Elliptic extension of Gustafson's $q$-integral of type $G_2$

TL;DR

This paper proves an elliptic extension of Gustafson's $q$-integral of type $G_2$, expressing it via elliptic gamma functions and connecting it to known $q$-beta integrals as a special case.

Contribution

It introduces a new elliptic beta integral of type $G_2$ and provides a proof using elliptic gamma functions and existing $BC_1$ integrals.

Findings

Evaluation formula expressed with elliptic gamma functions

Includes Gustafson's $q$-beta integral as a limit case

Utilizes elliptic beta integral of type $BC_1$ in proof

Abstract

The evaluation formula for an elliptic beta integral of type $G_2$ is proved. The integral is expressed by a product of Ruijsenaars' elliptic gamma functions, and the formula includes that of Gustafson's $q$-beta integral of type $G_2$ as a special limiting case as $p\to 0$. The elliptic beta integral of type $BC_1$ by van Diejen and Spiridonov is effectively used in the proof of the evaluation formula.

Equations

$$\frac{(q;q{)}_{\infty }}{2(2\pi \sqrt{-1})}{\int }_{\mathbb{T}}\frac{(x^2,x^{-2};q{)}_{\infty }}{{\prod }_{k=1}^4(a_{k}x,a_k{x}^{-1};q{)}_{\infty }}\frac{dx}{x}=\frac{(a_1{a}_2{a}_3{a}_4;q{)}_{\infty }}{{\prod }_{1\le i<j\le 4}(a_i{a}_j;q{)}_{\infty }},$$

$$\frac{(p;p{)}_{\infty }(q;q{)}_{\infty }}{2(2\pi \sqrt{-1})}{\int }_{\mathbb{T}}\frac{{\prod }_{k=1}^6\Gamma (a_{k}x,a_k{x}^{-1};p,q)}{\Gamma (x^2,x^{-2};p,q)}\frac{dx}{x}=\prod _{1\le i<j\le 6}\Gamma (a_i{a}_j;p,q),$$

$$\Gamma (u;p,q)=\frac{(pq{u}^{-1};p,q{)}_{\infty }}{(u;p,q{)}_{\infty }},\text{mbox}where(u;p,q{)}_{\infty }=\prod _{\mu ,\nu =0}^{\infty }(1-p^{\mu }{q}^{\nu }u).$$

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

$$\Gamma (pq{u}^{-1};p,q)=\frac{1}{\Gamma (u;p,q)},\frac{1}{\Gamma (u,u^{-1};p,q)}=-u^{-1}\theta (u;p)\theta (u;q).$$

$$\begin{array}{rl}& \frac{(q;q{)}_{\infty }^2}{12(2\pi \sqrt{-1}{)}^2}\int {\int }_{{\mathbb{T}}^2}\frac{{\prod }_{1\le i<j\le 3}(x_i{x}_j,x_i^{-1}{x}_j,x_i{x}_j^{-1},x_i^{-1}{x}_j^{-1};q{)}_{\infty }}{{\prod }_{i=1}^3{\prod }_{k=1}^4(a_k{x}_i,a_k{x}_i^{-1};q{)}_{\infty }}\frac{d{x}_1}{x_1}\frac{d{x}_2}{x_2}\\ & =\frac{(a_1^2{a}_2^2{a}_3^2{a}_4^2;q{)}_{\infty }}{(a_1{a}_2{a}_3{a}_4;q{)}_{\infty }}\prod _{i=1}^4\frac{(a_i;q{)}_{\infty }}{(a_i^2;q{)}_{\infty }}\prod _{1\le i<j\le 4}\frac{1}{(a_i{a}_j;q{)}_{\infty }}\prod _{1\le i<j<k\le 4}\frac{1}{(a_i{a}_j{a}_k;q{)}_{\infty }},\end{array}$$

$${\mathbb{T}}^2=\{(x_1,x_2)\in ({\mathbb{C}}^{\ast }{)}^2||x_i|=1(i=1,2)\}.$$

$$\begin{array}{rl}& \frac{(p;p{)}_{\infty }^2(q;q{)}_{\infty }^2}{12(2\pi \sqrt{-1}{)}^2}\int {\int }_{{\mathbb{T}}^2}\frac{{\prod }_{i=1}^3{\prod }_{k=1}^5\Gamma (a_k{x}_i,a_k{x}_i^{-1};p,q)}{{\prod }_{1\le i<j\le 3}\Gamma (x_i{x}_j,x_i^{-1}{x}_j,x_i{x}_j^{-1},x_i^{-1}{x}_j^{-1};p,q)}\frac{d{x}_1}{x_1}\frac{d{x}_2}{x_2}\\ & =\prod _{i=1}^5\frac{\Gamma (a_i^2;p,q)}{\Gamma (a_i;p,q)}\prod _{1\le i<j\le 5}\Gamma (a_i{a}_j;p,q)\prod _{\begin{array}{c}1\le i<j\\ <k\le 5\end{array}}\Gamma (a_i{a}_j{a}_k;p,q)\prod _{\begin{array}{c}1\le i<j\\ <k<l\le 5\end{array}}\Gamma (a_i{a}_j{a}_k{a}_l;p,q).\end{array}$$

$$\prod _{i=1}^5\frac{\Gamma (a_i^2;p,q)}{\Gamma (a_i;p,q)\Gamma (ϵp^{\frac{1}{2}}{q}^{\frac{1}{2}}{a}_i;p,q)}\prod _{1\le i<j\le 5}\frac{\Gamma (a_i{a}_j;p,q)}{\Gamma (ϵp^{\frac{1}{2}}{q}^{\frac{1}{2}}{a}_i{a}_j;p,q)}.$$

$$\prod _{i=1}^5\Gamma (-a_i,p^{\frac{1}{2}}{a}_i,-p^{\frac{1}{2}}{a}_i,q^{\frac{1}{2}}{a}_i,-q^{\frac{1}{2}}{a}_i,-ϵp^{\frac{1}{2}}{q}^{\frac{1}{2}}{a}_i;p,q)\prod _{1\le i<j\le 5}\Gamma (a_i{a}_j;p,q)\prod _{\begin{array}{c}1\le i<j\\ <k\le 5\end{array}}\Gamma (a_i{a}_j{a}_k;p,q).$$

$$\Phi (x)=\frac{{\prod }_{i=1}^3{\prod }_{k=1}^5\Gamma (a_k{x}_i,a_k{x}_i^{-1};p,q)}{{\prod }_{1\le i<j\le 3}\Gamma (x_i{x}_j,x_i^{-1}{x}_j,x_i{x}_j^{-1},x_i^{-1}{x}_j^{-1};p,q)}$$

$$\Phi (x)=\prod _{i=1}^3\frac{{\prod }_{k=1}^5\Gamma (a_k{x}_i,a_k{x}_i^{-1};p,q)}{\Gamma (x_i,x_i^{-1};p,q)}\prod _{1\le j<k\le 3}\frac{1}{\Gamma (x_j{x}_k^{-1},x_k{x}_j^{-1};p,q)}.$$

$$z_1=x_1/x_2,z_2=x_2.$$

$$R=\{\pm {\bar{\epsilon }}_1,\pm {\bar{\epsilon }}_2,\pm {\bar{\epsilon }}_3\}\cup \{\pm ({\bar{\epsilon }}_1-{\bar{\epsilon }}_2),\pm ({\bar{\epsilon }}_1-{\bar{\epsilon }}_3),\pm ({\bar{\epsilon }}_2-{\bar{\epsilon }}_3)\},$$

$${\alpha }_1={\bar{\epsilon }}_1-{\bar{\epsilon }}_2={\epsilon }_1-{\epsilon }_2,{\alpha }_2={\bar{\epsilon }}_2=(-{\epsilon }_1+2{\epsilon }_2-{\epsilon }_3)/3.$$

$$R^{+}$$

$$=\{{\alpha }_2,{\alpha }_1+{\alpha }_2,{\alpha }_1+2{\alpha }_2\}\cup \{{\alpha }_1,{\alpha }_1+3{\alpha }_2,2{\alpha }_1+3{\alpha }_2\}.$$

$${\varpi }_1=2{\alpha }_1+3{\alpha }_2,{\varpi }_2={\alpha }_1+2{\alpha }_2.$$

$$\begin{array}[]{|c|cc|}\hline\cr&\alpha_{1}&\alpha_{2}\\ \hline\cr s_{1}&-\alpha_{1}&\alpha_{1}+\alpha_{2}\\[3.0pt] s_{2}&\alpha_{1}+3\alpha_{2}&-\alpha_{2}\\[3.0pt] \hline\cr\end{array}$$

$$W=\{(s_1{s}_2{)}^k,(s_1{s}_2{)}^k{s}_2|k=0,1,\dots ,5\},$$

$$S_{c,\omega }f(\zeta )=f(\zeta +c\omega ).$$

$$w.f(\zeta )=f(w^{-1}\zeta )(w\in W).$$

$$\zeta \mapsto z=(e^{2\pi \sqrt{-1}(\zeta ,{\alpha }_1)},e^{2\pi \sqrt{-1}(\zeta ,{\alpha }_2)}).$$

$$w.z^{\lambda }=w.e^{2\pi \sqrt{-1}(\zeta ,\lambda )}=e^{2\pi \sqrt{-1}(w^{-1}\zeta ,\lambda )}=e^{2\pi \sqrt{-1}(\zeta ,w\lambda )}=z^{w\lambda },$$

