Generating Function of q- and Elliptic Multiple Polylogarithms of Hurwitz Type

Abstract

Ohno and Zagier (Indag Math 12:483–487, 2001) found that a generating function of sums of multiple polylogarithms can be written in terms of the Gauss hypergeometric function ${}_2F_1$. As a generalization of the Ohno and Zagier formula, Ihara et al. (Can J Math 76:1–17, 2022) showed that a generating function of sums of interpolated multiple polylogarithms of Hurwitz type can be expressed in terms of the generalized hypergeometric function ${}_{r+1}F_r$. In this paper, we establish q- and elliptic analogues of this result. We introduce elliptic q-multiple polylogarithms of Hurwitz type and study a generating function of sums of them. By taking the trigonometric and classical limits in the main theorem, we can obtain q- and elliptic generalizations of the Ihara, Kusunoki, Nakamura and Saeki formula.

1 Introduction

The Gauss hypergeometric function ${}_2F_1$ is a special function with various interesting properties and important applications. The function ${}_2F_1$ is defined by

$$\begin{aligned} {}_2F_1\begin{bmatrix} \begin{matrix} \alpha ,\beta \\ \gamma \end{matrix} ;z \end{bmatrix}:=\sum _{n=0}^{\infty } \frac{(\alpha )_n (\beta )_n}{(\gamma )_n n!}z^n \end{aligned}$$

with

$$\begin{aligned} (s)_n:= {\left\{ \begin{array}{ll} 1 &{} (n=0) \\ s (s+1) \cdots (s+n-1) &{} (n \ge 1). \end{array}\right. } \end{aligned}$$

For ${\varvec{k}}=(k_1, \dots , k_l) \in {{\mathbb {Z}}}_{> 0}^l$ and $a \in {{\mathbb {C}}}$ with $|a|<1$, the multiple polylogarithm is defined by

$$\begin{aligned} L_{\varvec{k}}(a):=\sum _{m_1,\ldots ,m_{l} \ge 1} \prod _{i=1}^l \frac{a^{m_i}}{(m_i+\cdots +m_l)^{k_i}}. \end{aligned}$$

In [20], Ohno and Zagier found that a generating function of sums of multiple polylogarithms can be expressed in terms of the Gauss hypergeometric function. Since then, various generalizations and analogous results have been obtained for the Ohno and Zagier formula. See for example [3,4,5, 8, 11, 14,15,16, 19, 21, 22].

In this paper, we focus on Ihara, Kusunoki, Nakamura and Saeki’s work [8]. They generalized the Ohno and Zagier’s result to interpolated multiple polylogarithms of Hurwitz type. The multiple polylogarithm of Hurwitz type is defined by

$$\begin{aligned} L_{\varvec{k}}(a,x):=&\sum _{\begin{array}{c} m_1,\ldots ,m_{l-1} \ge 1\\ m_l \ge 0 \end{array}} \prod _{i=1}^l \frac{a^{m_i}}{(m_i+\cdots +m_l+x)^{k_i}} \end{aligned}$$

and its interpolated version is defined by

$$\begin{aligned} L_{\varvec{k}}(a,\alpha ,x):=&(-1)^l \sum _{\widetilde{\varvec{k}}=(k_1 \Box \cdots \Box k_l)}L_{\widetilde{\varvec{k}}}(a,x)(-c_0(\alpha ))^{l-\textrm{dep}\, \widetilde{\varvec{k}}}, \end{aligned}$$

(1.1)

where we put $c_0(\alpha )=\frac{1}{\alpha -1}$ and the sum on the right-hand side of (1.1) runs over all $\widetilde{\varvec{k}}$ of the form $\widetilde{\varvec{k}}=(k_1 \Box \cdots \Box k_l)$ in which each $\Box$ filled by two candidates “,” or “$+$”. The expression $\textrm{dep}\, \widetilde{\varvec{k}}$ denotes the number of the components of $\widetilde{\varvec{k}}$.

It is shown in [8] that a generating function of sums of $L_{\varvec{k}}(a,\alpha ,x)$ of fixed weight, depth and all i-heights (see Sect. 3.1) can be written in terms of the generalized hypergeometric function ${}_{r+1}F_r$. The function ${}_{r+1}F_r$ is defined by

$$\begin{aligned} {}_{r+1}F_r \begin{bmatrix} \begin{matrix} a_1,\dots , a_{r+1} \\ b_1, \dots , b_r \end{matrix} ;z \end{bmatrix}:=\sum _{n=0}^{\infty } \frac{(a_1,\dots ,a_{r+1})_n}{(1,b_1,\dots ,b_r)_n}z^n. \end{aligned}$$

The purpose of this paper is to establish q- and elliptic analogues of the Ihara, Kusunoki, Nakamura and Saeki’s result. To treat both cases, we introduce elliptic q-multiple polylogarithms of Hurwitz type, denoted by $L_{\varvec{k}}(a,\alpha ,\beta ;p,q)$, and study a generating function of sums of $L_{\varvec{k}}(a,\alpha ,\beta ;p,q)$. By taking the trigonometric and classical limits in the main theorem (Theorem 4.9), we can obtain q- and elliptic generalizations of Theorem 3.5 of [8]. In particular, we show that a generating function of q-multiple polylogarithms of Hurwitz type can be written in terms of the q-hypergeometric function ${}_{r+1}\phi _{r}$. (See Theorem 5.3.)

The rest of this paper is organized as follows. In Sect. 2, we define elliptic q-multiple polylogarithms of Hurwitz type and establish basic properties of them that will be used in the later sections. In Sect. 3, we introduce generating functions $\Phi _j(a;p,q)$ of sums of $L_{\varvec{k}}(a,\alpha ,\beta ;p,q)$ with fixed weight, depth and i-height and deduce q-difference equations satisfied by them. In Sect. 4, we solve the q-difference equation (3.1) satisfied by $\Phi _{r-1}(a;p,q)$ by generalizing the idea used in the previous work [11]. In [11], we used Heine’s transformation formula ( [7])

$$\begin{aligned} {}_2\phi _1\begin{bmatrix} \begin{matrix} a,b \\ c \end{matrix} ;q,z \end{bmatrix}=\frac{(abz/c;q)_{\infty }}{(z;q)_{\infty }} {}_2\phi _1\begin{bmatrix} \begin{matrix} c/a,c/b \\ c \end{matrix} ;q,abz/c \end{bmatrix}. \end{aligned}$$

(1.2)

In this paper, Kajihara’s transformation formula for multiple basic hypergeometric series [9], which is a generalization of (1.2), plays important roles. In Sect. 5, we establish q- and elliptic analogues of the result due to Ihara, Kusunoki, Nakamura and Saeki [8] described above. When $\beta =1$, the q-analogue is essentially the same as Theorem 5.3 of Li-Wakabayashi [16].

2 q- and elliptic analogues of multiple polylogarithms of Hurwitz type

In this section, we introduce elliptic q-multiple polylogarithms of Hurwitz type and establish their basic properties.

2.1 q-Analogues of Apostol–Bernoulli Polynomials

Let q be a complex number with $0<|q|<1$. For $a \in {{\mathbb {C}}}$, we put

$$\begin{aligned} (a;q)_{\infty }&:=\prod _{j=0}^{\infty }(1-aq^j) \end{aligned}$$

and define the theta function $\theta (a;q)$ by

$$\begin{aligned} \theta (a;q):=(a;q)_{\infty }(qa^{-1};p)_{\infty }. \end{aligned}$$

For $a, \beta \in {{\mathbb {C}}}$, we define the Kronecker function $F(a,\beta ;q)$ by

$$\begin{aligned} F(a,\beta ;q):=\frac{\theta '(1;q)\theta (a\beta ;q)}{\theta (a;q)\theta (\beta ;q)}. \end{aligned}$$

The Kronecker function has the following Laurent series expansion (see [23]):

$$\begin{aligned} F(a,\beta ;q)=\sum _{n \in {{\mathbb {Z}}}}\frac{a^n}{q^n \beta -1} \ \ \ (|q|<|a|<1). \end{aligned}$$

(2.1)

For $k \in {{\mathbb {Z}}}_{\ge 0}$, we define the function $\psi _k (a, \beta ;q)$ by

$$\begin{aligned} \psi _{k}(a, \beta ;q):=-\frac{1}{(k-1)!} \frac{\partial ^{k-1}}{\partial \beta ^{k-1}} \left\{ \beta ^{k-1}\left( F(a,\beta ;q)-\frac{1}{\beta -1} \right) \right\} . \end{aligned}$$

By (2.1), the function $\psi _k(a,\beta ;q)$ has the following Laurent series expansion:

$$\begin{aligned} \psi _k(a,\beta ;q)=\sum _{n \in {{\mathbb {Z}}}\setminus \{0\}} \frac{a^n}{(1-q^n \beta )^k} \ \ (|q|^k<|a|<1). \end{aligned}$$

For $k=0$, we put $\psi _0(a,\beta ;q)=-1$.

In [12], we showed that $\psi _k(a;q):=\lim _{\beta \rightarrow 1} \psi _k(a,\beta ;q)$ can be regarded as q-analogues of periodic Bernoulli polynomials. We will show that the classical limits (the limits as $q \rightarrow 1$) of $\psi _k(a,\beta ;q)$ are given by Apostol-Bernoulli polynomials introduced in [2]. The Apostol-Bernoulli polynomials $B_k(x,w)$ are defined by

$$\begin{aligned} \sum _{k=0}^{\infty } B_k(x,w)\frac{t^k}{k!}=\frac{te^{xt}}{we^t-1}. \end{aligned}$$

When $w=1$, $B_n(x,1):=B_n(x)$ are classical Bernoulli polynomials.

For $-1<x<1, \ s \in {{\mathbb {R}}} \setminus {{\mathbb {Z}}}$ and $k \in {{\mathbb {Z}}}_{\ge 0}$, we now put

$$\begin{aligned} {\varvec{B}}_k(s,x)= {\left\{ \begin{array}{ll} -\dfrac{(2\pi \sqrt{-1})^k}{k!}e^{-2\pi \sqrt{-1}x\{s\}}B_k(\{s\},e^{-2\pi \sqrt{-1}x})-\dfrac{1}{x^k} &{} (x \ne 0) \vspace{1mm}\\ -\dfrac{(2 \pi \sqrt{-1})^k}{k!} B_k(\{s\}) &{} (x=0), \end{array}\right. } \end{aligned}$$

where $\{s \}$ denotes the fractional part of s. Then we have the following proposition:

Proposition 2.1

Let $k \in {{\mathbb {Z}}}_{>0}$. Then, for $0<t<k$ and $s \in {{\mathbb {R}}} {\setminus } {{\mathbb {Z}}}$,

$$\begin{aligned} \lim _{q \rightarrow 1}(1-q)^k \psi _k\left( q^t e^{2\pi \sqrt{-1}s},q^x;q\right) ={\varvec{B}}_k(s,x). \end{aligned}$$

Proof

The proof is similar to that of Proposition 4.2 of [12] by using the Fourier series expansions of Apostol–Bernoulli polynomials given in [6, 17]. $\square$

2.2 Elliptic q-Multiple Polylogarithms of Hurwitz Type

Let p be a complex number satisfying $0<|p|<1$. For ${\varvec{k}}=(k_i)_{i=1}^l \in {{\mathbb {Z}}}_{\ge 0}^l$, we define the elliptic q-multiple polylogarithm of Hurwitz type by

$$\begin{aligned} L_{\varvec{k}}(a,\alpha ,\beta ;p,q)&:=\frac{1}{(2\pi \sqrt{-1})^l} \int _{{{\mathbb {T}}}^l}\prod _{i=1}^l F(at_1\cdots t_i z_1\cdots z_i, \alpha ;p)\nonumber \\&\,\quad \times \psi _{k_i}\left( t_i^{-1}z_i^{-1}, \beta ;q\right) \frac{dz_1 \cdots dz_l}{z_1\cdots z_l}, \end{aligned}$$

(2.2)

where ${{\mathbb {T}}}$ is the unit circle $\{ z \in {{\mathbb {C}}}\mid |z|=1 \}$ traversed in the positive direction and $t_i \ (i=1,\dots , l)$ are complex numbers satisfying the following conditions:

$$\begin{aligned} |p|&<|at_1\cdots t_i|<1, \\ |q|^{k_i}&<|t_i^{-1}|<1 \ \ (k_i \ge 1). \end{aligned}$$

We put $k:=\min \{k_i \mid k_i \ge 1\}$. Then the function $L_{\varvec{k}}(a,\alpha ,\beta ;p,q)$ is holomorphic on the annulus $|pq^k|<|a|<1$ as a function of the variable a.

Remark 2.2

• (1) When $\beta =1$, the functions $L_{\varvec{k}}(a,\alpha ,1;p,q)$ become elliptic q-multiple polylogarithms introduced and studied in [11, 13].

• (2) When $k_1=0$, the function $L_{(0,k_2,\ldots .k_l)}(a,\alpha ,\beta ;p,q)$ is constant as a function of a. This follows from a similar argument to that given in Remark 2.1 of [11].

Let $T_a$ be the q-shift operator defined by

$$\begin{aligned}T_a f(a)=f(qa), \end{aligned}$$

where f(a) is a function of the variable a. The functions $L_{\varvec{k}}(a, \alpha , \beta ;p,q)$ satisfy the following q-difference relations:

Proposition 2.3

Let ${\varvec{k}}=(k_1,\ldots ,k_l) \in {{\mathbb {Z}}}_{\ge 0}^l$.

(a):

For $k_1 \ge 2$,

$$\begin{aligned} (1-\beta T_a)L_{\varvec{k}}(a, \alpha , \beta ;p,q)=L_{(k_1-1,k_2,\ldots ,k_l)} (a,\alpha , \beta ;p,q) . \end{aligned}$$

(b):

For $k_1=1$,

$$\begin{aligned} (1-\beta T_a)L_{\varvec{k}}(a, \alpha , \beta ;p,q)&=F(a,\alpha ;p)L_{(k_2,\ldots ,k_l)}(a,\alpha ,\beta ;p,q)\\&\quad +L_{(0,k_2,\ldots ,k_l)}(a,\alpha ,\beta ;p,q), \end{aligned}$$

where we put

$$\begin{aligned} L_{(k_2,\ldots ,k_l)}(a,\alpha ,\beta ;p,q)=1 \end{aligned}$$

for $l=1$.

