Connection Problem for an Extension of $q$-Hypergeometric Systems

Connection Problem for an Extension
of $\bm{q}$-Hypergeometric Systems

Takahiko NOBUKAWA

T. Nobukawa

Department of Mathematics, Kobe University, Rokko, Kobe 657-8501, Japan
<EMAIL_ADDRESS>
Received March 19, 2021, in final form October 14, 2022; Published online
October 21, 2022

We give an example of solutions of the connection problem associated with a
certain system of linear $q$-difference equations recently introduced by Park.
The result contains a connection formulas of the $q$-Lauricella hypergeometric
function $\varphi_{D}$ and those of the $q$-generalized hypergeometric
function ${}_{N+1}\varphi_{N}$ as special cases.

$q$-difference equations; $q$-hypergeometric series; connection matrices;
Yang–Baxter equation

33D70; 39A13

##  1 Introduction

In [24], a hypergeometric function $F_{L,N}$, which is a certain
generalization of hypergeometric functions, was defined by Tsuda. He also
obtained a Hamiltonian system $\mathcal{H}_{L,N}$, which describes an
isomonodromic deformation of an $L\times L$ Fuchsian system on
$\mathbb{P}^{1}$ with $N+3$ regular singularities, and which has particular
solutions in terms of the function $F_{L,N}$. In [20, 21], a $q$-analog of
Tsuda’s result was obtained by Park. Namely, she defined a $q$-hypergeometric
function $\mathcal{F}_{N,M}$, which is given by (2.2) below, and a system
$\mathcal{P}_{N,M}$ as a $q$-analog of the function $F_{N+1,M}$ and the system
$\mathcal{H}_{N+1,M}$, respectively. The function $\mathcal{F}_{N,M}$
converges locally and satisfies linear $q$-difference equations, given by
(2.5) and (2.6) below. Also the function $\mathcal{F}_{N,M}$ has the Euler-
type integral representation, given by (2.4). Thus the function
$\mathcal{F}_{N,M}$ must be continued analytically to $\mathbb{C}^{M}$.

Our aim is to solve the connection problem related with the function
$\mathcal{F}_{N,M}$, that is to give connection matrices among fundamental
solutions of the $q$-difference equations (2.5) and (2.6). The main result is
Theorem 3.8 in Section 3.2. Proposition 3.6 gives fundamental solutions for
the equation, and Theorem 3.8 gives connection matrices among these
fundamental solutions. Our method to calculate the connection matrices is to
use a connection formula of the generalized $q$-hypergeometric function
${}_{N+1}\varphi_{N}$ many times. The connection matrices are given by the
product of some matrices.

In the general theory of $q$-difference systems in several variables, local
solutions should be characterized by the asymptotic behavior near the
singularity in some prescribed sector. For the general theory of
$q$-difference systems in several variables, see [2, 3]. Our solutions are
defined in the region $\\{|t_{1}|\ll\cdots\ll|t_{L}|\ll
1\ll|t_{L+1}|\ll\cdots\ll|t_{M}|\\}$ where $0\leq L\leq M$, and characterized
by the asymptotic behavior of the following form

$\displaystyle t_{1}^{\delta_{1}}\cdots t_{M}^{\delta_{M}}(1+O(||x||)),$

at
$x=(t_{1}/t_{2},\dots,t_{L-1}/t_{L},t_{L},1/t_{L+1},t_{L+1}/t_{L+2},\dots,t_{M-1}/t_{M})=(0,\dots,0)$.
Here $\delta=(\delta_{1},\dots,\allowbreak\delta_{M})$ is a suitable
parameter. For more details, see Definition 3.1, Proposition 3.2, Remark 3.4.

The function $\mathcal{F}_{N,M}$ is a generalization both of the
$q$-Lauricella function $\varphi_{D}$ and of the generalized
$q$-hypergeometric function ${}_{N+1}\varphi_{N}$. Thus our results contain
solving the connection problem related with $\varphi_{D}$ and also related
with ${}_{N+1}\varphi_{N}$. The equality (3.12) below can be considered as a
part of the connection relations for ${}_{N+1}\varphi_{N}$, which was first
studied by Thomae [23] and was proved by Watson [25]. Also the $q$-Lauricella
function has a Jackson integral representation of Jordan–Pochhammer type, and
the connection problem related with the Jackson integral of Jordan–Pochhammer
type was solved by Mimachi [16]. In differential case, fundamental solutions
of the equation related with the Lauricella function $F_{D}$ were obtained by
Gelfand–Kapranov–Zelevinsky [9] as an application of the theory of the GKZ
hypergeometric function. Our solutions are generalizations of the
Gelfand–Kapranov–Zelevinsky’s solutions (see Remark 3.3).