$$w.f(z)=f(w.z_1,w.z_2)=f(z^{w{\alpha }_1},z^{w{\alpha }_2}),$$

$$s_1.f(z_1,z_2)=f(z_1^{-1},z_1{z}_2),s_2.f(z_1,z_2)=f(z_1{z}_2^3,z_2^{-1}),$$

$$w_0.f(z_1,z_2)=(s_1{s}_2{)}^3.f(z_1,z_2)=f(z_1^{-1},z_2^{-1}).$$

$$\frac{d(s_1.z_1)}{s_1.z_1}=-\frac{d{z}_1}{z_1},\frac{d(s_1.z_2)}{s_1.z_2}=\frac{d{z}_1}{z_1}+\frac{d{z}_2}{z_2}\text{mbox}and\frac{d(s_2.z_1)}{s_2.z_1}=\frac{d{z}_1}{z_1}+3\frac{d{z}_2}{z_2},\frac{d(s_2.z_2)}{s_2.z_2}=-\frac{d{z}_2}{z_2}.$$

$$T_{p,z_1}f(z)=f(p{z}_1,z_2),T_{p,z_2}f(z)=f(z_1,p{z}_2)$$

$$x_1=z_1{z}_2,x_2=z_2\text{mbox}and\frac{d{x}_1}{x_1}=\frac{d{z}_1}{z_1}+\frac{d{z}_2}{z_2},\frac{d{x}_2}{x_2}=\frac{d{z}_2}{z_2},$$

$$\Phi (z)={\Phi }_{+}(z){\Phi }_{+}(z^{-1}),$$

Full text

000 2010 Mathematics Subject Classification. Primary 33D67; Secondary 33D65, 39A13.000 Key words and phrases. elliptic beta integral, Gustafson’s $q$-beta integral, root system of type $G_{2}$000 This work is supported by JSPS Kakenhi Grants (B)15H03626 and (C)18K03339.

1 Introduction

The Askey–Wilson integral is a complex integral given by

[TABLE]

where $|a_{k}|<1$ $(k=1,\ldots,4)$ and $\mathbb{T}$ is the unit circle $\{x\in\mathbb{C}\,|\,|x|=1\}$ traversed in the positive direction. Hereafter, for a fixed $q\in\mathbb{C}^{*}$ satisfying $|q|<1$, we use the symbol $(u;q)_{\infty}=\prod_{\nu=1}^{\infty}(1-q^{\nu}u)$ and the abbreviation $(u_{1},\ldots,u_{m};q)_{\infty}=(u_{1};q)_{\infty}\cdots(u_{m};q)_{\infty}$. The infinite product on the right-hand side of (1.1) is expressed by a product of $q$-gamma functions. In this sense formula (1.1) can be regarded as a kind of beta integral, and in fact plays a fundamental role in the theory of Askey–Wilson $q$-orthogonal polynomials [1]. This type of $q$-beta integrals has been extended to multiple $q$-beta integrals in the framework of Macdonald theory of multivariable $q$-orthogonal polynomials associated with root systems. In this context the Askey–Wilson integral (1.1) is of type $BC_{1}$. In a series of pioneering works around 1990, Gustafson discovered various evaluation formulas for multiple $q$-beta integrals associated with root systems, including several remarkable identities which are not covered by the so-called Macdonald constant terms. In the cases of non-simply laced root systems, there are basically two types in Gustafson’s multiple $q$-beta integrals, which are later called type I and type II in the context of [15]. (See also [4] for their explicit forms.)

In the last two decades, several elliptic extensions of the $q$-beta integrals have been studied, especially for those of type $BC_{n}$ by van Diejen and Spiridonov [15], Spiridonov [14], Rains [11]. They include the elliptic extension of (1.1)

[TABLE]

where $|a_{k}|<1$ $(k=1,\ldots,6)$, under the balancing condition $a_{1}\cdots a_{6}=pq$. Here, for fixed $p$, $q\in\mathbb{C}^{*}$ satisfying $|p|<1$, $|q|<1$, we denote by $\Gamma(u;p,q)$ $(u\in\mathbb{C}^{\ast})$ the Ruijsenaars elliptic gamma function defined by

[TABLE]

We also use the notation $\Gamma(u_{1},\ldots,u_{m};p,q)=\Gamma(u_{1};p,q)\cdots\Gamma(u_{m};p,q)$. Note that $\Gamma(u;p,q)$ satisfies

[TABLE]

where $\theta(u;p)=(u,p/u;p)_{\infty}$ is a theta function satisfying $\theta(pu;p)=-\theta(u;p)/u$, and also satisfies

[TABLE]

The Askey–Wilson integral (1.1) is obtained from (1.2) as a special case, first by replacing $a_{6}$ with $pa_{6}$ and by taking the limit $p\to 0$ and $a_{5}\to 0$ consecutively.

Compared with the development in the cases of classical root systems, the elliptic extensions of the cases of exceptional root systems are not fully studied yet. The aim of this paper is to prove an elliptic extension of the following $q$-integral formula of type $G_{2}$ (of type I) due to Gustafson [2, p. 101, Theorem 8.1] and [3].

Proposition 1.1** (Gustafson)**

Suppose that $a_{k}\in\mathbb{C}^{*}\,(1\leq k\leq 4)$ satisfy $|a_{k}|<1$. Then we have

[TABLE]

where $x_{3}=x_{1}^{-1}x_{2}^{-1}$ and $\mathbb{T}^{2}$ is the 2-dimensional torus given by

[TABLE]

Our main result is

Theorem 1.2

Suppose that $x_{1}x_{2}x_{3}=1$ and $a_{k}\in\mathbb{C}^{*}\,(1\leq k\leq 5)$ satisfy $|a_{k}|<1$. Under the balancing condition $(a_{1}a_{2}a_{3}a_{4}a_{5})^{2}=pq$, we have

[TABLE]

In 2007 this formula was communicated as a conjecture by one of the authors (M. Ito) to V. P. Spiridonov. This conjecture was formulated by Spiridonov and Vartanov [12, 13] in the context of duality of superconformal indices. As far as we know, however, no proof of this formula has been given so far.

Remark 1. Gustafson’s formula (1.6) is included in (1.7) as a limiting case; first replace $a_{5}$ with $p^{1\over 2}a_{5}$, and then take the limit $p\to 0$.

Remark 2. By (1.5), under the condition $a_{1}a_{2}a_{3}a_{4}a_{5}=\epsilon p^{1\over 2}q^{1\over 2}$, where $\epsilon\in\{1,-1\}$, the right-hand side of (1.7) is also expressed as

[TABLE]

This coincides with the expression in the conjecture [13, p. 213, (36)] when $\epsilon=1$. Moreover, by the property $\Gamma(u^{2};p,q)=\Gamma(u,-u,p^{1\over 2}u,-p^{1\over 2}u,q^{1\over 2}u,-q^{1\over 2}u,p^{1\over 2}q^{1\over 2}u,-p^{1\over 2}q^{1\over 2}u;p,q)$, the right-hand side of (1.7) is rewritten into

[TABLE]

Remark 3. By the constraint $x_{1}x_{2}x_{3}=1$, the integrand

[TABLE]

of the left-hand side of (1.7) is also expressed as

[TABLE]

This expression consists of two parts, one depending on the short roots $\{x_{i},x_{i}^{-1}\,|\,1\leq i\leq 3\}$ and the other on the long roots $\{x_{j}x_{k}^{-1},x_{k}x_{j}^{-1}\,|\,1\leq j<k\leq 3\}$. In the proof of (1.7) we will use the coordinates $(z_{1},z_{2})$ associated with the simple roots, defined as

[TABLE]

Theorem 1.2 will be proved in two steps. The first step is to show that both sides of (1.7) satisfy a common system of $q$-difference equations, so that we can consequently confirm both sides coincide up to a constant. The second step is to analyze asymptotic behaviors of both sides at a singularity in order to determine the constant. This method is also applicable to other elliptic beta integrals. In particular, we refer to [8] for the $BC_{n}$ case including the formula (1.2), which might be simpler than the $G_{2}$ case of this paper.

This paper is organized as follows. After defining basic terminology of the root system $G_{2}$ in Section 2, we first present in Section 3 the explicit forms of the $q$-difference equations which the integral (1.7) satisfies (Proposition 3.1). In Section 4 we study the analytic continuation of the integral (1.7) as a meromorphic function of the parameters $a_{1},\ldots,a_{4}$ in a specific domain. We use this argument to show that the integral (1.7) is expressed as a product of elliptic gamma functions up to a constant. In Section 5 we explain a fundamental method, which corresponds to integration by parts in calculus, to deduce the $q$-difference equations for the contour integral (1.7). This method is formulated in terms of a $q$-difference coboundary operator $\nabla_{\rm sym}:\mathcal{G}_{\epsilon}\to\mathcal{F}$, where $\mathcal{G}_{\epsilon}$ and $\mathcal{F}$ are defined as spaces of theta functions specified by individual quasi-periodicities. Section 6 is a technical part; we investigate in detail the source and target spaces $\mathcal{G}_{\epsilon}$, $\mathcal{F}$ of the operator $\nabla_{\rm sym}$. We apply this argument to proving Lemma 3.2, which we used to derive the $q$-difference equations in Proposition 3.1. Section 7 is devoted to asymptotic analysis of the contour integral (1.7) along the singularity $a_{1}a_{2}=1$. It is used to determine the explicit value of the constant, which was indefinite at the stage of Section 4. It should be noted that the elliptic beta integral (1.2) of type $BC_{1}$ naturally arises in the process of calculation of the asymptotic behavior.

Lastly, we comment on our calculation of $\nabla_{\rm sym}$. For theta functions $\varphi\in\mathcal{G}_{\epsilon}$ we need to expand $\nabla_{\rm sym}\varphi\in\mathcal{F}$ as a linear combination of theta functions which belong to a particular basis of $\mathcal{F}$. In this paper we made use of the basis of $\mathcal{F}$ that consists of the Lagrange interpolation functions associated with the specific points ${\rm p}_{23}$, ${\rm p}_{13}$, ${\rm p}_{12}$ and ${\rm p}_{12}^{*}\in(\mathbb{C}^{*})^{2}$ defined in Section 6. (We constructed this basis in a heuristic way. See the set of theta functions $\{F_{1}(z),F_{2}(z),F^{\prime}_{3}(z),G(z)\}$, whose interpolation property is presented in the table below (6.16).) In the cases of $A_{n}$ and $BC_{n}$ root systems in [6] and [5, 7, 8, 9], respectively, we remark that Lagrange interpolation functions in a space of theta functions of particular quasi-periodicity are introduced by systematically specifying a set of reference points in $(\mathbb{C}^{*})^{n}$. It would be an interesting problem to find a universal way which produces adequate interpolation bases for general root systems.