Proof

The proof is similar to that of Proposition 2.4 of [11]. $\square$

2.3 Trigonometric Limit of $L_{\varvec{k}}(a,\alpha ,\beta ;p,q)$

Hereafter we put

$$\begin{aligned} c_0(a)=\frac{1}{a-1}, \ c_1(a)=\frac{a}{1-a}. \end{aligned}$$

We define the depth of ${\varvec{k}}=(k_i)_{i=1}^l \in {{\mathbb {Z}}}_{ > 0}^l$ by $\textrm{dep}\, {\varvec{k}}=l$.

For ${\varvec{k}} \in {{\mathbb {Z}}}_{>0}^l$, we define the functions $L_{\varvec{k}}(a,\beta ;q)$ and $L_{\varvec{k}}(a,\alpha ,\beta ;q)$ by

$$\begin{aligned} L_{\varvec{k}}(a,\beta ;q):=&\sum _{m_1,\ldots ,m_l \ge 1} \prod _{i=1}^l \frac{a^{m_i}}{(1-q^{m_i+\cdots +m_l}\beta )^{k_i}}, \nonumber \\ L_{\varvec{k}}(a,\alpha ,\beta ;q):=&(-1)^l \sum _{\widetilde{\varvec{k}}=(k_1 \Box \cdots \Box k_l)}L_{\widetilde{\varvec{k}}}(a,\beta ;q)(-c_0(\alpha ))^{l-\textrm{dep} \widetilde{\varvec{k}}}, \end{aligned}$$

(2.3)

where the sum on the right-hand side of (2.3) runs over all $\widetilde{\varvec{k}}$ of the form $\widetilde{\varvec{k}}=(k_1 \Box \cdots \Box k_l)$ in which each $\Box$ filled by two candidates “,” or “$+$”. Then we have

$$\begin{aligned} \lim _{q \rightarrow 1} (-1)^l (1-q)^{k_1+\cdots +k_l} L_{\varvec{k}}(a,\alpha ,q^{-1+x};q) =L_{\varvec{k}}(a,\alpha ,x), \end{aligned}$$

(2.4)

where $L_{\varvec{k}}(a,\alpha ,x)$ is the interpolated multiple Hurwitz polylogarithm (1.1).

The trigonometric limit (the limit as $p \rightarrow 0$) of $L_{\varvec{k}}(a,\alpha ,\beta ;p,q)$ is given by $L_{\varvec{k}}(a,\alpha ,\beta ;q)$, as follows:

Proposition 2.4

For ${\varvec{k}} \in {{\mathbb {Z}}}_{>0}^l$, we have

$$\begin{aligned} \lim _{p \rightarrow 0}L_{\varvec{k}}(a,\alpha ,\beta ;p,q)=L_{\varvec{k}}(a,\alpha ,\beta ;q). \end{aligned}$$

Proof

The proof is similar to that of Proposition 2.2 of [11]. $\square$

2.4 Classical Limit of $L_{\varvec{k}}(a,\alpha ,\beta ;p,q)$

We put $p=e^{2\pi \sqrt{-1} \tau }$ and denote the Kronecker function $F(e^{2\pi \sqrt{-1} u}, e^{2\pi \sqrt{-1}v};p)$ by $F(u,v;\tau )$. For ${\varvec{k}}=(k_i)_{i=1}^l \in {{\mathbb {Z}}}_{\ge 0}^l$ and $u, v \in {{\mathbb {C}}}$ with $0<\textrm{Im}\, u<\textrm{Im} \, \tau$, we define the function $L_{\varvec{k}}(u,v,x;\tau )$ by

$$\begin{aligned} L_{\varvec{k}}(u,v,x;\tau ):=(-1)^l \int _{[0, 1]^l} \prod _{i=1}^l F\left( u+\sum _{j=1}^i s_j,v;\tau \right) {\varvec{B}}_{k_i}(-s_i,x) \, ds_1 \cdots ds_l. \end{aligned}$$

Then, by Proposition 2.1, the following proposition holds:

Proposition 2.5

$$\begin{aligned} \lim _{q \rightarrow 1} (1-q)^{k_1+\cdots +k_l} L_{\varvec{k}}\left( e^{2\pi \sqrt{-1}u},e^{2\pi \sqrt{-1}v},q^x;p,q\right) =(-1)^l L_{\varvec{k}}(u,v,x;\tau ). \end{aligned}$$

Remark 2.6

When $x=0$, the functions $(-1)^l L_{\varvec{k}}(u,v,0;\tau )$ are multiple elliptic polylogarithms introduced in [13].

3 Generating Function of Elliptic q-Multiple Polylogarithms of Hurwitz Type

In this section, we introduce generating functions of $L_{\varvec{k}}(a,\alpha ,\beta ;p,q)$ and derive q-difference equations satisfied by them.

3.1 Sums of Fixed Weight, Depth, and i-Heights

For an index ${\varvec{k}}=(k_1,\dots ,k_l) \in {{\mathbb {Z}}}_{> 0}^l$, we define the weight and i-height of ${\varvec{k}}$, respectively by

$$\begin{aligned} |{\varvec{k}}|:=k_1+\cdots +k_l, \ i{-}\textrm{ht}({\varvec{k}}):=\#\{j \mid k_j \ge i+1\}. \end{aligned}$$

For nonnegative integers $k, l, h_1, \dots , h_r$ and $-2 \le j \le r-1$, we define the set of indices $I_j(k,l,h_1,\dots ,h_r)$ by

$$\begin{aligned}&I_j(k,l,h_1,\dots ,h_r)\\&:=\left\{ {\varvec{k}}=(k_i)_{i=1}^l \in {{\mathbb {Z}}}_{>0}^l \mid |{\varvec{k}}|=k, \ i{-}\textrm{ht}({\varvec{k}})=h_i \, (i=1,\dots ,r), \, k_1 \ge j+2 \right\} \end{aligned}$$

for $j \ge -1$ and

$$\begin{aligned} I_{-2}(k,l,h_1,\dots ,h_r):=\{{\varvec{k}}=(0,k_2,\dots ,k_l) \mid (k_2,\dots , k_l) \in I_{-1}(k,l-1,h_1,\dots ,h_r)\} . \end{aligned}$$

for $j=-2$. We define the sum $G_j(k,l,h_1,\dots ,h_r;a;p,q)$ by

$$\begin{aligned} G_j(k,l,h_1,\dots ,h_r;a,\alpha ,\beta ;p,q):=\sum _{{\varvec{k}} \in I_j(k,l,h_1,\dots ,h_r)}L_{\varvec{k}}(a,\alpha ,\beta ;p,q). \end{aligned}$$

Since $G_{-2}(k,l,h_1,\dots ,h_r;a,\alpha ,\beta ;p,q)$ is constant as a function of the variable a, we omit the variable a to denote it:

$$\begin{aligned} G_{-2}(k,l,h_1,\dots ,h_r;\alpha ,\beta ;p,q)&:=G_{-2}(k,l,h_1,\dots ,h_r ;a,\alpha ,\beta ;p,q). \end{aligned}$$

For $I_j(k,l,h_1,\dots ,h_r) =\emptyset$, we put

$$\begin{aligned}&G_j(k,l, h_1,\dots , h_r;a,\alpha ,\beta ;p,q)\\&:= {\left\{ \begin{array}{ll} 1 &{} \hbox {} (k,l,h_1,\dots ,h_r)=(0,0,0,\dots ,0) \, \hbox {and} \, j=-1 \\ 0 &{} \text{ otherwise. } \end{array}\right. } \end{aligned}$$

The functions $G_j(k,l,h_1,\dots ,h_r;a;p,q)$ satisfy the following q-difference relations:

Proposition 3.1

(a):

$$\begin{aligned}{} & {} (1-\beta T_a)G_{r-1}(k,l,h_1,\dots ,h_r;a,\alpha ,\beta ;p,q)\\{} & {} =G_{r-1}(k-1,l,h_1,\dots ,h_r;a,\alpha ,\beta ;p,q )\\{} & {} \quad +G_{r-2}(k-1,l,h_1,\dots ,h_{r-1},h_r-1;a,\alpha ,\beta ;p,q) \\{} & {} \quad -G_{r-1}(k-1,l,h_1,\dots ,h_{r-1},h_r-1;a,\alpha ,\beta ;p,q). \end{aligned}$$

(b):

For $0 \le j \le r-2$,

$$\begin{aligned}{} & {} (1-\beta T_a)(G_j(k,l,h_1,\dots ,h_r;a,\alpha ,\beta ;p,q)-G_{j+1}(k,l,h_1,\dots ,h_r;a,\alpha ,\beta ;p,q))\\{} & {} \quad =G_{j-1}(k-1,l,h_1,\dots ,h_j, h_{j+1}-1,h_{j+2},\dots ,h_r;a,\alpha ,\beta ;p,q)\\{} & {} \qquad -G_{j}(k-1,l,h_1,\dots ,h_j,h_{j+1}-1,h_{j+2},\dots ,h_r;a,\alpha ,\beta ;p,q). \end{aligned}$$

(c):

$$\begin{aligned}{} & {} (1-\beta T_a)(G_{-1}(k,l,h_1,\dots ,h_r;a,\alpha ,\beta ;p,q)-G_0(k,l,h_1,\dots ,h_r;a,\alpha ,\beta ;p,q)) \\{} & {} \quad =F(a,\alpha ;p)G_{-1}(k-1,l-1,h_1,\dots ,h_r;a,\alpha ,\beta ;p,q)\\{} & {} \qquad +G_{-2}(k-1,l,h_1,\dots ,h_r;\alpha ,\beta ;p,q). \end{aligned}$$

Proof

The proof of this proposition is similar to that of Lemma 6 of [16]. Thus we omit the detailed proof. $\square$

3.2 Generating Functions of $G_j$

We now define the generating functions $\Phi _j(a;p,q)$ of $G_j$ by

$$\begin{aligned}{} & {} \Phi _j(a;p,q)=\Phi _j(x_1,\ldots ,x_{r+2};a,\alpha ,\beta ;p,q)\\{} & {} =\sum _{k,l,h_1,\dots ,h_r \ge 0} G_j(k,l,h_1,\dots ,h_r;a,\alpha ,\beta ;p,q)\\{} & {} \quad \times x_1^{k-l-\sum _{i=1}^r h_i} x_2^{l-h_1}x_3^{h_1-h_2}\cdots x_{r+1}^{h_{r-1}-h_r}x_{r+2}^{h_r}. \end{aligned}$$

We also write $\Phi _{-2}(a;p,q)$ as $\Phi _{-2}(p,q)$. Note that $\Phi _{-2}(p,q)$ is constant as a function of the variable a.

The functions $\Phi _j(a;p,q)$ satisfy the following q-difference system:

Proposition 3.2

(a):

$$\begin{aligned}&(1-\beta T_a)\Phi _{r-1}(a;p,q)=x_1 \Phi _{r-1}(a;p,q)\\&+\frac{x_{r+2}}{x_{r+1}}(\Phi _{r-2}(a;p,q)-\Phi _{r-1}(a;p,q)-\delta _{r,1}), \end{aligned}$$

where $\delta$ stands for Kronecker’s delta.

(b):

For $0 \le j \le r-2$,

$$\begin{aligned}&(1-\beta T_a)(\Phi _j(a;p,q)-\Phi _{j+1}(a;p,q))\\&=\frac{x_{j+3}}{x_{j+2}}(\Phi _{j-1}(a;p,q)-\Phi _j(a;p,q)-\delta _{j,0}). \end{aligned}$$

(c):

$$\begin{aligned} (1-\beta T_a)(\Phi _{-1}(a;p,q)-\Phi _0(a;p,q))&=1-\beta +x_2F(a,\alpha ;p)\Phi _{-1}(a;p,q)\\&\,\,\quad +x_1\Phi _{-2}(p,q). \end{aligned}$$

Proof

The proof is similar to that of Proposition 7 of [16]. $\square$

We will now consider solving the q-difference system in Proposition 3.2 in a different way. We observe the following proposition:

Proposition 3.3

• (1) The functions $\Phi _j(a;p,q) (-1 \le j \le r-1)$ can be written as a linear combination of $(1-\beta T_a)^i \Phi _{r-1}(a;p,q) \ (i=0,\ldots ,r)$ as follows:

  $$\begin{aligned}&\Phi _j(a;p,q)=\frac{1}{x_{r+2}} \left\{ \sum _{i=0}^{r-2-j} (x_{r+2-i}-x_1 x_{r+1-i})(1-\beta T_a)^i\right. \\&\qquad \qquad \qquad \qquad \left. +x_{j+3} (1-\beta T_a)^{r-1-j} \right\} \Phi _{r-1} (a;p,q)+\delta _{j,-1}. \end{aligned}$$

• (2) The function $\Phi _{r-1}(a;p,q)$ satisfies the following higher order q-difference equation:

  $$\begin{aligned}{} & {} \{ (1-\beta T_a)^{r+1}-(x_1+x_2F(a,\alpha ;p))(1-\beta T_a)^r \nonumber \\{} & {} \qquad -F(a,\alpha ;p)\sum _{j=0}^{r-1}(x_{r+2-j}-x_1x_{r+1-j})(1-\beta T_a)^j \} \Phi _{r-1}(a;p,q)\nonumber \\{} & {} \quad =x_{r+2}F(a,\alpha ;p)+\frac{x_1x_{r+2}}{x_2}\Phi _{-2}(p,q). \end{aligned}$$

  (3.1)

Proof

We put $y_0=\Phi _{r-1}(a;p,q), \ y_j=\Phi _{r-j-1}(a;p,q)-\Phi _{r-j}(a;p,q) \ (1 \le j \le r)$. Then Proposition 3.2 implies

$$\begin{aligned} y_j=\frac{x_{r+2-j}}{x_{r+2}} \left\{ (1-\beta T_a)^j-x_1 (1-\beta T_a)^{j-1} \right\} y_0+ \delta _{j,r} \ (1 \le j \le r). \end{aligned}$$

(3.2)

By substituting the expressions (3.2) into

$$\begin{aligned} \Phi _j(a;p,q)=&y_0+\dots +y_{r-1-j}, \end{aligned}$$

we obtain the claim (1). The proof of the claim (2) is similar to that of Corollary 8 (iii) of [16]. $\square$

Proposition 3.3 implies solving the q-difference system reduces to the Eq. (3.1).