The contents of this paper are as follows. In Section 2, we give notations and
the properties of the function $\mathcal{F}_{N,M}$ and the system of
$q$-difference equations satisfied by $\mathcal{F}_{N,M}$. In Section 3, we
solve the connection problem of the system. In Section 3.1, we show
fundamental solutions of the $q$-difference system. In Section 3.2, we give
the matrices which connect the fundamental solution with the other fundamental
solution. In Section 4, we obtain an elliptic solution of the Yang–Baxter
equation as an application of Section 3.2.

##  2 Preliminaries

In this paper, we fix $q\in\mathbb{C}$ with $0<|q|<1$. We use the following
notations throughout the paper:

$\displaystyle(a)_{\infty}=\prod_{k=0}^{\infty}\big{(}1-aq^{k}\big{)},\qquad(a)_{m}=\frac{(a)_{\infty}}{(aq^{m})_{\infty}},$
$\displaystyle(a_{1},\dots,a_{n})_{m}=(a_{1})_{m}\cdots(a_{n})_{m},\qquad\theta(x)=(x,q/x)_{\infty}.$

The symmetric group of degree $n$ is denoted by $\mathfrak{S}_{n}$. In
addition, ${}^{\rm T}A$ is the transpose of a matrix $A$. We also use the
notation $abc/defg$ for the fraction $(abc)/(defg)$. Moreover, for a multi-
index $m=(m_{1},\dots,m_{M})$, we often use the notations

$\displaystyle|m|=\sum_{i=1}^{M}m_{i},\qquad
m(l)=\sum_{i=1}^{l}m_{i}-\sum_{i=l+1}^{M}m_{i},$ (2.1)

where $0\leq l\leq M$. Here, the empty sum is considered to be 0.

######  Definition 2.1 ([20, Definition 2.1]).

We assume $c_{j}\notin q^{\mathbb{Z}_{\leq 0}}=\\{q^{n};\ n=0,-1,-2,\dots\\}$
for $1\leq j\leq N$. We define the function $\mathcal{F}_{N,M}$ as

$\displaystyle\mathcal{F}_{N,M}\left(\\!\begin{matrix}\\{a_{j}\\}_{1\leq j\leq
N},\\{b_{i}\\}_{1\leq i\leq M}\\\ \\{c_{j}\\}_{1\leq j\leq
N}\end{matrix};\\{t_{i}\\}_{1\leq i\leq
M}\\!\right)=\sum_{m_{1},\dots,m_{M}\geq
0}\prod_{j=1}^{N}\frac{(a_{j})_{|m|}}{(c_{j})_{|m|}}\prod_{i=1}^{M}\frac{(b_{i})_{m_{i}}}{(q)_{m_{i}}}\prod_{i=1}^{M}t_{i}^{m_{i}}.\\!\\!\\!$
(2.2)

The series $\mathcal{F}_{N,M}$ converges in the region $|t_{i}|<1$.

When $N=1$ or $M=1$, we have

$\displaystyle\mathcal{F}_{1,M}\left(\begin{matrix}a,\\{b_{i}\\}_{1\leq i\leq
M}\\\ c\end{matrix};\\{t_{i}\\}_{1\leq i\leq
M}\right)=\varphi_{D}\left(\begin{matrix}a,\\{b_{i}\\}_{1\leq i\leq M}\\\
c\end{matrix};\\{t_{i}\\}_{1\leq i\leq M}\right),$
$\displaystyle\mathcal{F}_{N,1}\left(\begin{matrix}\\{a_{j}\\}_{1\leq j\leq
N},b\\\ \\{c_{j}\\}_{1\leq j\leq
N}\end{matrix};t\right)={}_{N+1}\varphi_{N}\left(\begin{matrix}\\{a_{j}\\}_{1\leq
j\leq N},\,b\\\ \\{c_{j}\\}_{1\leq j\leq N}\end{matrix};t\right),$

where $\varphi_{D}$ and ${}_{N+1}\varphi_{N}$ are the $q$-Lauricella function
and the generalized $q$-hypergeometric function defined by