2 Root system $G_{2}$

Let $\{\varepsilon_{1},\varepsilon_{2},\varepsilon_{3}\}$ be the standard basis of $\mathbb{R}^{3}$ with the inner product $(\cdot,\cdot)$ satisfying $(\varepsilon_{i},\varepsilon_{j})=\delta_{ij}$, and let $V$ be the hyperplane in $\mathbb{R}^{3}$ with equation $\xi_{1}+\xi_{2}+\xi_{3}=0$, i.e., $V=\{\xi\in\mathbb{R}^{3}\,|\,(\xi,\varepsilon_{1}+\varepsilon_{2}+\varepsilon_{3})=0\}.$

Let $R\subset V$ be the root system of type $G_{2}$ given by

[TABLE]

where $\bar{\varepsilon}_{i}=\varepsilon_{i}-(\varepsilon_{1}+\varepsilon_{2}+\varepsilon_{3})/3$. We refer the setting of the root system of type $G_{2}$ to Macdonald’s book [10]. We fix the set of simple roots $\{\alpha_{1},\alpha_{2}\}\subset R$ given by

[TABLE]

The set of positive roots is given by

[TABLE]

We also fix the set of fundamental weights $\{\varpi_{1},\varpi_{2}\}$ by $(\alpha_{i}^{\vee},\varpi_{j})=\delta_{ij}$, where $\alpha^{\vee}=2\alpha/(\alpha,\alpha)$. This implies that

[TABLE]

Let $P$ and $Q$ be the weight lattice and root lattice defined by $P=\mathbb{Z}\varpi_{1}+\mathbb{Z}\varpi_{2}$ and $Q=\mathbb{Z}\alpha_{1}+\mathbb{Z}\alpha_{2}$, respectively. For the root system $G_{2}$, the root lattice $Q$ coincides with the weight lattice $P$.

Let $W$ be the Weyl group of type $G_{2}$ generated by orthogonal reflections $s_{\alpha}$ $(\alpha\in R)$ with respect to the hyperplane perpendicular to $\alpha\in R$, which are given by $s_{\alpha}(\xi)=\xi-(\alpha^{\vee},\xi)\alpha$. The group $W$ is generated by the reflections $s_{i}=s_{\alpha_{i}}:V\to V$ $(i=1,2)$, and is isomorphic to the dihedral group of order 12.

[TABLE]

Moreover, $W$ is explicitly written as

[TABLE]

where $(s_{1}s_{2})^{k}$ coincide with the rotations around the origin through angle $k\pi/3$ on $V$ and $(s_{1}s_{2})^{k}s_{2}$ coincide with the reflections $s_{\alpha}$ $(\alpha\in R^{+})$ written as follows: $(s_{1}s_{2})^{0}s_{2}=s_{\alpha_{2}}$, $(s_{1}s_{2})^{1}s_{2}=s_{\alpha_{1}}$, $(s_{1}s_{2})^{2}s_{2}=s_{\alpha_{1}+\alpha_{2}}$, $(s_{1}s_{2})^{3}s_{2}=s_{2\alpha_{1}+3\alpha_{2}}$, $(s_{1}s_{2})^{4}s_{2}=s_{\alpha_{1}+2\alpha_{2}}$, $(s_{1}s_{2})^{5}s_{2}=s_{\alpha_{1}+3\alpha_{2}}$. We use the expression (2.1) of $W$ later. The element $w_{0}=(s_{1}s_{2})^{3}$ is the longest element of $W$. Note also that the inner product and the reflections are uniquely extended linearly to $V_{\mathbb{C}}=\mathbb{C}\otimes_{\mathbb{R}}V$.

We fix the set of fundamental coweights $\{\omega_{1},\omega_{2}\}$ by $(\omega_{i},\alpha_{j})=\delta_{ij}$, so that $\omega_{1}=\varpi_{1}$, $\omega_{2}=3\varpi_{2}.$ Let $P^{\vee}$ be the coweight lattice defined by $P^{\vee}=\mathbb{Z}\omega_{1}+\mathbb{Z}\omega_{2}$. For $c\in\mathbb{C}$ and $\omega\in P^{\vee}$ we denote by $S_{c,\omega}$ the $c$-shift operator with respect to $\omega$ for functions $f(\zeta)$ on $V_{\mathbb{C}}$ by

[TABLE]

We also define action of the Weyl group $W$ on $f(\zeta)$ by

[TABLE]

We consider the mapping from $V_{\mathbb{C}}$ to $(\mathbb{C}^{*})^{2}$ by

[TABLE]

If we write $\zeta\in V_{\mathbb{C}}$ with the fundamental coweights by $\zeta=\zeta_{1}\omega_{1}+\zeta_{2}\omega_{2}$, then the above mapping is written as $\zeta\mapsto z=(e^{2\pi\sqrt{-1}\zeta_{1}},e^{2\pi\sqrt{-1}\zeta_{2}})$. For $\lambda\in P$, we write $z^{\lambda}=e^{2\pi\sqrt{-1}(\zeta,\lambda)}$. In particular, for $\lambda=\lambda_{1}\alpha_{1}+\lambda_{2}\alpha_{2}\in Q=P$ we have the expression $z^{\lambda}=z_{1}^{\lambda_{1}}z_{2}^{\lambda_{2}}$, where $z_{i}=z^{\alpha_{i}}$. Through (2.3) and (2.4), for $w\in W$ we can define $w.z^{\lambda}=z^{w\lambda}$, i.e.,

[TABLE]

and we can also define $w.f(z)$ for functions $f(z)=f(z_{1},z_{2})$ on $(\mathbb{C}^{*})^{2}$ as

[TABLE]

so that, for instance, we have

[TABLE]

and

[TABLE]

We say that a function $f(z)$ is $W$-symmetric if $w.f(z)=f(z)$ for all $w\in W$. By chain rule for differential forms, we have

[TABLE]

We fix $p=e^{2\pi\sqrt{-1}\sigma}$ and $q=e^{2\pi\sqrt{-1}\tau}$, where ${\rm Im}\,\sigma>0$ and ${\rm Im}\,\tau>0$, respectively. If we consider a function $f(z)=f(z_{1},z_{2})$ on $(\mathbb{C}^{*})^{2}$ as the function on $V_{\mathbb{C}}$ through (2.4), then the $p$-shift operators for $f(z)$ with respect to $z_{i}$ $(i=1,2)$

[TABLE]

are induced by the $\sigma$-shift operators $S_{\sigma,\omega_{i}}$ with respect to $\omega_{i}\in P^{\vee}$ $(i=1,2)$, respectively. The $q$-shift operators for $f(z)$ with respect to $z_{i}$ $(i=1,2)$ are also defined by the $\tau$-shift operators with respect to $\omega_{i}$ $(i=1,2)$, respectively.

Using the notation $x_{i}=e^{2\pi\sqrt{-1}(\zeta,\bar{\varepsilon}_{i})}$ $(i=1,2,3)$, we have $x_{1}x_{2}x_{3}=e^{2\pi\sqrt{-1}(\zeta,\bar{\varepsilon}_{1}+\bar{\varepsilon}_{2}+\bar{\varepsilon}_{3})}=1$ and the variable change $(z_{1},z_{2})\mapsto(x_{1},x_{2})$ of $(\mathbb{C}^{*})^{2}$, where

[TABLE]

which we saw in (1.11). Though using the coordinates $(x_{1},x_{2},x_{3})$ with $x_{1}x_{2}x_{3}=1$ instead of $(z_{1},z_{2})$ we sometimes have simple expressions for functions on $(\mathbb{C}^{*})^{2}$ in appearance, like the integrands shown in (1.9) or (1.10) for instance, we use the coordinates $(z_{1},z_{2})$ of $(\mathbb{C}^{*})^{2}$ associated with simple roots in the succeeding sections.

3 $G_{2}$** elliptic Gustafson integral and its $q$-difference equations**

Let $\Phi(z)$ be function in $z=(z_{1},z_{2})\in(\mathbb{C}^{*})^{2}$ defined by

[TABLE]

where $z^{-1}=(z_{1}^{-1},z_{2}^{-1})$ and

[TABLE]

with complex parameters $a=(a_{1},\ldots,a_{5})\in\mathbb{C}^{*}$. We also use the notation $\Phi(a;z)=\Phi(a_{1},\ldots,a_{5};z)$ instead of $\Phi(z)$ when we need to make the dependence on the parameters $a=(a_{1},\ldots,a_{5})$ explicit. Through (1.11), $\Phi(z)$ coincides with (1.9) or (1.10). For the function $\Phi(z)$ we investigate the double integral

[TABLE]

over a 2-cycle $\sigma$. (2.8) implies $\varpi(x_{1},x_{2})=\varpi(z_{1},z_{2})$. (2.7) also implies $s_{1}.\varpi(z)=\varpi(z)$, $s_{2}.\varpi(z)=\varpi(z)$, so that $\varpi(z)$ is $W$-symmetric. If the parameters satisfy the condition $|a_{1}|<1,\ldots,|a_{5}|<1$, then $\Phi(z)$ is holomorphic in the neighborhood of the 2-dimensional torus

[TABLE]

and hence the integral

[TABLE]

defines a holomorphic function on the domain $\{(a_{1},\ldots,a_{5})\in(\mathbb{C}^{*})^{5}\,|\,|a_{i}|<1\ (i=1,\ldots,5)\}.$

We now formulate a system of $q$-difference equations for the integral $I(a_{1},\ldots,a_{5})$. Our goal is to establish the following proposition. We use the abbreviation $\theta(u_{1},\ldots,u_{m};p)=\theta(u_{1};p)\cdots\theta(u_{m};p)$.