4 Solving the q-Difference Equation (3.1)

In this section, we will solve the q-difference equation (3.1) based on a perturbation approach and the variation of parameters, which are also used in [11]. We first expand $\Phi _{r-1}(a;p,q)$, $\Phi _{-2}(p,q)$ and $F(a,\alpha ;p)$ into power series of the parameter p in (3.1). We define the coefficients $\phi _{0,n}(a;q), \ \phi _{-2,n}(q)$ of $\Phi _{r-1}(a;p,q), \ \Phi _{-2}(p,q)$, respectively by

$$\begin{aligned} \Phi _{r-1}(a;p,q)&=\sum _{n=0}^{\infty } \phi _{r-1,n}(a;q)p^n, \end{aligned}$$

(4.1)

$$\begin{aligned} \Phi _{-2}(p,q)&=\sum _{n=0}^{\infty } \phi _{-2,n}(q) p^n. \end{aligned}$$

(4.2)

The coefficients of $F(a,\alpha ;p)$ are given by the following:

Lemma 4.1

We define the Laurent polynomials $s_n(a,\alpha )$ by

$$\begin{aligned} s_n(a,\alpha )={\left\{ \begin{array}{ll} -c_1(a)+c_0(\alpha ) &{} (n=0) \\ -\sum _{d \mid n} (a^d \alpha ^{n/d}-a^{-d}\alpha ^{-n/d}) &{} (n \ge 1), \end{array}\right. } \end{aligned}$$

where $\sum _{d \mid n}$ denotes the sum running over all positive divisors of n. Then, for $|p|<|a|<1$ and $|p|<|\alpha |<1$, we have

$$\begin{aligned} F(a,\alpha ;p)=\sum _{n=0}^{\infty }s_n(a,\alpha )p^n. \end{aligned}$$

(4.3)

Proof

See Proposition 3.2 of [11]. $\square$

By (4.1), (4.2) and (4.3), comparing the coefficient of $p^n$ on both sides of (3.1) yields the following non-homogeneous q-difference equation:

$$\begin{aligned}{} & {} \bigg \{(1-\beta T_a)^{r+1}-(x_1+x_2 s_0(a,\alpha ))(1- \beta T_a)^r \nonumber \\{} & {} -s_0(a,\alpha )\sum _{j=0}^{r-1}(x_{r+2-j}-x_1 x_{r+1-j})(1-\beta T_a)^j \bigg \} \phi _{r-1,n}(a;q)=r_n(a), \end{aligned}$$

(4.4)

where we put

$$\begin{aligned} r_n (a){} & {} :=\sum _{k=0}^{n-1}s_{n-k}(a,\alpha )\left\{ x_2(1-\beta T_a)^r+ \sum _{j=0}^{r-1}(x_{r+2-j}-x_1 x_{r+1-j}) (1-\beta T_a)^j \right\} \phi _{r-1,k}(a;q) \\{} & {} +x_{r+2}s_n(a,\alpha )+\frac{x_1x_{r+2}}{x_2} \phi _{-2,n}(q). \end{aligned}$$

We will solve the equation (4.4) by using the method of variation of parameters. To do this, we introduce the q-hypergeometric function ${}_{r+1}\phi _r$. For $n \in {{\mathbb {Z}}}_{\ge 0}$, we define the q-shifted factorial $(a;q)_n$ by

$$\begin{aligned} (a;q)_n:&=\frac{(a;q)_{\infty }}{(aq^n;q)_{\infty }} \end{aligned}$$

and denote the product $(a_1;q)_n (a_2;q)_n \cdots (a_{m};q)_n$ by $(a_1,a_2,\dots ,a_m;q)_n$. The q-hypergeometric function ${}_{r+1}\phi _r \begin{bmatrix} \begin{matrix} a_1,\dots , a_{r+1} b_1, \dots , b_r \end{matrix} ;q,z \end{bmatrix}$ is defined by

$$\begin{aligned} {}_{r+1}\phi _r \begin{bmatrix} \begin{matrix} a_1,\dots , a_{r+1} \\ b_1, \dots , b_r \end{matrix} ;q,z \end{bmatrix}:=\sum _{n=0}^{\infty } \frac{(a_1,\dots ,a_{r+1};q)_n}{(q,b_1,\dots ,b_r;q)_n}z^n. \end{aligned}$$

In the following, we also use the notation

$$\begin{aligned} {}_{r+1} {\hat{\phi }}_r \begin{bmatrix} \begin{matrix} a_1,\dots ,a_{r+1} \\ b_1,\dots , b_{r+1} \end{matrix} ;q,z \end{bmatrix}={}_{r+1} \phi _r \begin{bmatrix} \begin{matrix} a_1,\dots ,a_{r+1} \\ b_1,\dots , {\widehat{q}}, \dots , b_{r+1} \end{matrix} ;q,z \end{bmatrix} \end{aligned}$$

when one of the $b_1,\dots ,b_{r+1}$ is q.

The homogeneous q-difference equation corresponding to (4.4) can be solved by using the function ${}_{r+1}\phi _r$, as follows:

Proposition 4.2

The solutions of the homogeneous q-difference equation

$$\begin{aligned}{} & {} \Biggl \{ (1-\beta T_a)^{r+1}-(x_1+x_2s_0(a,\alpha ))(1-\beta T_a)^r \nonumber \\{} & {} -s_0(a,\alpha )\sum _{j=0}^{r-1}(x_{r+2-j}-x_1x_{r+1-j})(1-\beta T_a)^j \Biggr \}f(a)=0 \end{aligned}$$

(4.5)

which have the form of

$$\begin{aligned} f(a)=a^{\lambda }\sum _{n=0}^{\infty }w_n a^n, \ w_0=1, \end{aligned}$$

are given by the following:

$$\begin{aligned} f_i(a)=a^{\lambda _i} {}_{r+1}{\hat{\phi }}_r \begin{bmatrix} \begin{matrix} q^{\lambda _i}\rho _1,\dots , q^{\lambda _i}\rho _{r+1} \\ q^{\lambda _i-\lambda _1+1},\dots , q^{\lambda _i-\lambda _{r+1}+1} \end{matrix} ;q, wa \end{bmatrix} \ \ (i=1,\dots , r+1), \end{aligned}$$

where we put

$$\begin{aligned} w=\frac{1-(x_1+x_2(1+c_0(\alpha ))-(1+c_0(\alpha ))\sum _{j=0}^{r-1}(x_{r+2-j}-x_1x_{r+1-j}) }{1-(x_1+x_2 c_0(\alpha ))-c_0(\alpha ) \sum _{j=0}^{r-1}(x_{r+2-j}-x_1 x_{r+1-j})}. \end{aligned}$$

Furthermore, $\lambda =\lambda _i \ (i=1, \dots , r+1)$ are the solutions of the equation

$$\begin{aligned}&(1-\beta q^{\lambda })^{r+1}-(x_1+x_2c_0(\alpha ))(1-\beta q^{\lambda })^r\\ {}&-c_0(\alpha ) \sum _{j=0}^{r-1}(x_{r+2-j}-x_1x_{r+1-j})(1-\beta q^{\lambda })^j=0 \end{aligned}$$

and $\rho _i \ (i=1,\dots , r+1)$ are the parameters determined by the following:

$$\begin{aligned}{} & {} (1-\beta t)^{r+1}-(x_1+x_2(1+c_0(\alpha )))(1-\beta t)^r\\{} & {} \qquad -(1+c_0(\alpha ))\sum _{j=0}^{r-1}(x_{r+2-j}-x_1x_{r+1-j})(1-\beta t)^j \\{} & {} \quad =\left( 1-(x_1+x_2(1+c_0(\alpha )) ) -(1+c_0(\alpha ))\sum _{j=0}^{r-1} (x_{r+2-j}-x_1 x_{r+1-j}) \right) \\{} & {} \quad \quad \times (1-t\rho _1)\dots (1-t\rho _{r+1}). \end{aligned}$$

Proof

We put

$$\begin{aligned} P(t)=&(1-\beta t)^{r+1}-(x_1+x_2c_0(\alpha ))(1-\beta t)^r\\&-c_0(\alpha ) \sum _{j=0}^{r-1}(x_{r+2-j}-x_1x_{r+1-j})(1-\beta t)^j, \\ Q(t)=&(1-\beta t)^{r+1}-(x_1+x_2(1+c_0(\alpha )))(1-\beta t)^r\\&-(1+c_0(\alpha )) \sum _{j=0}^{r-1}(x_{r+2-j}-x_1x_{r+1-j})(1-\beta t)^j. \end{aligned}$$

By putting $f(a)=a^{\lambda }\sum _{n=0}^{\infty } w_n a^n, \ w_0=1$ in (4.5) and then comparing the coefficients gives

$$\begin{aligned} P(q^{\lambda })=0 \end{aligned}$$

and

$$\begin{aligned} w_n=\frac{Q(q^{\lambda +n-1})}{P(q^{\lambda +n})} w_{n-1} \end{aligned}$$

for $n \ge 1$. Since

$$\begin{aligned} P(t)=&\left( 1-(x_1+x_2 c_0(\alpha )) -c_0(\alpha )\sum _{j=0}^{r-1} (x_{r+2-j}-x_1 x_{r+1-j})\right) \prod _{i=1}^{r+1}(1-tq^{-\lambda _i}), \\ Q(t)=&\left( 1-(x_1+x_2(1+c_0(\alpha )))-(1+c_0(\alpha ))\sum _{j=0}^{r-1}(x_{r+2-j}-x_1x_{r+1-j}) \right) \prod _{i=1}^{r+1}(1-t\rho _i), \end{aligned}$$

we have

$$\begin{aligned} w_n=&w^n\prod _{k=1}^n \frac{(1-q^{\lambda _i+k-1}\rho _1)\dots (1-q^{\lambda _i+k-1}\rho _{r+1})}{(1-q^{\lambda _i-\lambda _1+k})\dots (1-q^{\lambda _i-\lambda _{r+1}+k})} \\ =&w^n \frac{(q^{\lambda _i}\rho _1,\dots , q^{\lambda _i}\rho _{r+1};q)_n}{(q^{\lambda _i-\lambda _1+1},\dots , q^{\lambda _i-\lambda _{r+1}+1};q)_n} \end{aligned}$$

for $\lambda =\lambda _i$ and thus obtain the proposition. $\square$

We now assume that the vector

$$\begin{aligned} \textbf{u}(a)=\begin{pmatrix} \phi _{r-1,n}(a;q) \\ (1-\beta T_a)\phi _{r-1,n}(a;q) \\ \vdots \\ (1-\beta T_a)^r \phi _{r-1,n}(a;q) \end{pmatrix} \end{aligned}$$

is expressed, in terms of the vector

$$\begin{aligned} \textbf{v}(a)=\begin{pmatrix} c_{n,1}(a) \\ \vdots \\ c_{n,r+1}(a) \end{pmatrix} \end{aligned}$$

and the matrix

$$\begin{aligned} \Psi (a)=\begin{pmatrix} f_1(a) &{} \cdots &{} f_{r+1}(a) \\ (1-\beta T_a) f_1(a) &{} \cdots &{} (1-\beta T_a) f_{r+1}(a) \\ \vdots &{} \cdots &{} \vdots \\ (1-\beta T_a)^r f_1(a) &{} \cdots &{} (1-\beta T_a)^r f_{r+1}(a) \end{pmatrix}, \end{aligned}$$

as follows:

$$\begin{aligned} \textbf{u}(a)=\Psi (a) \textbf{v}(a) . \end{aligned}$$

(4.6)

Then we have the following:

Proposition 4.3

For $i=1,\dots , r+1$, $c_{n,i}(a)$ satisfies the following q-difference equation:

$$\begin{aligned}&(1-T_a)c_{n, i}(a)\nonumber \\&\quad =\frac{(-1)^r \beta ^{-r-1} q^{-\lambda _i}}{\prod _{ \begin{array}{c} 1 \le k \le r+1 \\ k \ne i \end{array}} (q^{\lambda _i}-q^{\lambda _k})} (1-a)a^{-\lambda _i} {}_{r+1}{\hat{\phi }}_r\begin{bmatrix} \begin{matrix} q^{1-\lambda _i} \rho _1^{-1} ,\dots , q^{1-\lambda _i}\rho _{r+1}^{-1} \\ q^{\lambda _1-\lambda _i+1}, \dots , q^{\lambda _{r+1}-\lambda _i+1} \end{matrix} ;q,a \end{bmatrix} r_n(a). \end{aligned}$$

(4.7)

In the following, we put

$$\begin{aligned} {\varvec{\lambda }}=(\lambda _1,\dots , \lambda _{r+1}), \ |{\varvec{\lambda }}|=\lambda _1+\dots +\lambda _{r+1}. \end{aligned}$$

Proposition 4.3 will be deduced from the following two lemmas:

Lemma 4.4

The determinant $W(a) =\begin{vmatrix} \Psi (a) \end{vmatrix}$ can be expressed as follows:

$$\begin{aligned} W(a)=\beta ^{\frac{1}{2}r (r+1)}\prod _{1\le k<l \le r+1}(q^{\lambda _k}-q^{\lambda _l}) \, a^{|{\varvec{\lambda }}|} \frac{(a;q)_{\infty }}{(wa;q)_{\infty }}. \end{aligned}$$

Lemma 4.5

Let $D_i(a)$ be the determinant of order r obtained by removing the r-th row and i-th column of $\Psi (a)$:

$$\begin{aligned} D_i(a):=\begin{vmatrix} f_1(a)&\cdots&f_{i-1}(a)&f_{i+1}(a)&\cdots&f_{r+1}(a) \\ \vdots{} & {} \vdots&\vdots{} & {} \vdots \\ (1-\beta T_a)^{r-1}f_1(a)&\cdots&(1-\beta T_a)^{r-1} f_{i-1}(a)&(1-\beta T_a)^{r-1} f_{i+1}(a)&\cdots&(1-\beta T_a)^{r-1}f_{r+1}(a) \end{vmatrix}. \end{aligned}$$

Then $D_i(a)$ can be expressed as follows:

$$\begin{aligned} D_i(a)&=\prod _{ \begin{array}{c} 1 \le k <l \le r+1\\ k,l \ne i \end{array} } (q^{\lambda _k}-q^{\lambda _l}) \beta ^{\frac{1}{2} r(r-1)}a^{|{\varvec{\lambda }}|-\lambda _i} \frac{(q^{-1}a;q)_{\infty }}{(wa;q)_{\infty }}\\&\quad \times {}_{r+1}{\hat{\phi }}_r \begin{bmatrix} \begin{matrix} q^{1-\lambda _i} \rho _1^{-1}, \dots , q^{1-\lambda _i} \rho _{r+1}^{-1}\\ q^{\lambda _1-\lambda _i+1}, \dots , q^{\lambda _{r+1}-\lambda _i+1} \end{matrix} ;q, q^{-1}a \end{bmatrix}. \end{aligned}$$

Lemmas 4.4 and 4.5 can be proved by using the following Kajihara’s transformation formula for multiple basic hypergeometric series [9] (see also [10]):