$\displaystyle\varphi_{D}\left(\begin{matrix}a,\\{b_{i}\\}_{1\leq i\leq M}\\\
c\end{matrix};\\{t_{i}\\}_{1\leq i\leq M}\right)=\sum_{m_{1},\dots,m_{M}\geq
0}\frac{(a)_{|m|}}{(c)_{|m|}}\prod_{i=1}^{M}\frac{(b_{i})_{m_{i}}}{(q)_{m_{i}}}\prod_{i=1}^{M}t_{i}^{m_{i}},$
$\displaystyle{}_{N+1}\varphi_{N}\left(\begin{matrix}\\{a_{j}\\}_{1\leq j\leq
N},\,b\\\ \\{c_{j}\\}_{1\leq j\leq N}\end{matrix};t\right)=\sum_{m\geq
0}\prod_{j=1}^{N}\frac{(a_{j})_{m}}{(c_{j})_{m}}\cdot\frac{(b)_{m}}{(q)_{m}}t^{m}.$

######  Proposition 2.2 ([20, Proposition 2.1]).

The series $\mathcal{F}_{N,M}$ satisfies the relation

$\displaystyle\mathcal{F}_{N,M}\left(\begin{matrix}\\{a_{j}\\}_{1\leq j\leq
N},\\{b_{i}\\}_{1\leq i\leq M}\\\ \\{c_{j}\\}_{1\leq j\leq
N}\end{matrix};\\{t_{i}\\}_{1\leq i\leq M}\right)$
$\displaystyle\qquad{}=\prod_{j=1}^{N}\frac{(a_{j})_{\infty}}{(c_{j})_{\infty}}\prod_{i=1}^{M}\frac{(b_{i}t_{i})_{\infty}}{(t_{i})_{\infty}}\cdot\mathcal{F}_{M,N}\left(\begin{matrix}\\{t_{i}\\}_{1\leq
i\leq M},\\{c_{j}/a_{j}\\}_{1\leq j\leq N}\\\ \\{b_{i}t_{i}\\}_{1\leq i\leq
M}\end{matrix};\\{a_{j}\\}_{1\leq j\leq N}\right).$ (2.3)

######  Remark 2.3.

When $N=1$, the relation (2.3) reduces to

$\displaystyle\varphi_{D}\left(\begin{matrix}a,\\{b_{i}\\}_{1\leq i\leq M}\\\
c\end{matrix};\\{t_{i}\\}_{1\leq i\leq
M}\right)=\frac{(a)_{\infty}}{(c)_{\infty}}\prod_{i=1}^{M}\frac{(b_{i}t_{i})_{\infty}}{(t_{i})_{\infty}}\cdot{}_{M+1}\varphi_{M}\left(\begin{matrix}\\{t_{i}\\}_{1\leq
i\leq M},c/a\\\ \\{b_{i}t_{i}\\}_{1\leq i\leq M}\end{matrix};a\right),$

which is relation (4.1) of [1], a generalization of Heine’s transformation for
${}_{2}\varphi_{1}$ [1, equation (1.4.1)].

It is well known that Heine’s transformation can be interpreted as a
$q$-integral form, so called a Jackson integral, of ${}_{2}\varphi_{1}$.
Similarly, the relation (2.3) can be rewritten in the Jackson integral form as
follows.

######  Corollary 2.4 ([20, Corollary 2.1]).

With $a_{j}=q^{\alpha_{j}}$, the relation (2.3) can be rewritten as

$\displaystyle\mathcal{F}_{N,M}\left(\begin{matrix}\\{q^{\alpha_{j}}\\}_{1\leq
j\leq N},\\{b_{i}\\}_{1\leq i\leq M}\\\ \\{c_{j}\\}_{1\leq j\leq
N}\end{matrix};\\{t_{i}\\}_{1\leq i\leq M}\right)$
$\displaystyle\qquad{}=\prod_{i=1}^{N}\frac{(q^{\alpha_{i}},c_{i}/q^{\alpha_{i}})_{\infty}}{(c_{i},q)_{\infty}}\cdot\int_{0}^{1}\cdots\int_{0}^{1}\prod_{j=1}^{N}\left\\{\frac{z_{j}^{\alpha_{j}-1}}{1-q}\frac{(qz_{j})_{\infty}}{(c_{j}z_{j}/q^{\alpha_{j}})_{\infty}}\right\\}$
$\displaystyle\qquad\quad{}\times\prod_{i=1}^{M}\frac{(b_{i}t_{i}z_{1}\cdots
z_{N})_{\infty}}{(t_{i}z_{1}\cdots z_{N})_{\infty}}{\rm d}_{q}z_{1}\cdots{\rm
d}_{q}z_{N},$ (2.4)

where the Jackson integral is defined as

$\displaystyle\int_{0}^{c}f(z){\rm d}_{q}z=c(1-q)\sum_{m\geq
0}f(cq^{m})q^{m},$

for $c\in\mathbb{C}$.