Proposition 3.1

Suppose that $|p|<|q|$. Under the balancing condition $(a_{1}a_{2}a_{3}a_{4}a_{5})^{2}=pq$, the integral $I(a)$ satisfies the system of $q$-difference equations

[TABLE]

for $k=1,\ldots,4$, provided that $|a_{1}|<1,\ldots,|a_{4}|<1$ and $|a_{5}|<|q|$.

Note that the condition $|a_{5}|<|q|$ is equivalent to $|a_{1}\cdots a_{4}|>|p|^{1\over 2}/|q|^{1\over 2}$ under the balancing condition. We need to assume that $p$ satisfies $|p|<|q|$ to guarantee that the above equations hold in a nonempty region.

For $u,v\in\mathbb{C}^{*}$ let $e(u,v;p)$ the function defined by

[TABLE]

which satisfies $e(u,v;p)=-e(v,u;p)$, $e(pu,v;p)=e(u,v;p)(pu^{2})^{-1}$ and the three-term relation

[TABLE]

We use the notation

[TABLE]

for any meromorphic function $\varphi(z)$ on $(\mathbb{C}^{*})^{2}$ such that $\varphi(z)\Phi(z)$ is holomorphic in a neighborhood of $\mathbb{T}^{2}$. Since $\Phi(z)$ satisfies that

[TABLE]

where

[TABLE]

the integral $I(a)=\langle 1\rangle$ satisfies

[TABLE]

Lemma 3.2

Under the condition $|p|<|q|$, $(a_{1}a_{2}a_{3}a_{4}a_{5})^{2}q=p$ and $|a_{i}|<1$ $(i=1,\ldots,5)$, we have

[TABLE]

In other words,

[TABLE]

Proof. The proof of this lemma will be given later in Section 6. $\square$

In general, we have the following.

Lemma 3.3

Under the condition $|p|<|q|$, $(a_{1}a_{2}a_{3}a_{4}a_{5})^{2}q=p$ and $|a_{i}|<1$ $(i=1,\ldots,5)$, the integral $I(a)$ satisfies the two-term relations

[TABLE]

for $k=1,\ldots,4$.

Proposition 3.1 is obtained from Lemma 3.3 replacing $a_{5}$ by $q^{-1}a_{5}$.

We now suppose that $(a_{1}\cdots a_{5})^{2}=pq$ and regard $a_{5}=\epsilon p^{1\over 2}q^{1\over 2}/a_{1}a_{2}a_{3}a_{4}$, where $\epsilon\in\{1,-1\}$, as a function of $(a_{1},\ldots,a_{4})$. Then the integral $I(a_{1},\ldots,a_{5})$ is defined on the nonempty open subset

[TABLE]

of $(\mathbb{C}^{*})^{4}$. The $q$-difference equations (3.3) for $I(a_{1},\ldots,a_{5})$ are defined on

[TABLE]

which is a nonempty open subset of $U_{0}$, if $|p|<|q|$.

4 Analytic continuation

The integral $I(a_{1},\ldots,a_{5})$, regarded as a holomorphic function in $(a_{1},\ldots,a_{4})\in U_{0}$, can be continued to a meromorphic function on $(\mathbb{C}^{*})^{4}$. We prove this fact by means the $q$-difference equations (3.3).

In view of Proposition 3.1 we consider the meromorphic function

[TABLE]

which is also written as (1.8) if $(a_{1}\ldots a_{5})^{2}=pq$, as is mentioned in the introduction. Then it turns out that $J(a_{1},\ldots,a_{5})$ satisfies the same $q$-difference equations as (3.3). In fact, from (1.4) one has

[TABLE]

for $k=1,\ldots,4$. In the following we regard $J(a_{1},\ldots,a_{5})$ as a meromorphic function in $(a_{1},\ldots,a_{4})$ through $a_{5}=\epsilon p^{1\over 2}q^{1\over 2}/a_{1}a_{2}a_{3}a_{4}$, where $\epsilon\in\{1,-1\}$, as before. Noting that the integral $I(a_{1},\ldots,a_{5})$ is a holomorphic function on $U_{0}$, we consider the meromorphic function

[TABLE]

on $U_{0}$. This ratio $f(a_{1},\ldots,a_{5})$ has poles in $U_{0}$ possibly along the divisors

[TABLE]

where $\mu,\nu=0,1,2,\ldots$. Also, $f(a_{1},\ldots,a_{5})$ is $q$-periodic with respect to $(a_{1},\ldots,a_{4})\in V_{0}$ in the sense that

[TABLE]

for $k=1,\ldots,4$.

Lemma 4.1

Suppose that $|p|<|q|^{9}$. Then, under the condition $(a_{1}a_{2}a_{3}a_{4}a_{5})^{2}=pq$, there exists an open subset $W_{0}\subset(\mathbb{C}^{\ast})^{4}$ of the form

[TABLE]

such that $W_{0}\subset V_{0}$ and that $f(a_{1},\ldots,a_{5})$ is holomorphic on $W_{0}$.

Proof. Under the assumption $|p|<|q|^{9}$, i.e., $|p/q|^{\frac{1}{8}}<|q|$, one can choose positive number $s$ such that

[TABLE]

From the condition $(a_{1}\cdots a_{5})^{2}=pq$ and (4.3) we have

[TABLE]

We first confirm that $W_{0}\subset V_{0}$. Suppose that $(a_{1},\ldots,a_{4})\in W_{0}$. Then, from (4.3) and (4.4) we have $|a_{1}\cdots a_{4}|>s^{4}\geq|p/q|^{\frac{1}{2}}$, which means that $W_{0}\subset V_{0}$. We next show that $f(a_{1},\ldots,a_{5})$ is holomorphic in $W_{0}$. For this purpose, from (4) we verify the following when $(a_{1},\ldots,a_{4})\in W_{0}$:

[TABLE]

and

[TABLE]

For (4.6), since $|pq|^{1\over 2}<|p|^{1\over 2}<|q|^{1\over 2}<1$, it suffices to show $|p^{1\over 2}q^{1\over 2}a_{i}|>|pq|$ $(1\leq i\leq 5)$, which is confirmed as follows. Using (4.4) and (4.5) we obtain

[TABLE]

and

[TABLE]

On the other hand, for (4.7), since $|a_{i}a_{j}|>|a_{i}a_{j}a_{k}|$, it suffices to show $|a_{i}a_{j}a_{k}|>|pq|$ $(1\leq i<j<k\leq 5)$, which is confirmed as follows. Using (4.3), (4.4) and (4.5) we obtain

[TABLE]

and

[TABLE]

This completes the proof. $\square$

Theorem 4.2

Suppose that $|p|<|q|^{9}$. Under the condition $(a_{1}a_{2}a_{3}a_{4}a_{5})^{2}=pq$, the integral $I(a_{1},\ldots,a_{5})$, regarded as a holomorphic function in $(a_{1},\ldots,a_{4})\in U_{0}$, is expressed as

[TABLE]

for some constant $b\in\mathbb{C}$ independent of $a_{1},\ldots,a_{5}$. In particular, $I(a_{1},\ldots,a_{5})$ is continued to a meromorphic function on $(\mathbb{C}^{\ast})^{4}$.

*Proof. * By Lemma 4.1, there exists an open subset $W_{0}\subset(\mathbb{C}^{\ast})^{4}$ of the form (4.3) where $f(a_{1},\ldots,a_{5})$ $=I(a_{1},\ldots,a_{5})/J(a_{1},\ldots,a_{5})$ is holomorphic and satisfies the $q$-difference equations

[TABLE]

for $(a_{1},\ldots,a_{4})\in W_{0}$. Note that $W_{0}$ is the product of $4$ copies of an annulus in which the ratio of the two radii is given by $s<|q|$. Hence, by the $q$-difference equations (4.8), the holomorphic function $f(a_{1},\ldots,a_{5})$ on $W_{0}$ is continued to a holomorphic function on the whole $(\mathbb{C}^{\ast})^{4}$. It must be a constant, however, since the continued function $f(a_{1},\ldots,a_{5})$ is $q$-periodic with respect to the variables $a_{1},\ldots,a_{4}$. If we denote this constant by $b$, we have $I(a_{1},\ldots,a_{5})=b\,J(a_{1},\ldots,a_{5})$ as a holomorphic function on $U_{0}$, and hence $I(a_{1},\ldots,a_{5})$ is continued to a meromorphic function on $(\mathbb{C}^{\ast})^{4}$. $\square$

We compute the constant $b$ in Section 7, and we eventually see that $b=12/(p;p)_{\infty}^{2}(q;q)_{\infty}^{2}$. Once this constant has been determined, we see that the statement above is valid for $|p|<1$ without any particular restriction.

5 Coboundary operator $\nabla_{\rm sym}$

In this section we explain a fundamental method for deriving $q$-difference equations of the contour integrals (3.6) based on an operator $\nabla_{\rm sym}$. This method corresponds to integration by parts in calculus, and will be used in the succeeding section for the proof of Lemma 3.2 presented in Section 3.

From the definition (3.1) of $\Phi(z)$ we have

[TABLE]

where

[TABLE]

From (2.5) and (2.6), we can immediately confirm that

[TABLE]

We remark that $f^{+}(z)$ and $f^{-}(z)$ have the quasi-periodicity

[TABLE]

[TABLE]

with respect to the $p$-shifts, respectively.

