) are given by using degenerate cases of the polynomials (6.1)

[TABLE]

which essentially coincide with the special cases of Okounkov’s interpolation polynomials [16, Theorem 5.2], [17] attached to single columns of partitions. (For the polynomials (6.5), see also [3, Introduction, p. 1073–p.1074] and [14].) Corollary 6.2 can be regarded as a $BC_{n}$ version of the determinant formula of Tarasov and Varchenko [20] for $q$-hypergeometric integrals of type $A_{n}$.

As a basis of the vector space $\mathsf{H}_{r-1,n}$, we can also take the symplectic Schur functions

[TABLE]

associated with the partitions $\lambda\in B_{r,n}=\{\lambda\in\mathbb{Z}^{n}\,|\,r-1\geq\lambda_{1}\geq\cdots\geq\lambda_{n}\geq 0\}$. These functions $\chi_{\lambda}(z)$ are expanded in terms of our interpolation polynomials as

[TABLE]

where $c_{\lambda\mu}=\chi_{\lambda}((x)_{t,\mu})$. The determinant of the matrix C=\big{(}c_{\lambda\mu}\big{)}_{\lambda\in B_{r,n},\mu\in Z_{r,n}} is given by

[TABLE]

as is proved in [3, Corollary 1.5] or [8, Theorem 3.2 (3.6)], for instance. We define the matrix

[TABLE]

Then we have $\mathsf{X}(a;y)=C\,\mathsf{K}(a;x,y)$, so that $\det\mathsf{X}(a;y)=\det C\det\mathsf{K}(a;x,y)$. Under the condition $a_{1}\cdots a_{2r+4}t^{2n-2}=q$, this implies

[TABLE]

Similarly we define

[TABLE]

which satisfies $\widetilde{\mathsf{X}}(a;y)=C\,\widetilde{\mathsf{K}}(a;x,y)$, so that $\det\widetilde{\mathsf{X}}(a;y)=\det C\,\det\widetilde{\mathsf{K}}(a;x,y)$. Then we have

[TABLE]

This determinant formula is a contour integral version of the formula ([13, Theorem 1.2] or [3, Theorem 1.3]) for Jackson integrals of type $BC_{n}$.

The elliptic version of the determinant formulas for Jackson integrals of type $BC_{n}$ has not been established yet. It would be an important problem to clarify the relationship between Jackson integrals and contour integrals in the context of elliptic hypergeometric pairings as in this paper.

Acknowledgment

This work was supported by JSPS Kakenhi Grants (B)15H03626 and (C)18K03339.