######  Proposition 2.5 ([20, Proposition 2.2]).

The series

$\mathcal{F}=\mathcal{F}_{N,M}\left(\begin{matrix}\\{a_{j}\\}_{1\leq j\leq
N},\\{b_{i}\\}_{1\leq i\leq M}\\\ \\{c_{j}\\}_{1\leq j\leq
N}\end{matrix};\\{t_{i}\\}_{1\leq i\leq M}\right)$

satisfies the $q$-difference equations

$\displaystyle\Biggl{\\{}t_{s}\prod_{j=1}^{N}(1-a_{j}T)\cdot(1-b_{s}T_{s})-\prod_{j=1}^{N}\big{(}1-c_{j}q^{-1}T\big{)}\cdot(1-T_{s})\Biggr{\\}}\mathcal{F}=0,\qquad
1\leq s\leq M,$ (2.5)
$\displaystyle\\{t_{r}(1-b_{r}T_{r})(1-T_{s})-t_{s}(1-b_{s}T_{s})(1-T_{r})\\}\mathcal{F}=0,\qquad
1\leq r<s\leq M,$ (2.6)

where $T_{s}$ is the $q$-shift operator $T_{s}f(t)=f(\dots,qt_{s},\dots)$ for
the variable $t_{s}$ and $T=\prod_{s=1}^{M}T_{s}$.

In this paper, we use the notation $E_{N,M}$ for the system of $q$-difference
equations (2.5) and (2.6).

######  Theorem 2.6 ([21, Theorem 4.1]).

The rank of $E_{N,M}$ is $NM+1$.

##  3 Connection problem of the $\bm{q}$-difference system $\bm{E_{N,M}}$

In this section, we consider the connection problem of the system $E_{N,M}$.
First, we show fundamental solutions of the system $E_{N,M}$, which converge
locally. Second, we calculate connection matrices. The connection formula can
be calculated by using Thomae–Watson’s formula (3.12) many times. We suppose
$a_{j}=q^{\alpha_{j}}$, $b_{i}=q^{\beta_{i}}$ and $c_{j}=q^{\gamma_{j}}$,
$1\leq i\leq M$, $1\leq j\leq N$.

###  3.1 Solutions of the $\bm{q}$-difference system $\bm{E_{N,M}}$

In this subsection, we show fundamental solutions of the $q$-difference system
$E_{N,M}$ which converge locally.

######  Definition 3.1.

We assume

$\displaystyle a_{j}/a_{k},c_{j}/c_{k},a_{j}/b_{i}\cdots
b_{M},c_{k}/b_{i}\cdots b_{M}\notin q^{\mathbb{Z}},$ (3.1)

for $1\leq i\leq M+1$, $1\leq j\neq k\leq N$. Here, $q^{D}=\\{q^{x};\,x\in
D\\}$. We define series $\mathcal{F}_{N,M}^{L}$, $\mathcal{F}_{N,M}^{L;k,l}$
and $\mathcal{G}_{N,M}^{L;k,l^{\prime}}$ as