We denote by $\mathcal{M}((\mathbb{C}^{*})^{2})$ the $\mathbb{C}$-vector space of meromorphic functions on $(\mathbb{C}^{*})^{2}$, and by $\mathcal{O}((\mathbb{C}^{*})^{2})$ the $\mathbb{C}$-vector space of holomorphic functions on $(\mathbb{C}^{*})^{2}$. For each function $\varphi(z)\in z_{2}^{1\over 2}\mathcal{M}((\mathbb{C}^{*})^{2})=\{z_{2}^{1\over 2}f(z)\,|\,f(z)\in\mathcal{M}((\mathbb{C}^{*})^{2})\}$ we define the function $\nabla\varphi(z)$ by

[TABLE]

and $\nabla_{\rm sym}\varphi(z)$ by the symmetrization of $\nabla\varphi(z)$:

[TABLE]

where $\mathcal{M}((\mathbb{C}^{*})^{2})^{W}$ denotes the $\mathbb{C}$-vector space of $W$-invariant meromorphic functions on $(\mathbb{C}^{*})^{2}$.

Lemma 5.1

Suppose that $|a_{k}|<1$ $(k=1,\ldots 5)$. For any $\varphi(z)\in z_{2}^{1\over 2}\mathcal{O}((\mathbb{C}^{*})^{2})$, we have $\langle\nabla\varphi(z)\rangle=0,$ and hence $\langle\nabla_{\rm sym}\varphi(z)\rangle=0.$

Proof. From (3.1) and (5.1), we have

[TABLE]

From (5.3) we have

[TABLE]

Since $z_{1}z_{2}^{3\over 2}T_{q,z_{1}}^{-{1\over 2}}\varphi(z)\in\mathcal{O}((\mathbb{C}^{*})^{2})$ if $\varphi(z)\in z_{2}^{1\over 2}\mathcal{O}((\mathbb{C}^{*})^{2})$, when $|a_{k}|<1$ $(k=1,\ldots 5)$ and $z_{2}$ is fixed as $|z_{2}|=1$, the right-hand side of (5.10) as a function of $z_{1}$ has no poles in the annulus $|q|\leq|z_{1}|\leq 1$. Hence, by Cauchy’s integral theorem we have

[TABLE]

Combining (5.9) and (5.11), we have

[TABLE]

Since $\Phi(z)$ and $\varpi(z)$ are $W$-symmetric, we therefore obtain

[TABLE]

Definition 5.2

We consider the $\mathbb{C}$-linear subspace $\mathcal{F}\subseteq\mathcal{O}((\mathbb{C}^{*})^{2})$ consisting of all $W$-invariant holomorphic functions $f(z)$ such that

[TABLE]

We set $W_{0}=\langle s_{2},w_{0}\rangle\subset W$, where $w_{0}=(s_{1}s_{2})^{3}$ is the longest element of $W$. For $\epsilon\in\{-1,1\}$ we consider the $\mathbb{C}$-subspace $\mathcal{G}_{\epsilon}\subseteq z_{2}^{1\over 2}\mathcal{O}((\mathbb{C}^{*})^{2})$ consisting of all $W_{0}$-invariant functions $f(z)\in z_{2}^{1\over 2}\mathcal{O}((\mathbb{C}^{*})^{2})$ such that

[TABLE]

Remark. It can be verified directly that if $f(z)\in\mathcal{M}((\mathbb{C}^{*})^{2})$ has the quasi-periodicity (5.12), so does the function $w.f(z)$ for any $w\in W$. Also if $f(z)\in z_{2}^{1\over 2}\mathcal{M}((\mathbb{C}^{*})^{2})$ satisfies (5.13), so does $w.f(z)$ for any $w\in W_{0}$.

Lemma 5.3

Under the condition $(a_{1}a_{2}a_{3}a_{4}a_{5})q^{1\over 2}=\epsilon p^{1\over 2}$, if $\varphi(z)\in\mathcal{G}_{\epsilon}$, then $\nabla_{\rm sym}\varphi(z)\in\mathcal{F}$.

Proof. Suppose that $\varphi(z)\in z_{2}^{1\over 2}\mathcal{M}((\mathbb{C}^{*})^{2})$ satisfies (5.13). Then we have

[TABLE]

Since $f^{+}(z)$ satisfies (5.5), we have

[TABLE]

Similarly, from (5.6) we have

[TABLE]

Hence, under the condition $(a_{1}a_{2}a_{3}a_{4}a_{5})q^{1\over 2}=\epsilon p^{1\over 2}$, both $f^{+}(z)T_{q,z_{1}}^{1\over 2}\varphi(z)$ and $f^{-}(z)T_{q,z_{1}}^{-{1\over 2}}\varphi(z)$ satisfy the same quasi-periodicity condition (5.12), as well as $\nabla\varphi(z)=f^{+}(z)T_{q,z_{1}}^{1\over 2}\varphi(z)+f^{-}(z)T_{q,z_{1}}^{-{1\over 2}}\varphi(z)$. This implies that $\nabla_{\rm sym}\varphi(z)=\sum_{w\in W}w.(\nabla\varphi(z))$ satisfies (5.12), since this quasi-periodicity is preserved by the action of $W$.

We introduce the elliptic version of the Weyl denominator

[TABLE]

which satisfies $w.\Delta(z)={\rm sgn}(w)\Delta(z)$ for all $w\in W$. Note that

[TABLE]

When $\varphi(z)$ belongs to $z_{2}^{1\over 2}\mathcal{O}((\mathbb{C}^{*})^{2})$, the numerator of the right-hand side is a quasi-periodic holomorphic function on $(\mathbb{C}^{*})^{2}$. Since it is alternating with respect to the action of $W$, it is divisible by $\Delta(z)$. This means that $\nabla_{\rm sym}\varphi(z)$ is holomorphic on $(\mathbb{C}^{*})^{2}$ and belongs to the space $\mathcal{F}$. $\square$

Lemma 5.4

For $\varphi(z)\in\mathcal{G}_{\epsilon}$, $\nabla_{\rm sym}\varphi(z)$ is expressed as

[TABLE]

where $f_{k}(z)$ and $\varphi_{k}(z)$ are given by

[TABLE]

for $k=0,1,\ldots,5$.

Proof. From the definition (5.3) of $f^{-}(z)$, we have

[TABLE]

Since $\varphi(z)\in\mathcal{G}_{\epsilon}$ is invariant under $w_{0}=(s_{1}s_{2})^{3}$, i.e., $\varphi(z)=\varphi(z^{-1})$, for $\varphi(z)\in\mathcal{G}_{\epsilon}$ we have

[TABLE]

Applying (5.17) and (5.18) to the definition (5.7) of $\nabla\varphi(z)$, for $\varphi(z)\in\mathcal{G}_{\epsilon}$ we have

[TABLE]

so that we have

[TABLE]

From the expression (2.1) of $W$, this implies that

[TABLE]

Since $\varphi(z)\in\mathcal{G}_{\epsilon}$ is invariant under $s_{2}$, i.e., $\varphi(z_{1},z_{2})=\varphi(z_{1}z_{2}^{3},z_{2}^{-1})$, for $\varphi(z)\in\mathcal{G}_{\epsilon}$ we have

[TABLE]

From (5.4), (5.20) and (5.19), we therefore obtain $\nabla_{\rm sym}\varphi(z)=4\sum_{k=0}^{5}(s_{1}s_{2})^{k}.(f_{0}(z)\varphi_{0}(z))$, which coincides with (5.15). $\square$

6 Proof of Lemma 3.2

The goal of this section is to give a proof of Lemma 3.2 investigating the $\mathbb{C}$-linear mapping $\nabla_{\rm sym}:\mathcal{G}_{\epsilon}\to\mathcal{F}$ defined in the previous section. For that purpose we first clarify the structure of the target space $\mathcal{F}$ in Definition 5.2.

Lemma 6.1

$\dim_{\mathbb{C}}\mathcal{F}\leq 4$.

Proof. For arbitrary $f(z)\in\mathcal{F}$, since $f(z)$ is a holomorphic function of $z=(z_{1},z_{2})\in(\mathbb{C}^{*})^{2}$, $f(z)$ can be expanded as Laurent series $\sum_{\lambda\in P}c_{\lambda}z^{\lambda}$, where $\lambda=\lambda_{1}\alpha_{1}+\lambda_{2}\alpha_{2}\in Q=P$.

Let $D$ be the set defined by $D=\{\lambda\in P\,|\,(\lambda,\alpha_{1})\geq 0,(\lambda,\alpha_{2})\geq 0,(\lambda-\varpi_{1},\varpi_{1})\leq 0\}$, which is the set of points in the triangle area consisting of three lines $(\lambda,\alpha_{1})=0$, $(\lambda,\alpha_{2})=0$ and $(\lambda-\varpi_{1},\varpi_{1})=0$. Then we can immediately confirm that $D=\{0,\varpi_{1},\varpi_{2},2\varpi_{2}\}$. (See Figure 2.) Let $H$ be the $W$-orbit of $D$, i.e., $H=\bigcup_{w\in W}wD$, which is the set of points in the regular hexagon area consisting of six lines $(\lambda-(s_{2}s_{1})^{k}\varpi_{1},(s_{2}s_{1})^{k}\varpi_{1})=0$ for $k=0,1,\ldots,5$. Since $f(z)\in\mathcal{F}$ is $W$-symmetric, i.e., $wf(z)=f(z)$ for $w\in W$, we have $c_{w\lambda}=c_{\lambda}$ for $w\in W$. This means that the coefficients $c_{\lambda}$ for $\lambda\in H$ are determined by $c_{\lambda}$ for $\lambda\in D$. On the other hand, since $f(z)\in\mathcal{F}$ satisfies that