Proposition 4.6

(Theorem 1.1 of [9]) For

$$\begin{aligned} {\varvec{x}}=(x_1,\dots , x_m), \ {\varvec{a}}=(a_1,\dots ,a_m), \\ {\varvec{y}}=(y_1,\dots ,y_n), \ {\varvec{b}}=(b_1,\dots ,b_n), \end{aligned}$$

we have

$$\begin{aligned}{} & {} \sum _{{\varvec{\mu }} \in {{\mathbb {Z}}_{\ge 0}^m}} z^{|{\varvec{\mu }}|} \frac{\Delta ({\varvec{x}}q^{\varvec{\mu }})}{\Delta ({\varvec{x}})} \prod _{1 \le i, j \le m} \frac{(a_j x_i/x_j;q)_{\mu _i}}{(qx_i/x_j;q)_{\mu _i}} \prod _{ \begin{array}{c} 1 \le i \le m \\ 1 \le l \le n \end{array} } \frac{(x_iy_l/b_l;q)_{\mu _i}}{(x_iy_l;q)_{\mu _i}} \\{} & {} \quad =\frac{(a_1\dots a_m b_1^{-1} \cdots b_n^{-1}z;q)_{\infty }}{(z;q)_{\infty }} \sum _{{\varvec{\nu }} \in {{\mathbb {Z}}_{\ge 0}^n}} \left( \frac{a_1 \cdots a_m}{b_1\cdots b_n} z \right) ^{|{\varvec{\nu }} |} \frac{\Delta ({\varvec{y}}q^{{\varvec{\nu }}})}{\Delta ({\varvec{y}})} \\{} & {} \qquad \times \prod _{ 1 \le k, l \le n } \frac{(b_ly_k/y_l;q)_{\nu _k}}{(qy_k/y_l;q)_{\nu _k}} \prod _{ \begin{array}{c} 1 \le k \le n \\ 1 \le j \le m \end{array} } \frac{(y_k x_j/a_j;q)_{\nu _k}}{(y_k x_j;q)_{\nu _k}}, \end{aligned}$$

where $\Delta ({\varvec{x}})=\Delta (x_1,\dots ,x_m)$ denotes the product of differences of $x_1,\dots x_m$:

$$\begin{aligned} \Delta ({\varvec{x}}):=\prod _{1\le i<j \le m}(x_i-x_j). \end{aligned}$$

Proof of Lemma 4.4

The determinant W(a) can be written as follows:

$$\begin{aligned} W(a)=&(-\beta )^{\frac{1}{2}r(r+1)} \det (f_j(q^{i-1}a))_{i,j=1}^{r+1} \\ =&(-\beta )^{\frac{1}{2}r(r+1)} \sum _{\sigma \in \mathfrak {S}_{r+1}} \textrm{sgn} (\sigma ) f_1(q^{\sigma (1)-1}a)\cdots f_{r+1}(q^{\sigma (r+1)-1}a) \\ =&(-\beta )^{\frac{1}{2}r(r+1)} a^{|{\varvec{\lambda }}|} \sum _{{\varvec{\mu }}=(\mu _1,\dots ,\mu _{r+1}) \in {{\mathbb {Z}}_{\ge 0}^{r+1}}} (wa)^{|{\varvec{\mu }}|}\prod _{1 \le i,j \le r+1}\frac{(q^{\lambda _j }\rho _j q^{\lambda _i-\lambda _j};q)_{\mu _i}}{(q^{1+\lambda _i-\lambda _j};q)_{\mu _i}}\\&\times \sum _{\sigma \in \mathfrak {S}_{r+1}} \textrm{sgn}(\sigma ) \prod _{j=1}^{r+1} (q^{\lambda _j+\mu _j})^{\sigma (j)-1} \\ =&(-\beta )^{\frac{1}{2}r(r+1)} a^{|{\varvec{\lambda }}|}\sum _{{\varvec{\mu }} \in {{\mathbb {Z}}_{\ge 0}^{r+1}}}(wa)^{|{\varvec{\mu }}|} \prod _{1 \le i,j \le r+1} \frac{(q^{\lambda _j}\rho _j q^{\lambda _i-\lambda _j};q)_{\mu _i}}{(q^{1+\lambda _i-\lambda _j};q)_{\mu _i}} \det ((q^{\lambda _j+\mu _j} )^{i-1})_{i,j=1}^{r+1} \\ =&\beta ^{\frac{1}{2}r(r+1)}a^{|{\varvec{\lambda }}|}\sum _{{\varvec{\mu }} \in {{\mathbb {Z}}_{\ge 0}^{r+1}}} (wa)^{|{\varvec{\mu }}|} \Delta (q^{\lambda _1+\mu _1},\dots ,q^{\lambda _{r+1}+\mu _{r+1}})\\&\times \prod _{1\le i,j \le r+1}\frac{(q^{\lambda _j}\rho _j q^{\lambda _i-\lambda _j};q)_{\mu _i}}{(q^{1+\lambda _i-\lambda _j};q)_{\mu _i}}. \end{aligned}$$

Thus, by putting

$$\begin{aligned} m=r+1, \ n=0, \ z=wa, \ {\varvec{a}}=(q^{\lambda _1}\rho _1,\dots ,q^{\lambda _{r+1}}\rho _{r+1}), \ {\varvec{x}}=(q^{\lambda _1},\dots ,q^{\lambda _{r+1}}) \end{aligned}$$

in Proposition 4.6, we obtain the lemma. $\square$

Remark 4.7

In the proof of Lemma 4.4, we have used the case $n=0$ of Proposition 4.6. This special case is due to Milne [18].

Proof of Lemma 4.5

We define the parameters $z_1,\dots , z_r$ and ${\bar{\rho }}_1, \dots , {\bar{\rho }}_r$ by

$$\begin{aligned} z_j= {\left\{ \begin{array}{ll} q^{\lambda _j} &{} j=1,\dots , i-1 \\ q^{\lambda _{j+1}} &{} j=i,\dots , r, \end{array}\right. } \ \ {\bar{\rho }}_j={\left\{ \begin{array}{ll} \rho _j &{} j=1,\dots , i-1 \\ \rho _{j+1} &{} j=i,\dots , r \end{array}\right. } \end{aligned}$$

and the functions $g_j(a)$ by

$$\begin{aligned} g_j(a)= {\left\{ \begin{array}{ll} a^{-\lambda _j} f_j(a) &{} j=1,\dots , i-1 \\ a^{-\lambda _{j+1}} f_{j+1}(a) &{} j=i,\dots , r. \end{array}\right. } \end{aligned}$$

Then we have

$$\begin{aligned} D_i(a)&=(-\beta )^{\frac{1}{2}r(r-1)} a^{|{\varvec{\lambda }}|-\lambda _i} \det \begin{pmatrix} z_l^{k-1}g_l(q^{k-1}a) \end{pmatrix}_{k,l=1}^r \\&=(-\beta )^{\frac{1}{2}r(r-1)} a^{|{\varvec{\lambda }}|-\lambda _i}\sum _{{\varvec{\mu }}=(\mu _1,\dots ,\mu _r) \in {{\mathbb {Z}}_{\ge 0}^r}} (wa)^{|{\varvec{\mu }}|} \prod _{1 \le k, l \le r} \frac{({\bar{\rho }}_l z_k ;q)_{\mu _k}}{(q z_k/z_l;q)_{\mu _k}}\\&\quad \times \prod _{1\le j \le r} \frac{(z_j \rho _i;q)_{\mu _j}}{(z_j q^{1-\lambda _i};q)_{\mu _j}} \sum _{\sigma \in \mathfrak {S}_r} \textrm{sgn}(\sigma ) \prod _{j=1}^r (z_j q^{\mu _j})^{\sigma (j)-1} \\&= (-\beta )^{\frac{1}{2}r(r-1)} a^{|{\varvec{\lambda }}|-\lambda _i}\sum _{{\varvec{\mu }} \in {{\mathbb {Z}}_{\ge 0}^r}} (wa)^{|{\varvec{\mu }}|} \prod _{1 \le k, l \le r} \frac{({\bar{\rho }}_l z_k;q)_{\mu _k}}{(q z_k/z_l;q)_{\mu _k}} \prod _{1\le j \le r} \frac{(z_j \rho _i;q)_{\mu _j}}{(z_j q^{1-\lambda _i};q)_{\mu _j}} \\&\quad \times \det \begin{pmatrix} (z_l q^{\mu _l})^{k-1} \end{pmatrix}_{k,l=1}^r\\&=\beta ^{\frac{1}{2}r(r-1)} a^{|{\varvec{\lambda }}|-\lambda _i}\sum _{{\varvec{\mu }} \in {{\mathbb {Z}}_{\ge 0}^r}} (wa)^{|{\varvec{\mu }}|} \Delta (z_1q^{\mu _1},\dots , z_r q^{\mu _r})\\&\quad \times \prod _{1 \le k, l \le r} \frac{({\bar{\rho }}_l z_k;q)_{\mu _k}}{(qz_k/z_l;q)_{\mu _k}} \prod _{1\le j \le r} \frac{(z_j \rho _i;q)_{\mu _j}}{(z_j q^{1-\lambda _i};q)_{\mu _j}}. \end{aligned}$$

Thus, by putting

$$\begin{aligned} m=r, \ n=1, \ z=wa, \ a_j={\bar{\rho }}_{j} z_j, \ b_1=\rho _i^{-1} q^{1-\lambda _i}, \ x_j=z_j, \ y_1= q^{1-\lambda _i} \end{aligned}$$

in Proposition 4.6, we have

$$\begin{aligned} D_i(a)&=\Delta (z_1,\ldots ,z_r) \beta ^{\frac{1}{2}r(r-1)}a^{|{\varvec{\lambda }}|-\lambda _i} \frac{(\rho _1 \cdots \rho _{r+1} q^{|{\varvec{\lambda }}|-1 }wa;q)_{\infty }}{(wa;q)_{\infty } } \\&\quad \times \sum _{\nu \in {{\mathbb {Z}}_{\ge 0}}} (\rho _1 \cdots \rho _{r+1}q^{|{\varvec{\lambda }}|-1 } wa)^{\nu } \frac{(q^{1-\lambda _i}\rho _1^{-1},\dots , q^{1-\lambda _i}\rho _{r+1}^{-1};q)_{\nu }}{(q^{\lambda _1-\lambda _i+1}, \dots , q^{\lambda _{r+1}-\lambda _i+1};q)_{\nu }} \\&= \Delta (z_1,\ldots ,z_r) \beta ^{\frac{1}{2}r(r-1)}a^{|{\varvec{\lambda }}|-\lambda _i} \frac{(q^{-1}a;q)_{\infty } }{(wa;q)_{\infty } }\\&\quad \times {}_{r+1}{\hat{\phi }}_r \begin{bmatrix} \begin{matrix} q^{1-\lambda _i} \rho _1^{-1}, \dots , q^{1-\lambda _i} \rho _{r+1}^{-1} \\ q^{\lambda _1-\lambda _i+1}, \dots , q^{\lambda _{r+1}-\lambda _i+1} \end{matrix} ;q, q^{-1}a \end{bmatrix}, \end{aligned}$$

which completes the proof. $\square$

We are now in a position to prove Proposition 4.3.

Proof of Proposition 4.3

We define the vector $\textbf{r}_n(a)$ by

$$\begin{aligned} \textbf{r}_n(a)=\begin{pmatrix} 0 \\ \vdots \\ 0 \\ r_n(a) \end{pmatrix} \end{aligned}$$

and matrix A by

$$\begin{aligned} A&=\begin{pmatrix} 0 &{} 1 &{} 0 &{} \cdots &{} 0 &{} 0 \\ 0 &{} 0 &{} 1 &{} \cdots &{} 0 &{} 0\\ \vdots &{} \vdots &{} \vdots &{} \ddots &{} \vdots &{} \vdots \\ 0 &{} 0 &{} 0 &{} \cdots &{} 0 &{} 1 \\ s_0(a, \alpha )(x_{r+2}-x_1x_{r+1}) &{} \cdots &{} \cdots &{} \cdots &{} s_0(a,\alpha )(x_3-x_1x_2) &{} x_1+x_2s_0(a,\alpha ) \end{pmatrix}. \end{aligned}$$

Then the q-difference equation (4.4) can be represented as

$$\begin{aligned} (1-\beta T_a)\textbf{u}(a)=A\textbf{u}(a)+\textbf{r}_n(a). \end{aligned}$$

(4.8)

Substituting (4.6) into (4.8) gives

$$\begin{aligned} (1-\beta T_a) (\Psi (a) \textbf{v}(a))=A \Psi (a) \textbf{v}(a)+\textbf{r}_n(a). \end{aligned}$$

(4.9)

By observing

$$\begin{aligned} (1-\beta T_a)\Psi (a)=A \Psi (a), \end{aligned}$$

the left-hand side of (4.9) can be written, as follows:

$$\begin{aligned} (1-\beta T_a)(\Psi (a) \textbf{v}(a))=&((1-\beta T_a)\Psi (a)) \textbf{v}(a)+\beta \Psi (qa)(1-T_a)\textbf{v}(a) \\ =&A\Psi (a) \textbf{v}(a) +\beta \Psi (qa) (1-T_a) \textbf{v}(a). \end{aligned}$$

Thus we have

$$\begin{aligned} \beta \Psi (qa)(1-T_a) \textbf{v}(a)=\textbf{r}_n(a). \end{aligned}$$

By Cramer’s rule, it holds that

$$\begin{aligned} (1-T_a) c_{n,i}(a)=(-1)^{i+r+1}\frac{D_i(qa)}{\beta W(qa)}r_n(a). \end{aligned}$$

Thus, by applying Lemmas 4.4 and 4.5, we finish the proof. $\square$

Proposition 4.3 implies that $\Phi _{r-1}(a)$ satisfies the following integral equation:

Proposition 4.8

We put ${\widetilde{F}}(a,\alpha ;p):=F(a,\alpha ;p)-s_0(a,\alpha )$ and define the function K(a, b; q) by

$$\begin{aligned} K(a,b;q)&:=(-\beta )^{-r-1}\sum _{i=1}^{r+1}\frac{q^{-\lambda _i}}{\prod _{ \begin{array}{c} 1 \le k \le r+1 \\ k \ne i \end{array}} (q^{\lambda _i}-q^{\lambda _k})}{}_{r+1}{\hat{\phi }}_r \begin{bmatrix} \begin{matrix} q^{\lambda _i} \rho _1 ,\dots , q^{\lambda _i}\rho _{r+1} \\ q^{\lambda _i-\lambda _1+1}, \dots , q^{\lambda _{i}-\lambda _{r+1}+1} \end{matrix} ;q,wa \end{bmatrix} \\&\quad \times {}_{r+1} {\hat{\phi }}_r \begin{bmatrix} \begin{matrix} q^{1-\lambda _i}\rho _1^{-1},\dots , q^{1-\lambda _i}\rho _{r+1}^{-1} \\ q^{\lambda _1-\lambda _i+1}, \dots , q^{\lambda _{r+1}-\lambda _i+1} \end{matrix} ;q,b \end{bmatrix}(1-b)F(q^{-\lambda _i},ab^{-1};q). \end{aligned}$$