$\displaystyle\mathcal{F}_{N,M}^{L}\left(\begin{matrix}\\{a_{j}\\}_{1\leq
j\leq N},\\{b_{i}\\}_{1\leq i\leq M}\\\ \\{c_{j}\\}_{1\leq j\leq
N}\end{matrix};\\{t_{i}\\}_{1\leq i\leq M}\right)$
$\displaystyle\qquad{}=\sum_{m_{1},\dots,m_{M}\geq
0}\prod_{j=1}^{N}\frac{(a_{j}/b_{L+1}\cdots
b_{M})_{m(L)}}{(c_{j}/b_{L+1}\cdots
b_{M})_{m(L)}}\prod_{i=1}^{M}\frac{(b_{i})_{m_{i}}}{(q)_{m_{i}}}\prod_{i=1}^{L}t_{i}^{m_{i}}\prod_{i=L+1}^{M}\left(\frac{q}{b_{i}t_{i}}\right)^{m_{i}},$
(3.2)
$\displaystyle\mathcal{F}_{N,M}^{L;k,l}\left(\begin{matrix}\\{a_{j}\\}_{1\leq
j\leq N},\\{b_{i}\\}_{1\leq i\leq N}\\\ \\{c_{j}\\}_{1\leq j\leq
N}\end{matrix};\\{t_{i}\\}_{1\leq i\leq M}\right)=\sum_{m_{1},\dots,m_{M}\geq
0}\Biggl{\\{}\prod_{j=1}^{N}\frac{(qa_{k}/c_{j})_{m_{L+1}}}{(qa_{k}/a_{j})_{m_{L+1}}}$
$\displaystyle\qquad{}\times\prod_{i=1}^{L}\frac{(b_{i})_{m_{i}}}{(q)_{m_{i}}}\prod_{i=L+1}^{l-1}\frac{(b_{i})_{m_{i+1}}}{(q)_{m_{i+1}}}\prod_{i=l+1}^{M}\frac{(b_{i})_{m_{i}}}{(q)_{m_{i}}}\frac{(a_{k}/b_{l+1}\cdots
b_{M})_{m(l)}}{(qa_{k}/b_{l}\cdots b_{M})_{m(l)}}$
$\displaystyle\qquad{}\times\prod_{i=1}^{L}\left(\frac{qt_{i}}{b_{l}t_{l}}\right)^{m_{i}}\prod_{i=L+1}^{l-1}\left(\frac{qt_{i}}{b_{l}t_{l}}\right)^{m_{i+1}}\prod_{i=l+1}^{M}\left(\frac{b_{l}t_{l}}{b_{i}t_{i}}\right)^{m_{i}}\left(\prod_{j=1}^{N}\frac{c_{j}}{a_{j}}\cdot\frac{q}{b_{l}t_{l}}\right)^{m_{L+1}}\Biggr{\\}},$
(3.3)
$\displaystyle\mathcal{G}_{N,M}^{L;k,l^{\prime}}\left(\begin{matrix}\\{a_{j}\\}_{1\leq
j\leq N},\\{b_{i}\\}_{1\leq i\leq M}\\\ \\{c_{j}\\}_{1\leq j\leq
N}\end{matrix};\\{t_{i}\\}_{1\leq i\leq M}\right)=\sum_{m_{1},\dots,m_{M}\geq
0}\Biggl{\\{}\prod_{j=1}^{N}\frac{(qa_{j}/c_{k})_{m_{L}}}{(qc_{j}/c_{k})_{m_{L}}}$
$\displaystyle\qquad{}\times\prod_{i=1}^{l^{\prime}-1}\frac{(b_{i})_{m_{i}}}{(q)_{m_{i}}}\prod_{i=l^{\prime}+1}^{L}\frac{(b_{i})_{m_{i-1}}}{(q)_{m_{i-1}}}\prod_{i=L+1}^{M}\frac{(b_{i})_{m_{i}}}{(q)_{m_{i}}}\frac{(c_{k}/qb_{l^{\prime}+1}\cdots
b_{M})_{m(l^{\prime}-1)}}{(c_{k}/b_{l^{\prime}}\cdots
b_{M})_{m(l^{\prime}-1)}}$
$\displaystyle\qquad\times\prod_{i=1}^{l^{\prime}-1}\left(\frac{qt_{i}}{b_{l^{\prime}}t_{l^{\prime}}}\right)^{m_{i}}\prod_{i=l^{\prime}+1}^{L}\left(\frac{b_{l^{\prime}}t_{l^{\prime}}}{b_{i}t_{i}}\right)^{m_{i-1}}\prod_{i=L+1}^{M}\left(\frac{b_{l^{\prime}}t_{l^{\prime}}}{b_{i}t_{i}}\right)^{m_{i}}\cdot\left(\frac{b_{l^{\prime}}t_{l^{\prime}}}{q}\right)^{m_{L}}\Biggr{\\}},$
(3.4)

where $0\leq L\leq M$, $1\leq k\leq N$, $L+1\leq l\leq M$ and $1\leq
l^{\prime}\leq L$. Here, as mentioned in preliminaries (2.1),