[TABLE]

we have $\sum_{\lambda\in P}c_{\lambda}\,p^{\lambda_{1}}z^{\lambda}=\sum_{\lambda\in P}c_{\lambda}\,p^{-2}z^{\lambda-2\varpi_{1}}.$ Equating the coefficients of $z_{\lambda}$ on both sides, we have $c_{\lambda}\,p^{\lambda_{1}}=c_{\lambda+2\varpi_{1}}p^{-2}$, i.e.,

[TABLE]

Moreover, applying (6.1) to $f(z)=s_{1}.f(z)=\sum_{\lambda\in P}c_{s_{1}\lambda}z^{\lambda}$, we also have $c_{s_{1}\lambda}\,p^{\lambda_{1}}=c_{s_{1}(\lambda+2\varpi_{1})}p^{-2}$, i.e.,

[TABLE]

where $s_{1}\varpi_{1}=\alpha_{1}+3\alpha_{2}$. Combining (6.2) and (6.3), for arbitrary $n_{1},n_{2}\in\mathbb{Z}$ we have

[TABLE]

so that the coefficients $c_{\lambda}$ for $\lambda\in H_{n_{1},n_{2}}=\{\lambda+2n_{1}\varpi_{1}+2n_{2}(s_{1}\varpi_{1})\,|\,\lambda\in H\}$ are determined by $c_{\lambda}$ for $\lambda\in H$. Since the lattice $P$ is covered by the sets $H_{n_{1},n_{2}}$ for $(n_{1},n_{2})\in\mathbb{Z}^{2}$, i.e., $P=\bigcup_{(n_{1},n_{2})\in\mathbb{Z}^{2}}H_{n_{1},n_{2}}$, the coefficients $c_{\lambda}$ for $\lambda\in P$ are also determined by $c_{\lambda}$ for $H$, and consequently, determined by $c_{\lambda}$ for $\lambda\in D$. Therefore we obtain $\dim_{\mathbb{C}}\mathcal{F}\leq|D|=4$. $\square$

Remark. In the next proposition, we prove that $\dim_{\mathbb{C}}\mathcal{F}=|D|=4$. In general, we can consider the $\mathbb{C}$-linear subspace $\mathcal{F}_{n}\subseteq\mathcal{O}((\mathbb{C}^{*})^{2})$ consisting of all $W$-invariant holomorphic functions $f(z)$ such that

[TABLE]

In the same way as above, it is actually confirmed that

[TABLE]

where

[TABLE]

Proposition 6.2

Let $F_{k}(z)$ be functions defined in (3.8). For generic $a_{1},a_{2},a_{3},a_{4}\in\mathbb{C}^{*}$, the set $\{F_{1}(z),\ldots,F_{4}(z)\}$ is a $\mathbb{C}$-basis of the space $\mathcal{F}$, i.e.,

[TABLE]

In particular, $\dim_{\mathbb{C}}\mathcal{F}=4$.

In the rest of this section, we omit the base $p$ in the notation $e(u,v;p)$ defined in (3.4), so that

[TABLE]

In order to prove Proposition 6.2, we define the points $\mathrm{p}_{ij}$ $(1\leq i<j\leq 5)$ in $(\mathbb{C}^{*})^{2}$ given by

[TABLE]

i.e., $z_{2}=a_{j}$, $z_{1}z_{2}=a_{i}$, $z_{1}z_{2}^{2}=a_{i}a_{j}$ if $z={\rm p}_{ij}$. By definition the values $F_{k}(z)$ at $z=\mathrm{p}_{ij}$ are given by

[TABLE]

and hence

[TABLE]

This already implies that $F_{k}(z)$ $(k=1,2,3)$ are linearly independent for generic $a_{1},a_{2},a_{3}$. In fact, their values at $z=\mathrm{p}_{23},\mathrm{p}_{13},\mathrm{p}_{12}$ are given as follows.

[TABLE]

Let $G(z)$ be function defined by

[TABLE]

Lemma 6.3

For $z=\mathrm{p}_{ij}\in(\mathbb{C}^{*})^{2}$ $(1\leq i<j\leq 3)$ the function $G(z)$ satisfies

[TABLE]

Moreover for $\{i,j,k\}=\{1,2,3\}$ and $x\in\mathbb{C}^{*},$ if $z=(a_{k}/x,x),$ then we have

[TABLE]

which is independent of $a_{4}$. In particular, we have

[TABLE]

Proof. From (6.6) and (6.7), we have (6.8). Without loss of generality we prove (6.9) for $k=1$. Using (6.5) we also have

[TABLE]

Denoting $g(x)=G(a_{1}/x,x)$ as a function of $x$, from (6) we immediately see that $g(x)$ is holomorphic on $\mathbb{C}^{*}$ and satisfies

[TABLE]

and

[TABLE]

This implies that $g(x)$ is divisible by $\theta(a_{1}a_{2}x,a_{1}a_{3}x,a_{2}/x,a_{3}/x;p)$, i.e.,

[TABLE]

where $c$ is a constant independent of $x$. We evaluate $g(a_{2}/a_{1})$ in two ways. From (6) we have

[TABLE]

On the other hand, (6.12) implies $g(a_{2}/a_{1})=c\,\theta(a_{2}^{2},a_{2}a_{3},a_{1},a_{1}a_{3}/a_{2};p),$ so that we have

[TABLE]

We therefore obtain

[TABLE]

which coincides with (6.9) for $k=1$. $\square$

Proof of Proposition 6.2. From (6.8) and (6.10) of Lemma 6.3, we see that $F_{1}(z),F_{2}(z),F_{3}(z),G(z)$ are linearly independent for generic $a_{1},a_{2},a_{3},a_{4}$.

[TABLE]

Hence $F_{1}(z),F_{2}(z),F_{3}(z),F_{4}(z)$ are also linearly independent for generic $a_{1},a_{2},a_{3},a_{4}$. From Lemma 6.1, we therefore obtain that $\{F_{1}(z),\ldots,F_{4}(z)\}$ is a $\mathbb{C}$-basis of $\mathcal{F}$. This completes the proof of Proposition 6.2. $\square$

By the definition (6.7) of $G(z)$ and Proposition 6.2, the set $\{F_{1}(z),F_{2}(z),F_{3}(z),G(z)\}$ is also a $\mathbb{C}$-basis of the space $\mathcal{F}$, i.e., $\mathcal{F}=\mathbb{C}F_{1}(z)\oplus\mathbb{C}F_{2}(z)\oplus\mathbb{C}F_{3}(z)\oplus\mathbb{C}G(z)$. Here we slightly deform $F_{3}(z)$ in the set $\{F_{1}(z),F_{2}(z),F_{3}(z),G(z)\}$ as follows. Let ${\rm p}_{ij}^{*}$ $(1\leq i<j\leq 5)$ be points in $(\mathbb{C}^{*})^{2}$ given by

[TABLE]

so that $z_{2}=a_{j}/a_{i}$, $z_{1}z_{2}=a_{i}$, $z_{1}z_{2}^{2}=a_{j}$ if $z={\rm p}_{ij}^{*}$. We remark that $F_{1}({\rm p}_{12}^{*})=F_{2}({\rm p}_{12}^{*})=0$ and $G({\rm p}_{12}^{*})$ is evaluated as

[TABLE]

which is already confirmed in (6.13). We now define $F^{\prime}_{3}(z)$ as

[TABLE]

which satisfies

[TABLE]

[TABLE]

Therefore the set $\{F_{1}(z),F_{2}(z),F^{\prime}_{3}(z),G(z)\}$ is also a $\mathbb{C}$-basis of the space $\mathcal{F}$, i.e.,

[TABLE]

which will be used in the proof of Lemma 6.5.

Let $\phi_{ij}(z)$ and $\phi^{\prime}_{ij}(z)$ be functions defined by

[TABLE]

From direct calculation we can immediately confirm that

[TABLE]

Proposition 6.4

For generic $a_{1},a_{2},a_{3}$, the set $\{\phi_{12}(z),\phi_{13}(z),\phi_{23}(z),\phi^{\prime}_{12}(z),\phi^{\prime}_{13}(z),\phi^{\prime}_{23}(z)\}$ is a $\mathbb{C}$-basis of the space $\mathcal{G}_{\epsilon}$. In particular, $\dim_{\mathbb{C}}\mathcal{G}_{\epsilon}=6$.

Remark. We omit the proof of this proposition since we need the fact (6.20) only for the succeeding discussions.

Lemma 6.5

Under the condition $(a_{1}a_{2}a_{3}a_{4}a_{5})q^{1\over 2}=\epsilon p^{1\over 2}$, $\nabla_{\rm sym}\phi_{12}(z)$ and $\nabla_{\rm sym}\phi^{\prime}_{12}(z)$ are expanded as

[TABLE]

where the coefficients are given by

[TABLE]

Proof. For $\varphi(z)\in\mathcal{G}_{\epsilon}$ we simply use notation

[TABLE]

where $f_{k}(z)$ and $\varphi_{k}(z)$ are given in (5.16). Our aim is to confirm that $\widetilde{\phi}_{12}(z)$ and $\widetilde{\phi}^{\prime}_{12}(z)$ are expanded as the right-hand side of (6.21) and (6.22), respectively. Before that we evaluate the special values $\widetilde{\phi}_{12}(\mathrm{p}_{1j})$, $\widetilde{\phi}_{12}^{\prime}(\mathrm{p}_{1j})$ and $\widetilde{\phi}_{12}(\mathrm{p}_{12}^{*})$, $\widetilde{\phi}^{\prime}_{12}(\mathrm{p}_{12}^{*})$.