Then, for $|pq|<|a|<1$, we have the following:

$$\begin{aligned} \Phi _{r-1}(a;p,q)&=\frac{1}{2\pi \sqrt{-1}} \int _{{\mathbb {T}}}K(a,tz;q) {\widetilde{F}}(tz,\alpha ;p) \Bigl (x_2 (1-\beta T_z)^r \nonumber \\&\quad + \sum _{j=0}^{r-1} (x_{r+2-j}-x_1 x_{r+1-j}) (1-\beta T_z)^j \Bigr )\Phi _{r-1}(tz;p,q) \ \frac{dz}{z} \nonumber \\&\quad +\frac{1}{2\pi \sqrt{-1}} \int _{{\mathbb {T}}} K(a,tz;q) \left( x_{r+2}F(tz,\alpha ;p)+\frac{x_1x_{r+2}}{x_2} \Phi _{-2}(p,q) \right) \frac{dz}{z} \nonumber \\&\quad (|a|<|t|<|q^{-1}a|, \ \ |p|<|t|<1). \end{aligned}$$

(4.10)

Proof

By definition, the right-hand side of (4.7) can be expressed as

$$\begin{aligned} a^{-\lambda _i}\sum _{k=-n}^{\infty } d_{n,i,k}a^k. \end{aligned}$$

Meanwhile, by (4.6), we can put

$$\begin{aligned} c_{n,i}(a)=a^{-\lambda _i} \sum _{k=-n}^{\infty } c_{n,i,k}a^k. \end{aligned}$$

By substituting these expressions into (4.7), we obtain

$$\begin{aligned} c_{n,i,k}=\frac{d_{n,i,k}}{1-q^{-\lambda _i+k}}. \end{aligned}$$

(4.11)

By (4.11) and (2.1), we have

$$\begin{aligned} c_{n,i,k}a^k=-\frac{1}{2\pi \sqrt{-1}} \int _{{\mathbb {T}}} d_{n,i,k} \, (tz)^k F(q^{-\lambda _i}, at^{-1}z^{-1};q) \frac{dz}{z}. \ \ \ \end{aligned}$$

Thus it holds that

$$\begin{aligned} c_{n,i}(a)&=-a^{-\lambda _i}\sum _{k=-n}^{\infty } \frac{1}{2\pi \sqrt{-1}} \int _{{\mathbb {T}}}d_{n,i,k}(tz)^k F(q^{-\lambda _i},at^{-1}z^{-1};q)\frac{dz}{z} \nonumber \\&=-\frac{a^{-\lambda _i}}{2\pi \sqrt{-1}} \int _{{\mathbb {T}}}\left( \sum _{k=-n}^{\infty }d_{n,i,k}(tz)^k \right) F(q^{-\lambda _i},at^{-1}z^{-1};q)\frac{dz}{z} \nonumber \\&=\frac{(-\beta )^{-r-1}q^{-\lambda _i}a^{-\lambda _i}}{\prod _{\begin{array}{c} 1\le k \le r+1 \\ k \ne i \end{array} }(q^{\lambda _i}-q^{\lambda _k})}\nonumber \\ {}&\quad \times \frac{1}{\,2\pi \sqrt{-1}} \int _{{\mathbb {T}}} {}_{r+1}{\hat{\phi }}_r \begin{bmatrix} \begin{matrix} q^{1-\lambda _i} \rho _1^{-1} ,\dots , q^{1-\lambda _i}\rho _{r+1}^{-1} \\ q^{\lambda _1-\lambda _i+1}, \dots , q^{\lambda _{r+1}-\lambda _i+1} \end{matrix} ;q,tz \end{bmatrix}(1-tz) \nonumber \\&\quad \times F(q^{-\lambda _i}, at^{-1}z^{-1};q)r_n(tz) \frac{dz}{z}. \end{aligned}$$

(4.12)

By (4.12) and (4.6), we have

$$\begin{aligned} \phi _{r-1,n}(a;q)=&\sum _{i=1}^{r+1}c_{n,i}(a) f_i(a) \\ =&\frac{1}{2\pi \sqrt{-1}} \int _{{\mathbb {T}}} K(a,tz;q)r_n(tz) \frac{dz}{z}. \end{aligned}$$

By observing

$$\begin{aligned}&\sum _{n=0}^{\infty } r_n(tz)p^n\\&\quad = \sum _{n=0}^{\infty } \left( \sum _{k=0}^{n-1} s_{n-k}(tz,\alpha ) \left( x_2(1-\beta T_z)^r +\sum _{j=0}^{r-1} (x_{r+2-j}-x_1x_{r+1-j}) (1-\beta T_z)^j \right) \phi _{r-1,k}(tz;q) \right. \\&\qquad \left. +x_{r+2} s_n(tz,\alpha ) +\frac{x_1x_{r+2}}{x_2} \phi _{-2,n}(q)\right) p^n \\&\quad =\left( \sum _{n=1}^{\infty }s_{n}(tz,\alpha )p^n \right) \left( x_2(1-\beta T_z)^r + \sum _{j=0}^{r-1} (x_{r+2-j}-x_1 x_{r+1-j}) (1-\beta T_z)^j \right) \\&\qquad \times \left( \sum _{k=0}^{\infty } \phi _{r-1,k}(tz;q)p^k \right) \\&\qquad +x_{r+2} \sum _{n=0}^{\infty } s_{n}(tz,\alpha )p^n +\frac{x_1x_{r+2}}{x_2} \sum _{n=0}^{\infty } \phi _{-2,n}(q)p^n \\&\quad ={\widetilde{F}}(tz,\alpha ;p) \left( x_2(1-\beta T_z)^r+\sum _{j=0}^{r-1} (x_{r+2-j}-x_1x_{r+1-j}) (1-\beta T_z)^j \right) \Phi _{r-1}(tz;p,q)\\&\qquad +x_{r+2}F(tz,\alpha ;p) +\frac{x_1x_{r+2}}{x_2}\Phi _{-2}(p,q), \end{aligned}$$

we find that

$$\begin{aligned} \Phi _{r-1}(a;p,q)&=\sum _{n=0}^{\infty } \phi _{r-1,n}(a;q) p^n \\&=\frac{1}{2\pi \sqrt{-1}} \int _{{\mathbb {T}}} K(a,tz;q) \left( \sum _{n=0}^{\infty }r_n(tz) p^n \right) \frac{dz}{z} \\&=\frac{1}{2\pi \sqrt{-1}} \int _{{\mathbb {T}}}K(a,tz;q) {\widetilde{F}}(tz,\alpha ;p)\Bigl \{x_2 (1-\beta T_z)^r \\&\quad + \sum _{j=0}^{r-1} (x_{r+2-j}-x_1 x_{r+1-j}) (1-\beta T_z)^j \Bigr \}\Phi _{r-1}(tz;p,q) \ \frac{dz}{z} \\&\quad +\frac{1}{2\pi \sqrt{-1}} \int _{{\mathbb {T}}} K(a,tz;q) \left( x_{r+2}F(tz,\alpha ;p)+\frac{x_1x_{r+2}}{x_2} \Phi _{-2}(p,q) \right) \frac{dz}{z} \end{aligned}$$

and thus obtain the proposition. $\square$

Let us consider solving the integral equation (4.10) in terms of formal power series in p. We rewrite the q-difference equation (3.1) as

$$\begin{aligned} (1-\beta T_a)^{r+1}\Phi _{r-1}(a;p,q)=\sum _{j=0}^r \xi _{j}(a)(1-\beta T_a)^{r-j} \Phi _{r-1}(a;p,q)+\xi _{r+1}(a). \end{aligned}$$

(4.13)

The coefficients $\xi _j(a)$ can be explicitly expressed, as follows:

$$\begin{aligned} \xi _j(a)= {\left\{ \begin{array}{ll} x_1+x_2 F(a,\alpha ;p) &{} (j=0) \\ (x_{j+2}-x_1x_{j+1})F(a,\alpha ;p) &{} (1 \le j \le r) \\ x_{r+2}F(a,\alpha ;p)+\dfrac{x_1x_{r+2}}{x_2}\Phi _{-2}(p,q) &{} (j=r+1). \end{array}\right. } \end{aligned}$$

For integers i, j, n with $0 \le i \le r, \ 0 \le j \le r, \ n \ge 1$, we define $K_{i,j}(a,b;p,q)$ as

$$\begin{aligned}&K_{i,j}(a,b;p,q)\\ :=&{\left\{ \begin{array}{ll} x_2 K(a,b;q){\widetilde{F}}(b,\alpha ;p) &{} (i=0, \ j=r) \\ (x_{r+2-j}-x_1 x_{r+1-j})K(a,b;q){\widetilde{F}}(b,\alpha ;p) &{} (i=0, \ 0 \le j \le r-1) \\ (1-T_a T_b)K_{i-1,0}(a,b;p,q)+K_{i-1,r}(qa,qb;p,q)\xi _r(b) &{} (i \ge 1, \ j=0) \\ (1-T_a T_b)K_{i-1,j}(a,b;p,q)+K_{i-1,j-1}(qa,qb;p,q)\\ +K_{i-1,r}(qa,qb;p,q)\xi _{r-j}(b) &{} (i \ge 1, \ 1 \le j \le r) \end{array}\right. } \end{aligned}$$

and $K_{j}^{(n)}(a,b;p,q)$ as

$$\begin{aligned} K_j^{(n)}(a,b;p,q):=&{\left\{ \begin{array}{ll} K_{0,j}(a,b;p,q) &{} (n=1) \\ \displaystyle \sum \nolimits _{i=0}^r \frac{1}{2\pi \sqrt{-1}} \int _{{\mathbb {T}}}K_{i}^{(n-1)}(a,tz;p,q)K_{i,j}(tz,b;p,q)\frac{dz}{z } &{} (n \ge 2) \end{array}\right. } \\&(\max \{ |a|, |qb|, \ |p| \}<|t|<\min \{ |q^{-n+2}|, \ |w^{-1}|, \ \ |q^{-n+1} a|, \ |b|\}). \end{aligned}$$

Under the condition

$$\begin{aligned} |p|<|w^{-1}|, \ |pq|<|a|<\min \{1, \ |w^{-1}|\}, \end{aligned}$$

$K_{j}^{(n)}(a,b;p,q)$ is holomorphic as a function of b on the domain given by

$$\begin{aligned} |a|<|b|<\min \{|q^{-n+1}|, \ |q^{-1}w^{-1}|, \ |q^{-n} a| \}. \end{aligned}$$

It is observed that $K_{j}^{(n)}(a,b;p,q)$ is a power series in p expressed as

$$\begin{aligned} K_j^{(n)}(a,b;p,q)=\sum _{m=n}^{\infty } k_{j,m}^{(n)}(a,b;q) p^m. \end{aligned}$$

(4.14)

For $j=0,\dots , r$, we define $\Gamma _j(a,b;p,q)$ as formal power series in p given by

$$\begin{aligned} \Gamma _{j}(a,b;p,q):=\sum _{m=1}^{\infty } \left\{ \sum _{n=1}^{m} k_{j,m}^{(n)} (a,b;q) \right\} p^m \in {{\mathbb {C}}}[[p]]. \end{aligned}$$

Since

$$\begin{aligned} \Gamma _j(a,b;p,q) \equiv \sum _{n=1}^{L} K_j^{(n)}(a,b;p,q) \pmod {p^N} \end{aligned}$$

for $N \ge 1$ and $L \ge N-1$, we have

$$\begin{aligned} \Gamma _{j}(a,b;p,q)=\sum _{n=1}^{\infty } K_j^{(n)} (a,b;p,q) \end{aligned}$$

(4.15)

in ${{\mathbb {C}}}[[p]]$. For integers i with $0 \le i \le r$, we define $h_i (a;p,q)$ by

$$\begin{aligned} h_i(a;p,q):=&{\left\{ \begin{array}{ll} \displaystyle \frac{1}{2\pi \sqrt{-1}} \int _{{\mathbb {T}}} K(a,tz;q) \xi _{r+1} (tz) \frac{dz}{z}&{} (i=0) \vspace{1mm} \\ \displaystyle \frac{1}{2\pi \sqrt{-1}}\int _{{\mathbb {T}}} K_{i-1,r}(qa,qtz;p,q) \xi _{r+1}(tz) \frac{dz}{z}\\ +(1-\beta T_a )h_{i-1}(a;p,q) &{} (1 \le i \le r) \end{array}\right. } \\&( \max \{ |a|, \ |pq^{-i}| \}<|t|<\min \{1,|q^{-1}a| \}). \end{aligned}$$

Then $h_i(a;p,q)$ are holomorphic on the domain given by $|pq^{1-i}|<|a|<\min \{1, \ |w^{-1}|\}$.

The main theorem of this paper is as follows:

Theorem 4.9

Assume that the following condition holds:

$$\begin{aligned} |a|< \min \{1, \ |w^{-1}| \}. \end{aligned}$$

We consider $\Phi _{r-1}(a;p,q)$ and $h_i(a;p,q) \ (i=0,\ldots ,r)$ as formal power series in p by (4.1) and (4.3). Then we have the following equality:

$$\begin{aligned} \Phi _{r-1}(a;p,q)&= \sum _{i=0}^r \frac{1}{2\pi \sqrt{-1}} \int _{{\mathbb {T}}} \Gamma _{i}(a,tz;p,q) h_i(tz;p,q) \frac{dz}{z}+h_0(a;p,q) \\&\quad (|a|<|t|<\min \{1, \ |q^{-1}a|, \ |w^{-1}|\}). \nonumber \end{aligned}$$

(4.16)

Proof

We first show that the following identity holds for $0 \le i \le r$ and $|pq^{1-i}|<|a|<\min \{1, \ |w^{-1}| \}$:

$$\begin{aligned}&(1-\beta T_a)^i \Phi _{r-1}(a;p,q)=\sum _{j=0}^r \frac{1}{2\pi \sqrt{-1}} \int _{{\mathbb {T}}} K_{i,j}(a,tz;p,q)\nonumber \\&\qquad \qquad \qquad \qquad \qquad \qquad \qquad \times (1-\beta T_{z})^j\Phi _{r-1}(tz;p,q) \frac{dz}{z} +h_i(a;p,q) \\&( \max \{|a|, \ |p| \}<|t|<\min \{1, \ |q^{-1}a|, |w^{-1}| \} ) \nonumber . \end{aligned}$$

(4.17)

When $i=0$, (4.17) follows immediately from (4.10) and the definition of $K_{0,j}(a,b;p,q)$.