First we evaluate $\widetilde{\phi}_{12}(\mathrm{p}_{1j})$ and $\widetilde{\phi}_{12}^{\prime}(\mathrm{p}_{1j})$. From (6.4), if $z=\mathrm{p}_{ij}$, then $z_{2}=a_{j}$, $z_{1}z_{2}=a_{i}$. Hence, since $f_{2}(z)$, $f_{3}(z)$ have factors $\theta(a_{k}z_{1}^{-1}z_{2}^{-1};p)$ $(1\leq k\leq 5)$ and $f_{4}(z)$, $f_{5}(z)$ have factors $\theta(a_{k}z_{2}^{-1};p)$ $(1\leq k\leq 5)$, we have

[TABLE]

[TABLE]

From (6.29), this implies that

[TABLE]

By definition we have $\varphi_{0}(z)=\varphi(q^{1\over 2}z_{1},z_{2})$ and $\varphi_{1}(z)=\varphi(q^{1\over 2}z_{1}^{2}z_{2}^{3},z_{1}^{-1}z_{2}^{-1})$. If we put $\varphi(z)=\phi_{12}(z)$, then the explicit forms of $\varphi_{0}(z)$ and $\varphi_{1}(z)$ are given as

[TABLE]

so that we have

[TABLE]

On the other hand, in the same way as above, if we put $\varphi(z)=\phi_{12}^{\prime}(z)$, then we have

[TABLE]

Therefore, if $\varphi(z)=\phi_{12}(z),\phi_{12}^{\prime}(z)$, then we eventually obtain $\widetilde{\varphi}(\mathrm{p}_{1j})=f_{0}(\mathrm{p}_{1j})\varphi_{0}(\mathrm{p}_{1j})$, i.e.,

[TABLE]

for $j=2,3,4,5$, in particular, $\widetilde{\phi}_{12}(\mathrm{p}_{12})=\widetilde{\phi}_{12}^{\prime}(\mathrm{p}_{12})=0$.

Next we evaluate $\widetilde{\phi}_{12}(\mathrm{p}_{12}^{*})$ and $\widetilde{\phi}^{\prime}_{12}(\mathrm{p}_{12}^{*})$. From (6.14), if $z=\mathrm{p}_{12}^{*}$, then $z_{1}z_{2}=a_{1}$, $z_{1}z_{2}^{2}=a_{2}$. Since $f_{2}(z)$, $f_{3}(z)$ have factors $\theta(a_{k}z_{1}^{-1}z_{2}^{-1};p)$ $(1\leq k\leq 5)$ and $f_{3}(z)$, $f_{4}(z)$ have factors $\theta(a_{k}a_{1}^{-1}z_{2}^{-2};p)$ $(1\leq k\leq 5)$, we have

[TABLE]

From (6.29), this implies that

[TABLE]

When $\varphi(z)=\phi_{12}(z)$ or $\phi_{12}^{\prime}(z)$ the functions $\varphi_{0}(z)$, $\varphi_{1}(z)$ and $\varphi_{5}(z)$ have factors $e(a_{1},z_{2})e(a_{2},z_{2})$, $e(a_{1},z_{1}z_{2})e(a_{2},z_{2}z_{2})$ and $e(a_{1},z_{1}z_{2}^{2})e(a_{2},z_{1}z_{2}^{2})$, respectively. This implies $\varphi_{1}(\mathrm{p}_{12}^{*})=0$ and $\varphi_{5}(\mathrm{p}_{12}^{*})=0$. We therefore obtain $\widetilde{\varphi}(\mathrm{p}_{12}^{*})=f_{0}(\mathrm{p}_{12}^{*})\varphi_{0}(\mathrm{p}_{12}^{*})$, i.e.,

[TABLE]

We now calculate the expansions of $\widetilde{\phi}_{12}(z)$ and $\widetilde{\phi}^{\prime}_{12}(z).$ From (6.17), $\widetilde{\phi}_{12}(z),\widetilde{\phi}^{\prime}_{12}(z)\in\mathcal{F}$ are expanded in terms of $F_{1}(z),F_{2}(z),F^{\prime}_{3}(z),G(z)$, i.e.,

[TABLE]

where $c_{1},c_{2},c_{3},c_{12},c^{\prime}_{1},c^{\prime}_{2},c^{\prime}_{3},c^{\prime}_{12}$ are some constants independent of $z$. Our aim is to show that $c_{3}=c_{3}^{\prime}=0$ and the others are given by (6.23)–(6.28). Applying (6.6), (6.8) and (6.16) to (6.34), we have the expressions

[TABLE]

In particular, we immediately have $c_{3}=0$ because $\widetilde{\phi}_{12}(\mathrm{p}_{12})=0$. Furthermore, from (6.5) and (6.32) we obtain

[TABLE]

which is equal to (6.24). Since $\widetilde{\phi}_{12}(z)$ is symmetric with respect to the transposition of $a_{1}$ and $a_{2}$, we obtain the expression (6.23) of $c_{1}$ from (6.24) of $c_{2}$ interchanging $a_{1}$ and $a_{2}$. From (6.15) and (6.33) we have

[TABLE]

which is equal to (6.25).

On the other hand, for $c^{\prime}_{1}$, $c^{\prime}_{2}$ and $c^{\prime}_{12}$ we shall omit the process to calculate the expressions (6.26), (6.27) and (6.28) since it is almost the same way as above for $c_{1}$, $c_{2}$ and $c_{12}$. $\square$

Proposition 6.6

Under the condition $(a_{1}a_{2}a_{3}a_{4}a_{5})^{2}q=p$, for $k=1,2,3$ we have

[TABLE]

where the constant $C_{k}$ is given as

[TABLE]

when $\{i,j,k\}=\{1,2,3\}$. In particular, we have

[TABLE]

Proof. Since (6.37) follows from (6.36) by Lemma 5.1, we show (6.36). By the symmetry with respect to the indices $1,2,3$, it suffices to consider the case where $k=1$ with

[TABLE]

From the condition $(a_{1}a_{2}a_{3}a_{4}a_{5})^{2}q=p$, we suppose that $(a_{1}a_{2}a_{3}a_{4}a_{5})q^{1\over 2}=\epsilon p^{1\over 2}$, where $\epsilon\in\{-1,1\}$. Then, by Lemma 6.5, we have

[TABLE]

[TABLE]

where $c_{1},c_{2},c_{12}$, $c^{\prime}_{1},c^{\prime}_{2},c^{\prime}_{12}$ are explicitly given by (6.23)–(6.28). Eliminating $F_{2}(z)$ from the above equations, we have

[TABLE]

since one can verify $c_{1}c^{\prime}_{2}-c^{\prime}_{1}c_{2}\neq 0$ in the course of computation. Therefore $C_{1}$ in (6.37) is calculated as

[TABLE]

where $N$ and $D$ are expressed as

[TABLE]

Here we can confirm that $N$ and $D$ are factorized as

[TABLE]

respectively. In fact, (6.42) is confirmed as follows. Denoting by $g(u)$ the right-hand side of (6.40) as a function of $u=a_{1}$, $g(u)$ satisfies

[TABLE]

and $g(q^{-{1\over 2}})=0$. This implies that $g(u)$ is divisible by $\theta(q^{{1\over 2}}u,a_{3}u/a_{2};p)$, i.e.,

[TABLE]

where $c$ is a constant independent of $u$. Evaluating $g(-\epsilon p^{1\over 2}a_{2})$ in two ways, we have

[TABLE]

so that we have $c=a_{2}\theta(\epsilon p^{1\over 2},\epsilon p^{1\over 2}q^{1\over 2}a_{2}a_{3};p).$ We therefore obtain

[TABLE]

which is equivalent to the right-hand side of (6.42). For the case (6.43) of $D$, we can actually deduce it from (6.41) in the same way as (6.42) of $N$ above.

From (6), (6.42) and (6.43), we consequently obtain

[TABLE]

which coincides with (6.38) by using the identities

[TABLE]

and the relations

[TABLE]

under the condition $(a_{1}a_{2}a_{3}a_{4}a_{5})q^{1\over 2}=\epsilon p^{1\over 2}$. Thus we finally obtain the expression (6.38) of $C_{1}$ independent of $\epsilon$. $\square$

Proof of Lemma 3.2. By Proposition 6.6, under the condition $(a_{1}a_{2}a_{3}a_{4}a_{5})^{2}q=p$, we have the relation between $\langle F_{2}(z)\rangle$ and $\langle G(z)\rangle$ as $\langle F_{2}(z)\rangle=C_{2}\langle G(z)\rangle$, where

[TABLE]

Therefore we obtain $\langle F_{1}(z)\rangle=(C_{1}/C_{2})\langle F_{2}(z)\rangle$ and from (6.38) and (6.44) the coefficient $C_{1}/C_{2}$ coincides with that of (3.9) in Lemma 3.2. $\square$

*Remark. * The quotient space $\mathcal{H}=\mathcal{F}/\nabla_{\rm sym}\mathcal{G}_{\epsilon}$ can be interpreted as the $q$-difference de Rham cohomology associated with our integral $I(a)$. Note that Proposition 6.6 means $\dim_{\mathbb{C}}\mathcal{H}\leq 1$. Although one can directly verify that $\dim_{\mathbb{C}}\mathcal{H}=1$, we omit the proof since Theorem 1.2 should eventually imply $\mathcal{H}\neq 0$.

7 Computation of the constant $b$

In this section we use the double-sign symbol like $\Gamma(uv^{\pm 1};p,q)=\Gamma(uv,uv^{-1};p,q)$ for abbreviation.

As before, we assume that the parameters satisfy the balancing condition $(a_{1}\cdots a_{5})^{2}=pq$, and regard $a_{5}=\epsilon p^{1\over 2}q^{1\over 2}/a_{1}\cdots a_{4}$, where $\epsilon\in\{-1,1\}$, as a function of $(a_{1},\ldots,a_{4})$. By Theorem 4.2 we showed that the meromorphic functions $I(a_{1},\ldots,a_{5})$ and $J(a_{1},\ldots,a_{5})$ are related by the formula

[TABLE]

provided that $|p|$ is sufficiently small. To determine the constant $b$, we investigate the behavior of these two functions along the divisor $a_{1}a_{2}=1$.

We first consider the limit of $J(a_{1},\ldots,a_{5})$ as $a_{1}\to a_{2}^{-1}$. Since $\Gamma(a_{1}a_{2};p,q)$ is written as

[TABLE]

we have

[TABLE]

Using this, from (4) we have

[TABLE]

where $a_{5}$ in the right-hand side should be understood as $a_{5}=\epsilon p^{1\over 2}q^{1\over 2}/a_{1}a_{2}a_{3}a_{4}$. Since we have $a_{5}\to\epsilon\,p^{1\over 2}q^{1\over 2}/a_{3}a_{4}$ in the limit $a_{1}\to a_{2}^{-1}$, from the definition (1.3) of $\Gamma(u;p,q)$ we obtain

[TABLE]

as $a_{1}\to a_{2}^{-1}$. Using this, (7.3) implies

Lemma 7.1

In the limit as $a_{1}\to a_{2}^{-1}$, the function $J(a)$ satisfies that

[TABLE]

We next investigate the behavior of $I(a_{1},\ldots,a_{5})$ as $a_{1}\to a_{2}^{-1}$, assuming that $|p|$ is sufficiently small so that equality (7.1) holds. Here, for convenience, we suppose that $|p|<|q|^{9}$ as in Theorem 4.2. We denote by $C_{r}(c)$ the positively oriented circle with center $c$ and radius $r$, i.e.,

[TABLE]

Under the condition $a_{5}=\epsilon p^{1\over 2}q^{1\over 2}/a_{1}a_{2}a_{3}a_{4}$ with $|a_{k}|<1$ $(k=1,\ldots,5)$, we consider $I(a)$ defined by (3.2) as the iterated integral

[TABLE]

where the integrand $\Phi(a;z)$ defined by (3.1) is written as

[TABLE]

The function $\Phi(a;z)$ is rewritten as

[TABLE]

where $\Delta(z;p,q)$ denotes the reciprocal of the denominator of $\Phi(a;z)$ in the expression (7.4). In fact, $\Delta(z;p,q)$ turns out to be a holomorphic function of $z\in(\mathbb{C}^{*})^{2}$, since it is computed as

[TABLE]

where $\Delta(z;p)=z_{1}^{-3}z_{2}^{-5}\theta(z_{2},z_{1}z_{2},z_{1}z_{2}^{2},z_{1},z_{1}z_{2}^{3},z_{1}^{2}z_{2}^{3};p)$ is just the elliptic Weyl denominator introduced by (5.14) in Section 5, and (7.6) is immediately confirmed from (1.5). We also write $I(a)$ as

[TABLE]

Since $\Phi(a;z)$ is rewritten from (7.4) as

[TABLE]

where

[TABLE]

$I_{1}(a;z_{2})$ is written as

[TABLE]

We now suppose $|a_{i}|<|a_{1}|<1$ $(i=2,\ldots,5)$. Then, the integral

[TABLE]

defines a holomorphic function of $z_{2}$ on the domain $|a_{1}|^{1\over 2}<|z_{2}|<|a_{1}|^{-{1\over 2}}$. For any $z_{2}\in\mathbb{C}$ satisfying $|a_{1}|^{2}<|z_{2}|<|a_{1}|^{-2}$, it is easy to see that there exists $R>0$ such that the circle $C_{R}(0)$ in the $z_{1}$-plane keeps the points $z_{1}=a_{1}z_{2}^{-1}$, $a_{1}z_{2}^{-2}$ inside and $z_{1}=a_{1}^{-1}z_{2}^{-1}$, $a_{1}^{-1}z_{2}^{-2}$ outside. Then, the integral in (7.9) is continued analytically to the domain $|a_{1}|^{2}<|z_{2}|<|a_{1}|^{-2}$ by deforming $\mathbb{T}=C_{1}(0)$ to $C_{R}(0)$. Hence

[TABLE]

defines a meromorphic function on the domain $|a_{1}|^{2}<|z_{2}|<|a_{1}|^{-2}$.

Lemma 7.2

Suppose $|a_{i}|<|a_{1}|<1$ $(i=2,\ldots,5)$. The integral $I_{1}(a;z_{2})$ is also expressed as

[TABLE]

using a cycle $\mathcal{T}_{1}(z_{2})$ defined by the homology equivalence

[TABLE]

where $\varepsilon>0$ is sufficiently small.

Proof. Since $C_{R}(0)\sim\mathcal{T}_{1}(z_{2})+C_{\varepsilon}(a_{1}z_{2}^{-1})-C_{\varepsilon}(a_{1}^{-1}z_{2}^{-1})+C_{\varepsilon}(a_{1}z_{2}^{-2})-C_{\varepsilon}(a_{1}^{-1}z_{2}^{-2})$ by (7.11), from (7.7) we have

[TABLE]

Since $\Phi(a;z)$ is written as (7.5) with (7.6), using the formula (7.2), we have

[TABLE]

and

[TABLE]

Moreover, in the same way as above, we also have

[TABLE]

which coincides with (7) up to sign, and

[TABLE]

which also coincides with (7) up to sign. Hence, applying (7)–(7.16) to (7.12), we see that $I_{1}(a;z_{2})$ is expressed as (7.10) in Lemma 7.2. $\square$

Lemma 7.3

Suppose $|a_{i}|<|a_{1}|<1$ $(i=2,\ldots,5)$. Then, it follows that

[TABLE]

using a cycle $\mathcal{T}_{2}$ defined by the homological equivalence

[TABLE]

where $\varepsilon>0$ is sufficiently small.

Proof. From Lemma 7.2, (7.7) implies that

[TABLE]

Since $\mathbb{T}\sim\mathcal{T}_{2}+C_{\varepsilon}(a_{1})-C_{\varepsilon}(a_{1}^{-1})$ by (7.18), the initial term of the right-hand side of (7.19) is

[TABLE]

Since $\Phi(a;z)$ is written as (7.8), using (7.2), the second term in the right-hand side of (7.20) is calculated as

[TABLE]

Since the function

[TABLE]

of $z_{1}$ is now holomorphic at the points $z_{1}=1,a_{1}^{-1},a_{1}^{-2},a_{1}^{-3}$, which are avoided by the contour $\mathcal{T}_{1}(a_{1})$ of the integral, we can deform the contour $\mathcal{T}_{1}(a_{1})$ to the circle $\{z_{1}\in\mathbb{C}\,|\,|z_{1}|=|a_{1}|^{-1}\}$ across these points, provided $|a_{i}|<|a_{1}|<1$ $(i=2,\ldots,5)$. This means that

[TABLE]

From (7.21) and (7.22), we therefore obtain

[TABLE]

In the same way as above, the third term in the right-hand side of (7.20) is also calculated as

[TABLE]

Hence, combining (7.19), (7.20), (7.23) and (7.24), we obtain the expression (7.17) of $I(a)$ in Lemma 7.3. $\square$

Lemma 7.4

Suppose that $|a_{i}|<|a_{2}|<1$ $(i=3,\ldots,5)$ and $|q|<|a_{2}|<1$. Then it follows that

[TABLE]

Proof. Before taking the limit $a_{1}\to a_{2}^{-1}$ for $I(a)$, we need to extend $I(a)$ by analytic continuation to the function of $a_{1}$ on the domain $1\leq|a_{1}|<|q|^{-1}$, provided $1<|a_{2}^{-1}|<|q|^{-1}$. For $a_{1}$ satisfying $1<|a_{1}|<|q|^{-1}$, the integral of the first term of (7.17) in Lemma 7.3 can be extended by

[TABLE]

deforming the cycle $\mathcal{T}_{2}\times\mathcal{T}_{1}(z_{2})$ to $\mathbb{T}\times\mathbb{T}$ of the integral as Figure 5.

Since the integral (7.26) as a function of $a_{1}$ is regular at $a_{1}=a_{2}^{-1}$, the integral (7.26) itself has a finite limit as $a_{1}\to a_{2}^{-1}$. Hence, using (7.2), (7.17) implies that

[TABLE]

where $a_{5}$ in the right-hand side should be understood as $a_{5}=\epsilon p^{1\over 2}q^{1\over 2}/a_{1}a_{2}a_{3}a_{4}\to\epsilon p^{1\over 2}q^{1\over 2}/a_{3}a_{4}$. Applying the variable changes

[TABLE]

to the integrals in (7.27), respectively from the top, we have

[TABLE]

whose contour $|w|=1$ is deformed from the contours $|w|=|a_{2}|^{{1\over 2}}$ or $|w|=|a_{2}|^{-{1\over 2}}$, since its integrand has no poles in the annulus $|a_{2}|^{{1\over 2}}\leq|w|\leq|a_{2}|^{-{1\over 2}}$, provided $|a_{i}|<|a_{2}|<1$ $(i=3,\ldots,5)$. Here the integral

[TABLE]

coincides with the elliptic integral (1.2) of type $BC_{1}$ for specific parameters

[TABLE]

satisfying the balancing condition $\prod_{k=3}^{5}(a_{2}^{1\over 2}a_{k})(a_{2}^{-{1\over 2}}a_{k})=(a_{3}a_{4}a_{5})^{2}=pq$, and is evaluated as

[TABLE]

which is confirmed from the right-hand side of (1.2). Therefore, applying this to (7.28), we eventually obtain (7.25) in Lemma 7.4. This completes the proof. $\square$

Lastly we determine the constant $b$ independent of $a$ appearing in (7.1). Comparing both sides of (7.1) as $a_{1}\to a_{2}^{-1}$ using Lemmas 7.1 and 7.4, we obtain

[TABLE]

Acknowledgements

The authors are grateful to the anonymous referee for valuable comments that have helped them improve the manuscript. This work is supported by JSPS Kakenhi Grants (B)15H03626 and (C)18K03339.