We now assume that (4.17) holds for some i with $0\le i \le r-1$. When $|pq^{-i}|<|a|<1$, (4.17) implies

$$\begin{aligned}&(1-\beta T_a)^{i+1} \Phi _{r-1}(a;p,q)\\&=\sum _{j=0}^r \frac{1}{2\pi \sqrt{-1}} \int _{{\mathbb {T}}}(1-\beta T_a T_z) K_{i,j}(a,tz;p,q) (1-\beta T_{z})^j \Phi _{r-1}(tz;p,q) \frac{dz}{z}\\&\quad +(1-\beta T_a)h_i(a;p,q) \\&\quad (|a|<|t|<\min \{1, \ |q^{-1}a|, \ |w^{-1}| \}). \end{aligned}$$

Since

$$\begin{aligned} (1-\beta T_a T_b) f(a,b)g(a,b)&=\{(1-T_a T_b)f(a,b)\}g(a,b)\\&\quad +f(qa,qb) \{(1-\beta T_a T_b)g(a,b) \} \end{aligned}$$

for functions f(a, b) and g(a, b) of a and b, we have

$$\begin{aligned}&(1-\beta T_a)^{i+1}\Phi _{r-1}(a;p,q)\\&\quad =\sum _{j=0}^{r} \frac{1}{2\pi \sqrt{-1}}\int _{{\mathbb {T}}} \bigl [ (1-T_a T_z) K_{i,j}(a,tz;p,q) (1-\beta T_z)^{j} \\&\qquad +K_{i,j}(qa,qtz;p,q)(1-\beta T_z)^{j+1} \bigr ]\Phi _{r-1}(tz;p,q)\frac{dz}{z} +(1-\beta T_a)h_i(a;p,q) \\&\quad = \frac{1}{2\pi \sqrt{-1}} \int _{{\mathbb {T}}} (1-T_a T_z)K_{i,0}(a,tz;p,q) \Phi _{r-1}(tz;p,q) \frac{dz}{z}\\&\qquad +\sum _{j=1}^r \frac{1}{2\pi \sqrt{-1}}\int _{{\mathbb {T}}} \bigl [ (1-T_a T_z) K_{i,j}(a,tz;p,q)\\&\qquad +K_{i,j-1} (^qa,qtz;p,q) ](1-\beta T_z)^{j} \Phi _{r-1}(tz;p,q) \frac{dz}{z} \\&\qquad +\frac{1}{2\pi \sqrt{-1}} \int _{{\mathbb {T}}}K_{i,r}(qa,qtz;p,q)\\&\qquad \times (1-\beta T_z)^{r+1} \Phi _{r-1}(tz;p,q) \frac{dz}{z}+(1-\beta T_a) h_i(a;p,q). \end{aligned}$$

By applying (4.13) to $(1-\beta T_z)^{r+1} \Phi _{r-1}(tz;p,q)$, we find that (4.17) holds for $i+1$, which completes the proof of (4.17) for $i=0,\ldots ,r$.

We next assume that the condition

$$\begin{aligned} |pq^{1-r}|<\min \{1, \ |w^{-1}| \} , \ |p q^{1-r}|<|a|< \min \{1, \ |w^{-1}| \}. \end{aligned}$$

(4.18)

holds and show that the following identity holds for $n \in {{\mathbb {Z}}}_{>0}$:

$$\begin{aligned} \Phi _{r-1}(a;p,q)&= \sum _{j=0}^r \frac{1}{2\pi \sqrt{-1}} \int _{{\mathbb {T}}} K_{j}^{(n)} (a,tz;p,q)(1-\beta T_z)^j \Phi _{r-1}(tz;p,q) \frac{dz}{z} \nonumber \\&\quad +\sum _{j=0}^r \sum _{l=1}^{n-1} \frac{1}{2\pi \sqrt{-1}} \int _{{\mathbb {T}}} K_{j}^{(l)}(a,tz;p,q) h_j(tz;p,q) \frac{dz}{z} +h_0(a;p,q) \nonumber \\&\quad (|a|<|t|<\min \{1, \ |q^{-1}a|, \ |w^{-1}| \}) . \end{aligned}$$

(4.19)

When $n=1$, (4.19) follows immediately from (4.17) with $i=0$. We assume that (4.19) holds for some n. The condition (4.18) allows us to apply (4.17) to $(1-\beta T_{z})^j\Phi _{r-1}(tz;p,q)$ in (4.19) for n. By (4.17) and the induction hypothesis, we have

$$\begin{aligned}&\Phi _{r-1}(a;p,q) \\&= \sum _{i=0}^r \frac{1}{2\pi \sqrt{-1}} \int _{{\mathbb {T}}} K_i^{(n)} (a,t'z';p,q) \biggl \{ \sum _{j=0}^r \frac{1}{2\pi \sqrt{-1}}\int _{{\mathbb {T}}} K_{i,j}(t'z',tz;p,q) \\&\quad \times (1-\beta T_z)^j \Phi _{r-1}(tz;p,q) \frac{dz}{z}+h_i(t'z';p,q) \biggr \} \frac{dz'}{z'} \\&\quad + \sum _{j=0}^r \sum _{l=1}^{n-1} \frac{1}{2\pi \sqrt{-1}}\int _{{\mathbb {T}}}K_j^{(l)} (a,t'z';p,q) h_j(t'z';p,q) \frac{dz'}{z'}+h_0(a;p,q) \\&\quad (|a|<|t'|<\min \{1, \ |q^{-1}a|, \ |w^{-1}| \}, \ \ |t'|<|t|<\min \{1, \ |q^{-1}t'|, \ |w^{-1}|). \end{aligned}$$

Since

$$\begin{aligned} \sum _{i=0}^r \frac{1}{2\pi \sqrt{-1}}\int _{{\mathbb {T}}}K_i^{(n)} (a,t'z';p,q)K_{i,j}(t'z',tz;p,q) \frac{dz'}{z'}=K_{j}^{(n+1)}(a,tz;p,q), \end{aligned}$$

it holds that

$$\begin{aligned} \Phi _{r-1}(a;p,q)&= \sum _{j=0}^r \frac{1}{2\pi \sqrt{-1}}\int _{{\mathbb {T}}} K_{j}^{(n+1)}(a,tz;p,q) (1-\beta T_z)^j \Phi _{r-1}(tz;p,q) \frac{dz}{z} \\&\quad +\sum _{j=0}^r \sum _{l=1}^n \frac{1}{2\pi \sqrt{-1}} \int _{{\mathbb {T}}} K_j^{(l)} (a,tz;p,q) h_j(tz;p,q) \frac{dz}{z}+h_0(a;p,q) \\&\quad (|a|<|t|<\min \{1, \ |q^{-1}a|, \ |w^{-1}| \}), \end{aligned}$$

which implies that (4.19) holds for $n+1$. Thus we find that (4.19) holds for all $n \in {{\mathbb {Z}}}_{>0}$.

Finally we consider (4.19) as the equality of formal power series in p. By (4.15), letting $n \rightarrow \infty$ in (4.19) gives (4.16). Thus we finish the proof of the theorem. $\square$

Remark 4.10

By the definition of the functions $K_{j}^{(n)}(a,b;p,q)$, the formal power series $\Gamma _i(a,b;p,q) \ (i=0,\ldots ,r)$ satisfy the following integral equations:

$$\begin{aligned} \Gamma _i(a,b;p,q)&= K_{0,i}(a,b;p,q)+\sum _{j=0}^r \frac{1}{2\pi \sqrt{-1}} \int _{{\mathbb {T}}}\Gamma _j(a,tz;p,q)K_{j,i}(tz,b;p,q) \frac{dz}{z} \\&\quad (\max \{|a|, |qb| \}< |t|<\min \{|w^{-1}|, 1, |q^{-1}a|, |b| \}). \end{aligned}$$

5 Trigonometric and Classical Limits of the Main Theorem

In this section, we examine what Theorem 4.9 can be reduced to by taking the trigonometric and classical limits.

5.1 Trigonometric Limit

For $-1 \le j \le r-1$, we put

$$\begin{aligned} G_j(k,l,h_1,\dots ,h_r;a,\alpha ,\beta ;q):=\sum _{{\varvec{k}} \in I_j(k,l,h_1,\dots ,h_r)} L_{\varvec{k}}(a,\alpha ,\beta ;q) \end{aligned}$$

and define the function $\Phi _j(a;q)$ by

$$\begin{aligned} \Phi _j(a;q)&=\Phi _j(x_1,\ldots ,x_{r+2};a,\alpha ,\beta ;q) \\&:=\sum _{k,l,h_1,\dots ,h_r \ge 0} G_j(k,l,h_1,\dots ,h_r ;a,\alpha ,\beta ;q)\\&\qquad \times x_1^{k-l-\sum _{i=1}^r h_i }x_2^{l-h_1} x_3^{h_1-h_2} \times \cdots \times x_{r+1}^{h_{r-1}-h_r} x_{r+2}^{h_r}. \end{aligned}$$

Then, by Proposition 2.4, we have

$$\begin{aligned} \lim _{p \rightarrow 0}\Phi _j(a;p,q)=\Phi _{j}(a;q) \end{aligned}$$

for $j \ge -1$. The limit of $\Phi _{-2}(p,q)$ as $p \rightarrow 0$ is given by the following:

Lemma 5.1

$$\begin{aligned} \lim _{p \rightarrow 0}\Phi _{-2}(p,q)=-\frac{x_2}{x_1}c_0(\alpha ). \end{aligned}$$

Proof

For $t \in {{\mathbb {C}}}$ satisfying $|p|<|at|<1$ and $|q|<|t^{-1}|<1$, we have

$$\begin{aligned}&G_{-2}(k,l,h_1,\dots ,h_r;\alpha ,\beta ;p,q)\nonumber \\&\quad =-\frac{1}{2\pi \sqrt{-1}} \int _{{\mathbb {T}}}F(atz,\alpha ;p) G_{-1}(k.l-1,h_1,\dots ,h_r;at z,\alpha ,\beta ;p,q) \frac{dz}{z}. \end{aligned}$$

Thus we obtain

$$\begin{aligned} \Phi _{-2}(p,q)&=\sum _{k,l,h_1,\dots ,h_r \ge 0} G_{-2}(k,l,h_1,\dots ,h_r;\alpha ,\beta ;p,q)\\&\quad \times x_1^{k-l-\sum _{i=1}^r h_i} x_2^{l-h_1}x_3^{h_1-h_2}\cdots x_{r+1}^{h_{r-1}-h_r} x_{r+2}^{h_r} \\&= -\frac{x_2}{x_1 2 \pi \sqrt{-1}} \int _{{\mathbb {T}}}F(atz,\alpha ;p) \Phi _{-1} (atz;p,q) \frac{dz}{z}. \end{aligned}$$

Hence the limit of $\Phi _{-2}(p,q)$ can be calculated as

$$\begin{aligned} \lim _{p \rightarrow 0} \Phi _{-2}(p,q)=&-\frac{x_2}{x_1 2\pi \sqrt{-1}} \int _{{\mathbb {T}}} (-c_1(atz)+c_0(\alpha )) \Phi _{-1}(atz;q) \frac{dz}{z} \\ =&-\frac{x_2}{x_1}c_0(\alpha ). \end{aligned}$$

$\square$

As the trigonometric degeneration of (4.16), we obtain the following:

Theorem 5.2

$$\begin{aligned}{} & {} \Phi _{r-1 }(a;q)\nonumber \\{} & {} =(-\beta )^{-r-1} x_{r+2} \, a \sum _{i=1}^{r+1} \frac{1}{\prod _{\begin{array}{c} 1 \le l \le r+1\\ l \ne i \end{array} }(q^{\lambda _i}-q^{\lambda _l}) \ (q^{\lambda _i}-q) } \nonumber \\{} & {} \qquad \times {}_{r+1}{\hat{\phi }}_r \begin{bmatrix} \begin{matrix} q^{\lambda _i} \rho _1, \dots , q^{\lambda _i} \rho _{r+1} \\ q^{\lambda _i-\lambda _1+1},\dots , q^{\lambda _i-\lambda _{r+1}+1} \end{matrix} ;q, wa \end{bmatrix} \nonumber \\{} & {} \qquad \times {}_{r+2} {\hat{\phi }}_{r+1}\begin{bmatrix} \begin{matrix} q^{1-\lambda _i} \rho _1^{-1}, \dots , q^{1-\lambda _i}\rho _{r+1}^{-1}, q^{1-\lambda _i} \\ q^{\lambda _1-\lambda _i+1}, \dots , q^{\lambda _{r+1}-\lambda _i+1}, q^{2-\lambda _i} \end{matrix} ;q,a \end{bmatrix}. \end{aligned}$$

(5.1)

Proof

By Lemma 5.1, comparing the constant terms of the formal power series in p on both sides of (4.16) yields

$$\begin{aligned} \Phi _{r-1}(a;q)=-\frac{x_{r+2}}{2\pi \sqrt{-1}} \int _{{\mathbb {T}}} K(a,tz;q) c_1(tz) \frac{dz}{z} \ \ (|a|<|t|<\min \{1, \ |q^{-1}a|). \end{aligned}$$

By observing

$$\begin{aligned}&\frac{1}{2\pi \sqrt{-1}}\int _{{\mathbb {T}}}{}_{r+1}{\hat{\phi }}_r \begin{bmatrix} \begin{matrix} q^{1-\lambda _i} \rho _1^{-1},\dots , q^{1-\lambda _i}\rho _{r+1}^{-1} \\ q^{\lambda _1-\lambda _i+1}, \dots , q^{\lambda _{r+1}-\lambda _i+1} \end{matrix} ;q ,tz \end{bmatrix} (1- tz)F(q^{-\lambda _i},at^{-1}z^{-1};q) c_1( tz) \frac{dz}{z} \\&\quad = \sum _{m=0}^{\infty } \sum _{n \in {{\mathbb {Z}}}} \frac{(q^{1-\lambda _i} \rho _1^{-1}, \dots , q^{1-\lambda _i}\rho _{r+1}^{-1} ;q)_m}{(q^{\lambda _1-\lambda _i+1},\dots ,q^{\lambda _{r+1}-\lambda _i+1};q)_m (q^{n-\lambda _i}-1)} a^n \times \frac{1}{2\pi \sqrt{-1}} \int _{{\mathbb {T}}} (tz)^{m+1-n} \frac{dz}{z} \\&\quad = \sum _{m=0}^{\infty } \frac{(q^{1-\lambda _i}\rho _1^{-1},\dots ,q^{1-\lambda _i}\rho _{r+1}^{-1};q)_m}{(q^{\lambda _1-\lambda _i+1},\dots ,q^{\lambda _{r+1}-\lambda _i+1};q)_m (q^{m+1-\lambda _i}-1)} a^{m+1} \\&\quad = -\frac{a}{1-q^{1-\lambda _i}} {}_{r+2} {\hat{\phi }}_{r+1} \begin{bmatrix} \begin{matrix} q^{1-\lambda _i} \alpha _1^{-1}, \dots , q^{1-\lambda _i}\alpha _{r+1}^{-1}, q^{1-\lambda _i} \\ q^{\lambda _1-\lambda _{i}+1}, \dots , q^{\lambda _{r+1}-\lambda _i+1}, q^{2-\lambda _i} \end{matrix} ;q, a \end{bmatrix}, \end{aligned}$$

we obtain the theorem. $\square$

By applying Kajihara’s transformation (Proposition 4.6) to Theorem 5.2, Theorem 5.2 is equivalent to the following theorem:

Theorem 5.3

$$\begin{aligned} \Phi _{r-1}(a;q)=-\frac{\beta ^{-r-1}x_{r+2}a}{\prod _{k=1}^{r+1} (q^{\lambda _k}-q)} {}_{r+2}\phi _{r+1}\begin{bmatrix} \begin{matrix} q, q\rho _1,\dots ,q\rho _{r+1} \\ q^{2-\lambda _1},\dots ,q^{2-\lambda _{r+1}} \end{matrix} ;q, wa \end{bmatrix}. \end{aligned}$$

Proof

By applying Proposition 4.6 with

$$\begin{aligned}&m=1, \ n=r, \ {\varvec{x}}=(q^{\lambda _i+1}), \ {\varvec{y}}=(q^{-\lambda _1}, \dots , q^{-\lambda _{i-1}}, q^{-\lambda _{i+1}}, \dots , q^{-\lambda _{r+1}}), \\&{\varvec{a}}=(q^{\lambda _i}\rho _i), \ {\varvec{b}}=(q^{-\lambda _1+1}\rho _1^{-1}, \dots , q^{-\lambda _{i-1}+1} \rho _{i-1}^{-1},q^{-\lambda _{i+1}+1}\rho _{i+1}^{-1},\dots , q^{-\lambda _{r+1}+1}\rho _{r+1}^{-1}), \\&z=wa \end{aligned}$$

to ${}_{r+1}{\hat{\phi }}_r \begin{bmatrix} \begin{matrix} q^{\lambda _i} \rho _1, \dots , q^{\lambda _i}\rho _{r+1} \\ q^{\lambda _i-\lambda _1+1},\dots ,q^{\lambda _i-\lambda _{r+1}+1} \end{matrix} ;q, wa \end{bmatrix}$ on the right-hand side of (5.1), we have

$$\begin{aligned} \Phi _{r-1}(a;q)&=-\beta ^{-r-1} x_{r+2}q^{-|{\varvec{\lambda }}|}a \frac{(q^{-r}a;q)_{\infty }}{( wa;q)_{\infty }} \sum _{{\varvec{\nu }}=(\nu _i)_{i=1}^{r+1} \in {{\mathbb {Z}}_{\ge 0}^{r+1}}} (q^{-r}a)^{|{\varvec{\nu }}|}\\&\quad \times \prod _{1 \le k <l \le r+1}\frac{q^{\nu _k-\lambda _k} -q^{\nu _l-\lambda _l}}{q^{-\lambda _k}-q^{-\lambda _l}} \\&\quad \times \prod _{1 \le k,l \le r+1} \frac{(q^{1-\lambda _k}\rho _l^{-1};q)_{\nu _k}}{(q^{1-\lambda _k+\lambda _l};q)_{\nu _k}} \sum _{i=1}^{r+1} \frac{q^{-1}}{q^{-1}-q^{\nu _i-\lambda _i}} \prod _{ \begin{array}{c} 1 \le l \le r+1 \\ l \ne i \end{array} } \frac{q^{\nu _i-\lambda _i}}{q^{\nu _i-\lambda _i}-q^{\nu _l-\lambda _l}}, \end{aligned}$$

where we have used the identity

$$\begin{aligned} w \rho _1 \cdots \rho _{r+1} q^{\lambda _1+\cdots +\lambda _{r+1}}=1. \end{aligned}$$

(5.2)

We now recall Lagrange’s interpolation formula: For a polynomial p(x) of degree less than or equal to r,

$$\begin{aligned} p(x)= \sum _{i=1}^{r+1} p(y_i) \prod _{ \begin{array}{c} 1 \le l \le r+1 \\ l \ne i \end{array} } \frac{x-y_l}{y_i-y_l}. \end{aligned}$$

(5.3)

By putting $p(x)=x^r, \ \ y_i=q^{\nu _i-\lambda _i} \ (1 \le i \le r+1)$ in (5.3) and then divide the both sides by $\prod _{k=1}^{r+1} (x-q^{\nu _k-\lambda _k})$, we have

$$\begin{aligned} \frac{x^r}{\prod _{k=1}^{r+1} (x-q^{\nu _k-\lambda _k})}=\sum _{i=1}^{r+1} \frac{1}{x-q^{\nu _i-\lambda _i}} \prod _{ \begin{array}{c} 1 \le l \le r+1 \\ l \ne i \end{array} } \frac{q^{\nu _i-\lambda _i}}{q^{\nu _i-\lambda _i}-q^{\nu _l-\lambda _l}}. \end{aligned}$$

(5.4)

Setting $x=q^{-1}$ in (5.4) gives

$$\begin{aligned} \sum _{i=1}^{r+1} \frac{q^{-1}}{q^{-1}-q^{\nu _i-\lambda _i}} \prod _{ \begin{array}{c} 1 \le l \le r+1 \\ l \ne i \end{array} } \frac{q^{\nu _i-\lambda _i}}{q^{\nu _i-\lambda _i}-q^{\nu _l-\lambda _l}} = \frac{q^{|{\varvec{\lambda }}|}}{\prod _{k=1}^{r+1} (q^{\lambda _k}-q)} \prod _{k=1}^{r+1} \frac{(q^{-\lambda _k+1};q)_{\nu _k}}{(q^{-\lambda _k+2};q)_{\nu _k}}. \end{aligned}$$

Thus we find that

$$\begin{aligned} \Phi _{r-1}(a;q)=&-\frac{\beta ^{-r-1} x_{r+2}a}{\prod _{k=1}^{r+1} (q^{\lambda _k}-q)} \times \frac{(q^{-r}a;q)_{\infty }}{(wa;q)_{\infty }} \sum _{{\varvec{\nu }} =(\nu _i)_{i=1}^{r+1} \in {{\mathbb {Z}}_{\ge 0}^{r+1}}}(q^{-r}a)^{|{\varvec{\nu }}|}\\&\times \prod _{1 \le k<l \le r+1}\frac{q^{\nu _k-\lambda _k} -q^{\nu _l-\lambda _l}}{q^{-\lambda _k}-q^{-\lambda _l}} \\&\times \prod _{1 \le k,l \le r+1} \frac{(q^{1-\lambda _k}\rho _l^{-1};q)_{\nu _k}}{(q^{1-\lambda _k+\lambda _l};q)_{\nu _k}} \prod _{1 \le k \le r+1}\frac{(q^{-\lambda _k+1};q)_{\nu _k}}{(q^{-\lambda _k+2};q)_{\nu _k}}. \end{aligned}$$

By putting

$$\begin{aligned} m=r+1, \ n=1, \ {\varvec{x}}=(q^{-\lambda _1},\dots ,q^{-\lambda _{r+1}}), \ {\varvec{y}}=(q^2), \\ {\varvec{a}}=(q^{1-\lambda _1}\rho _1^{-1}, \dots , q^{1-\lambda _{r+1}} \rho _{r+1}^{-1}), \ {\varvec{b}}=(q), \ z=q^{-r+k}a \end{aligned}$$

in Proposition 4.6, we see that

$$\begin{aligned}&\sum _{\nu \in {{\mathbb {Z}}_{\ge 0}^{r+1}}}(q^{-r}a)^{|{\varvec{\nu }}|}\prod _{1 \le k<l \le r+1}\frac{q^{\nu _k-\lambda _k} -q^{\nu _l-\lambda _l}}{q^{-\lambda _k}-q^{-\lambda _l}}\\&\qquad \times \prod _{1 \le k,l \le r+1} \frac{(q^{1-\lambda _k}\rho _l^{-1};q)_{\nu _k}}{(q^{1-\lambda _k+\lambda _l};q)_{\nu _k}} \prod _{1 \le k \le r+1}\frac{(q^{-\lambda _k+1};q)_{\nu _k}}{(q^{-\lambda _k+2};q)_{\nu _k}} \\&\quad = \frac{(wa;q)_{\infty }}{(q^{-r}a;q)_{\infty }} \sum _{\mu \in {{\mathbb {Z}}_{\ge 0}}}(wa)^{\mu } \prod _{1 \le j \le r+1} \frac{(q\rho _j;q)_{\mu }}{(q^{2-\lambda _j};q)_{\mu }} \\&\quad = \frac{(wa;q)_{\infty }}{(q^{-r}a;q)_{\infty }} {}_{r+2}\phi _{r+1} \begin{bmatrix} \begin{matrix} q, q\rho _1,\dots ,q\rho _{r+1} \\ q^{2-\lambda _1},\dots ,q^{2-\lambda _{r+1}} \end{matrix} ;q, wa \end{bmatrix}, \end{aligned}$$

where we have used the identity (5.2) again. Hence we obtain the theorem. $\square$

Since

$$\begin{aligned} \prod _{k=1}^{r+1}(q^{\lambda _k}-q)&=\beta ^{-r-1} \Bigl \{(1-q\beta )^{r+1}-(x_1+x_2 c_0(\alpha ))(1-q\beta )^r \\&\quad -c_0(\alpha ) \sum _{j=0}^{r-1} (x_{r+2-j}-x_1 x_{r+1-j})(1-q\beta )^j \Bigr \}, \end{aligned}$$

Theorem 5.3 with $\beta =1$ is essentially the same as Theorem 9 of [16].

We will show that Theorem 5.3 can be considered to be a q-generalization of Theorem 3.5 of [8]. For nonnegative integers $k.k,h_1,\ldots ,h_r$, we put

$$\begin{aligned} G_{r-1}(k,l,h_1,\ldots ,h_r;a,\alpha ,x):=\sum _{{\varvec{k}} \in I_{r-1}(k,l,h_1,\dots ,h_r)}L_{\varvec{k}}(a,\alpha ,x) \end{aligned}$$

and define the function $\Phi _{r-1}(a)$ by

$$\begin{aligned} \Phi _{r-1}(a)&=\Phi _{r-1}(x_1,\ldots ,x_{r+2};a,\alpha ,x)\\&:=\sum _{k,l,h_1,\dots ,h_r \ge 0} G_{r-1}(k,l,h_1,\dots ,h_r;a,\alpha ,x)\\&\qquad \times x_1^{k-l-\sum _{i=1}^r h_i} x_2^{l-h_1}x_3^{h_1-h_2}\cdots x_{r+1}^{h_{r-1}-h_r}x_{r+2}^{h_r}. \end{aligned}$$

We put

$$\begin{aligned} {\widetilde{x}}_1=(1-q)x_1, \ \ {\widetilde{x}}_i=-(1-q)^{i-1}x_i \ (2 \le i \le r+2). \end{aligned}$$

(5.5)

Then, by (2.4), we have

$$\begin{aligned} \lim _{q \rightarrow 1} \Phi _{r-1}({\widetilde{x}}_1,\ldots ,{\widetilde{x}}_{r+2};a,\alpha ,q^{x-1};q)=\Phi _{r-1}(x_1,\ldots ,x_{r+2};a,\alpha ,x). \end{aligned}$$

(5.6)

For $n \in {{\mathbb {Z}}}_{\ge 0}$, we define the shifted factorial $(a)_n$ by

$$\begin{aligned} (a)_n:&={\left\{ \begin{array}{ll} 1 &{} (n=0) \\ a (a+1) \cdots (a+n-1) &{} (n \ge 1) \end{array}\right. } \end{aligned}$$

and denote the product $(a_1)_n (a_2)_n \cdots (a_{m})_n$ by $(a_1,a_2,\dots ,a_m)_n$. The generalized hypergeometric function ${}_{r+1}F_r \begin{bmatrix} \begin{matrix} a_1,\dots , a_{r+1} \\ b_1, \dots , b_r \end{matrix} ;z \end{bmatrix}$ is defined by

$$\begin{aligned} {}_{r+1}F_r \begin{bmatrix} \begin{matrix} a_1,\dots , a_{r+1} \\ b_1, \dots , b_r \end{matrix} ;z \end{bmatrix}:=\sum _{n=0}^{\infty } \frac{(a_1,\dots ,a_{r+1})_n}{(1,b_1,\dots ,b_r)_n}z^n. \end{aligned}$$

The following limit formula holds:

$$\begin{aligned} \lim _{q \rightarrow 1}{}_{r+1}\phi _r \begin{bmatrix} \begin{matrix} q^{a_1},\dots , q^{a_{r+1}} \\ q^{b_1}, \dots , q^{b_r} \end{matrix} ;q,z \end{bmatrix}={}_{r+1}F_r \begin{bmatrix} \begin{matrix} a_1,\dots , a_{r+1} \\ b_1, \dots , b_r \end{matrix} ;z \end{bmatrix}. \end{aligned}$$

(5.7)

We also use the notation

$$\begin{aligned} {}_{r+1} {\hat{F}}_r \begin{bmatrix} \begin{matrix} a_1,\dots ,a_{r+1} \\ b_1,\dots , b_{r+1} \end{matrix} ;z \end{bmatrix}={}_{r+1} F_r \begin{bmatrix} \begin{matrix} a_1,\dots ,a_{r+1} \\ b_1,\dots , {\widehat{1}}, \dots , b_{r+1} \end{matrix} ;z \end{bmatrix} \end{aligned}$$

when one of the $b_1,\dots ,b_{r+1}$ is 1.

Let ${\widetilde{\rho }}_i$ and ${\widetilde{\lambda }}_i \ (i=1,\ldots ,r+1)$ be complex numbers defined by the following relations:

$$\begin{aligned}&(t+x-1)^{r+1}-(x_1-(c_0(\alpha )+1)x_2)(t+x-1)^r \nonumber \\&+(c_0(\alpha )+1)\sum _{j=0}^{r-1}(x_{r+2-j}-x_1x_{r+1-j})(t+x-1)^j =\prod _{i=1}^{r+1}(t+{\widetilde{\rho }}_i), \end{aligned}$$

(5.8)

$$\begin{aligned}&(\lambda +x-1)^{r+1}-(x_1-c_0(\alpha )x_2)(\lambda +x-1)^r \nonumber \\&+c_0(\alpha )\sum _{j=0}^{r-1}(x_{r+2-j}-x_1x_{r+1-j})(\lambda +x-1)^j=\prod _{i=1}^{r+1}(\lambda -{\widetilde{\lambda }}_i). \end{aligned}$$

(5.9)

By Theorems 5.3, (5.6) and (5.7), we have the following theorem.

Theorem 5.4

$$\begin{aligned} \Phi _{r-1}(a)=&\frac{x_{r+2}a}{ x^{r+1}-(x_1-c_0(\alpha ) x_2)x^r +c_0(\alpha )\sum _{j=0}^{r-1} (x_{r+2-j}-x_1 x_{r+1-j})x^j} \\&\times {}_{r+2}F_{r+1} \begin{bmatrix} \begin{matrix} 1, 1+{\widetilde{\rho }}_1, \dots , 1+{\widetilde{\rho }}_{r+1} \\ 2-{\widetilde{\lambda }}_1, \dots , 2-{\widetilde{\lambda }}_{r+1} \end{matrix} ;a \end{bmatrix}. \end{aligned}$$

Theorem 5.4 is essentially the same as Theorem 3.5 of [8]. Meanwhile, by Theorem 5.2, we obtain the following theorem.

Theorem 5.5

$$\begin{aligned} \Phi _{r-1}(a)=&\sum _{i=1}^{r+1} \frac{x_{r+2} a}{\displaystyle \prod _{\begin{array}{c} 1 \le l \le r+1 \\ l \ne i \end{array}}({\widetilde{\lambda }}_i-{\widetilde{\lambda }}_l) (1-{\widetilde{\lambda }}_i) } {}_{r+1}{\hat{F}}_{r} \begin{bmatrix} \begin{matrix} {\widetilde{\lambda }}_i+{\widetilde{\rho }}_1, \dots , {\widetilde{\lambda }}_i+{\widetilde{\rho }}_{r+1} \\ {\widetilde{\lambda }}_i-{\widetilde{\lambda }}_1+1, \dots , {\widetilde{\lambda }}_i-{\widetilde{\lambda }}_{r+1}+1 \end{matrix} ;a \end{bmatrix} \nonumber \\&\times {}_{r+2}{\hat{F}}_{r+1} \begin{bmatrix} \begin{matrix} 1-{\widetilde{\lambda }}_i-{\widetilde{\rho }}_1, \dots , 1-{\widetilde{\lambda }}_i-{\widetilde{\rho }}_{r+1} , 1-{\widetilde{\lambda }}_{i} \\ {\widetilde{\lambda }}_1-{\widetilde{\lambda }}_i+1, \dots , {\widetilde{\lambda }}_{r+1}-{\widetilde{\lambda }}_i+1, 2-{\widetilde{\lambda }}_{i} \end{matrix} ;a \end{bmatrix}. \end{aligned}$$

(5.10)

By using the Gauss summation formula (Theorem 2.2.2 of [1])

$$\begin{aligned} {}_{2}F_{1} \begin{bmatrix} \begin{matrix} a, b \\ c \end{matrix} ;1 \end{bmatrix}=\frac{\Gamma (c)\Gamma (c-a-b)}{\Gamma (c-a)\Gamma (c-b)} \end{aligned}$$

and the Pfaff–Saalschutz identity (Corollary 3.3.5 of [1])

$$\begin{aligned} {}_{3}F_{2} \begin{bmatrix} \begin{matrix} a, b, c \\ d,e \end{matrix} ;1 \end{bmatrix}=\frac{\Gamma (e)\Gamma (d+e-a-b-c)}{\Gamma (e-a)\Gamma (d+e-b-c)} {}_{3}F_{2} \begin{bmatrix} \begin{matrix} a, d-b, d-c \\ d,d+e-b-c \end{matrix} ;1 \end{bmatrix}, \end{aligned}$$

the formula (5.10) with $r=1, \ c_0(\alpha )=-1, x=1$ and $a=1$ can be written as follows:

$$\begin{aligned} \Phi _{0}(x_1,x_2,x_3;1,0,1)&=x_3 \int _0^1 \Biggl \{{s}^{-{\widetilde{\lambda }}_1} (1-s)^{x_2-1} \frac{\Gamma ({\widetilde{\lambda }}_1-{\widetilde{\lambda }}_2)\Gamma (1-x_2)}{\Gamma (1-{\widetilde{\lambda }}_2) \Gamma (1-{\widetilde{\lambda }}_2+x_1)} {}_{2}F_{1} \begin{bmatrix} \begin{matrix} {\widetilde{\lambda }}_2, {\widetilde{\lambda }}_2-x_1 \\ {\widetilde{\lambda }}_2-{\widetilde{\lambda }}_1+1 \end{matrix} ;s \end{bmatrix} \nonumber \\&\quad +s^{-{\widetilde{\lambda }}_2}(1-s)^{x_2-1} \frac{\Gamma ({\widetilde{\lambda }}_2-{\widetilde{\lambda }}_1)\Gamma (1-x_2)}{\Gamma (1-{\widetilde{\lambda }}_1) \Gamma (1-{\widetilde{\lambda }}_1+x_1)} {}_{2}F_{1} \begin{bmatrix} \begin{matrix} {\widetilde{\lambda }}_1, {\widetilde{\lambda }}_1-x_1 \\ {\widetilde{\lambda }}_1-{\widetilde{\lambda }}_2+1 \end{matrix} ;s \end{bmatrix} \Biggr \} ds. \end{aligned}$$

(5.11)

The expression (5.11) is given in [4].

5.2 Classical Limit

Let u be a complex variable with $0<\textrm{Im}\, u<\textrm{Im}\, \tau$. For $-2 \le j \le r-1$, we put

$$\begin{aligned} G_j(k,l,h_1,\dots ,h_r;u,v,x; \tau ):=\sum _{{\varvec{k}} \in I_j(k,l,h_1,\dots ,h_r)} L_{\varvec{k}}(u,v,x; \tau ) \end{aligned}$$

and define the function $\Phi _j(u;\tau )$ by

$$\begin{aligned} \Phi _j(u;\tau )&=\Phi _j(x_1,\ldots ,x_{r+2};u,v,x; \tau ) \\&:=\sum _{k,l,h_1,\dots ,h_r \ge 0} G_j(k,l,h_1,\dots ,h_r ;u,v,x; \tau )\\&\quad \times x_1^{k-l-\sum _{i=1}^r h_i } x_2^{l-h_1} x_3^{h_1-h_2} \times \cdots \times x_{r+1}^{h_{r-1}-h_r} x_{r+2}^{h_r}. \end{aligned}$$

We also write $\Phi _{-2}(u;\tau )$ as $\Phi _{-2}(\tau )$. Then, by Proposition 2.5, we have

$$\begin{aligned} \lim _{q \rightarrow 1}\Phi _j({\widetilde{x}}_1,\ldots ,{\widetilde{x}}_{r+2}; e^{2\pi \sqrt{-1} u}, e^{2\pi \sqrt{-1} v}, q^x;p,q) =\Phi _{j}(x_1,\ldots ,x_{r+2};u,v,x;\tau ), \end{aligned}$$

(5.12)

where ${\widetilde{x}}_1,\ldots ,{\widetilde{x}}_{r+2}$ are parameters defined as (5.5). By proposition 3.3 (2) and (5.12), $\Phi _{r-1}(u;\tau )$ satisfies the following differential equation:

$$\begin{aligned}{} & {} \Biggl \{ \left( x+\frac{1}{2\pi \sqrt{-1}}\frac{d}{du}\right) ^{r+1}-(x_1-x_2 F(u,v;\tau ))\left( x+\frac{1}{2\pi \sqrt{-1}}\frac{d}{du} \right) ^r \\{} & {} \qquad +F(u,v;\tau ) \sum _{j=0}^{r-1} (x_{r+2-j}-x_1 x_{r+1-j})\left( x+\frac{1}{2\pi \sqrt{-1}} \frac{d}{du} \right) ^j \Biggr \} \Phi _{r-1}(u;\tau )\\{} & {} \quad = -x_{r+2} F(u,v;\tau )+\frac{x_1 x_{r+2}}{x_2} \Phi _{-2}(\tau ). \end{aligned}$$

We now put

$$\begin{aligned} K(u,s)&:=(-1)^{-r-1} (1-e^{2\pi \sqrt{-1}(s+u)}) \sum _{i=1}^{r+1}\frac{1}{\prod _{\begin{array}{c} 1 \le l \le r+1 \\ l \ne i \end{array}} (\widetilde{\lambda _l}-{\widetilde{\lambda }}_i)} \\&\quad \times {}_{r+1}{\hat{F}}_{r} \begin{bmatrix} \begin{matrix} {\widetilde{\lambda }}_i+{\widetilde{\rho }}_1, \dots , {\widetilde{\lambda }}_i+{\widetilde{\rho }}_{r+1} \\ {\widetilde{\lambda }}_i-{\widetilde{\lambda }}_1+1, \dots , {\widetilde{\lambda }}_i-{\widetilde{\lambda }}_{r+1}+1 \end{matrix} ;e^{2\pi \sqrt{-1} u} \end{bmatrix} \\&\quad \times {}_{r+1}{\hat{F}}_{r} \begin{bmatrix} \begin{matrix} 1-{\widetilde{\lambda }}_i-{\widetilde{\rho }}_1, \dots , 1-{\widetilde{\lambda }}_i-{\widetilde{\rho }}_{r+1} \\ {\widetilde{\lambda }}_1-{\widetilde{\lambda }}_i+1, \dots , {\widetilde{\lambda }}_{r+1}-{\widetilde{\lambda }}_{i}+1 \end{matrix} ;e^{2\pi \sqrt{-1}(u+s)} \end{bmatrix} \left( -{\varvec{B}}_1(-s,-{\widetilde{\lambda }}_i)+\frac{1}{{\widetilde{\lambda }}_i} \right) , \\ {\widetilde{\xi }}_j(u)&:= {\left\{ \begin{array}{ll} x_1-x_2 F(a,\alpha ;p) &{} (j=0) \\ (x_1x_{j+1}-x_{j+2})F(u,v;\tau ) &{} (1 \le j \le r) \\ -x_{r+2}F(u,v;\tau ) +\dfrac{x_1x_{r+2}}{x_2}\Phi _{-2}(\tau )&{} (j=r+1), \end{array}\right. } \end{aligned}$$

where ${\widetilde{\rho }}_i$ and ${\widetilde{\lambda }}_i \ (i=1,\ldots ,r+1)$ are parameters defined as (5.8) and (5.9).

For integers k, j, n with $0 \le k \le r, \ 0 \le j \le r$ and $n \ge 1$, we define the functions $K_{i,j}(u,s;\tau )$ and $K_{j}^{(n)}(u,s;\tau )$ as follows:

$$\begin{aligned}&K_{i,j}(u,s;\tau )\\ {}&\quad := {\left\{ \begin{array}{ll} -x_2 K(u,s){\widetilde{F}}(u+s,v;\tau ) &{} (i=0, \ j=r) \vspace{1mm} \\ (x_1 x_{r+1-j}-x_{r+2-j})K(u,s){\widetilde{F}}(u+s,v;\tau ) &{} (i=0, \ 0 \le j \le r-1) \vspace{1mm} \\ \dfrac{1}{2\pi \sqrt{-1}} \dfrac{d}{du} K_{i-1,0}(u,s;\tau )+K_{i-1,r}(u,s;\tau ){\widetilde{\xi }}_r(u+s) &{} (i \ge 1, \ j=0) \vspace{1mm} \\ \dfrac{1}{2 \pi \sqrt{-1}} \dfrac{d}{du}K_{i-1,j}(u,s;\tau )&{}\\ \qquad +K_{i-1,j-1}(u,s;\tau )+K_{i-1,r}(u,s;\tau ){\widetilde{\xi }}_{r-j}(s) &{} (i \ge 1, \ 1 \le j \le r), \end{array}\right. } \\&K_j^{(n)}(u,s;\tau )\\ {}&\quad := {\left\{ \begin{array}{ll} K_{0,j}(u,s;\tau ) &{} (n=1) \vspace{1mm} \\ \displaystyle \sum _{i=0}^r \int _{0}^1 K_{i}^{(n-1)}(u,s';\tau )K_{i,j}(u+s',s-s';\tau ) \, ds' &{} (n \ge 2). \end{array}\right. } \end{aligned}$$

It is observed that $K_{j}^{(n)}(a,b;p,q)$ is a power series in p expressed as

$$\begin{aligned} K_j^{(n)}(u,s;\tau )=\sum _{m=n}^{\infty } k_{j,m}^{(n)}(u,s) p^m. \end{aligned}$$

We also put

$$\begin{aligned} \Gamma _{j}(u,s;\tau ):=\sum _{m=1}^{\infty } \left\{ \sum _{n=1}^{m} k_{j,m}^{(n)} (u,s) \right\} p^m \in {{\mathbb {C}}}[[p]] \end{aligned}$$

for $j=0,\dots , r$.

For integers i with $0 \le i \le r$, we define the functions $h_i (u;\tau )$ by

$$\begin{aligned} h_i(u;\tau ):=&{\left\{ \begin{array}{ll} \displaystyle \int _0^1 K(u,s) {\widetilde{\xi }}_{r+1} (u+s) \, ds&{} (i=0) \vspace{1mm} \\ \displaystyle \int _0^1 K_{i-1,r}(u,s;\tau ) {\widetilde{\xi }}_{r+1}(u+s) \, ds+\left( x+\frac{1}{2\pi \sqrt{-1}} \frac{d}{du} \right) h_{i-1}(u;\tau ) &{} (1 \le i \le r). \end{array}\right. } \end{aligned}$$

By Theorem 4.9, we obtain the following theorem:

Theorem 5.6

We consider $\Phi _{r-1}(u;\tau )$ and $h_i(u;\tau ) \ (i=0,\ldots ,r)$ as formal power series in p. Then we have the following equality:

$$\begin{aligned} \Phi _{r-1}(u;\tau )=\sum _{i=0}^r \int _0^1 \Gamma _i(u,s;\tau )h_i(u+s;\tau ) \, ds+h_0(u;\tau ). \end{aligned}$$