$\displaystyle m(l)=\sum_{i=1}^{l}m_{i}-\sum_{i=l+1}^{M}m_{i}.$

We find that the series (3.2) converges in

$\left\\{|t_{i}|<1,\,1\leq i\leq L,\,\left|\frac{c_{1}\cdots
c_{N}q}{a_{1}\cdots a_{N}b_{i}t_{i}}\right|<1,\,L+1\leq i\leq M\right\\},$

the series (3.3) converges in

$\left\\{\left|\frac{c_{1}\cdots c_{N}q}{a_{1}\cdots
a_{N}b_{l}t_{l}}\right|<1,\,\left|\frac{qt_{i}}{b_{l}t_{l}}\right|<1,\,1\leq
i\leq l-1,\,\left|\frac{qt_{l}}{b_{i}t_{i}}\right|<1,\,l+1\leq i\leq
M\right\\},$

and the series (3.4) converges in

$\left\\{|t_{l}|<1,\,\left|\frac{qt_{i}}{b_{l}t_{l}}\right|<1,\,1\leq i\leq
l-1,\,\left|\frac{qt_{l}}{b_{i}t_{i}}\right|<1,\,l+1\leq i\leq M\right\\}.$

######  Proposition 3.2.

We assume the conditions (3.1). For $0\leq L\leq M$, the functions

$\displaystyle\prod_{i=L+1}^{M}t_{i}^{-\beta_{i}}\cdot\mathcal{F}_{N,M}^{L}\left(\begin{matrix}\\{a_{j}\\}_{1\leq
j\leq N},\\{b_{i}\\}_{1\leq i\leq M}\\\ \\{c_{j}\\}_{1\leq j\leq
N}\end{matrix};\\{t_{i}\\}_{1\leq i\leq M}\right),$ (3.5) $\displaystyle
t_{l}^{1+\sum_{i=l+1}^{M}\beta_{i}-\gamma_{k}}\prod_{i=l+1}^{M}t_{i}^{-\beta_{i}}\cdot\mathcal{G}_{N,M}^{L;k,l}\left(\begin{matrix}\\{a_{j}\\}_{1\leq
j\leq N},\\{b_{i}\\}_{1\leq i\leq M}\\\ \\{c_{j}\\}_{1\leq j\leq
N}\end{matrix};\\{t_{i}\\}_{1\leq i\leq M}\right),$ (3.6) $\displaystyle\hskip
227.62204pt1\leq k\leq N,\ 1\leq l\leq L,$ $\displaystyle
t_{l}^{-\alpha_{k}+\sum_{i=l+1}^{M}\beta_{i}}\prod_{i=l+1}^{M}t_{i}^{-\beta_{i}}\cdot\mathcal{F}_{N,M}^{L;k,l}\left(\begin{matrix}\\{a_{j}\\}_{1\leq
j\leq N},\\{b_{i}\\}_{1\leq i\leq M}\\\ \\{c_{j}\\}_{1\leq j\leq
N}\end{matrix};\\{t_{i}\\}_{1\leq i\leq M}\right),$ (3.7) $\displaystyle\hskip
227.62204pt1\leq k\leq N,\ L+1\leq l\leq M,$

satisfy the $q$-difference system $E_{N,M}$.

###### Proof.

We can check them easily. Here, we check that the function (3.5) satisfies the
$q$-difference equations (2.5). First, we have

$\displaystyle T_{i}t_{i}^{\alpha}=q^{\alpha}t_{i}^{\alpha}T_{i},$

as an operator. Thus we obtain

$\displaystyle\left(\prod_{i=L+1}^{M}t_{i}^{-\beta_{i}}\right)^{-1}\left\\{t_{s}\prod_{j=1}^{N}(1-a_{j}T)\cdot(1-b_{s}T_{s})-\prod_{j=1}^{N}\big{(}1-c_{j}q^{-1}T\big{)}\cdot(1-T_{s})\right\\}\prod_{i=L+1}^{M}t_{i}^{-\beta_{i}}$
$\displaystyle=\begin{cases}\displaystyle\left\\{t_{s}\prod_{j=1}^{N}\left(1-\frac{a_{j}}{b_{L+1}\cdots
b_{M}}T\right)\cdot(1-b_{s}T_{s})-\prod_{j=1}^{N}\left(1-\frac{c_{j}q^{-1}}{b_{L+1}\cdots
b_{M}}T\right)\cdot(1-T_{s})\right\\},\\\ \hskip 312.9803pt1\leq s\leq L,\\\
\displaystyle\left\\{t_{s}\prod_{j=1}^{N}\left(1-\frac{a_{j}}{b_{L+1}\cdots
b_{M}}T\right)\cdot(1-T_{s})-\prod_{j=1}^{N}\left(1-\frac{c_{j}q^{-1}}{b_{L+1}\cdots
b_{M}}T\right)\cdot\left(1-\frac{1}{b_{s}}T_{s}\right)\right\\},\\\ \hskip
312.9803ptL+1\leq s\leq M,\end{cases}$

as an operator. For $1\leq s\leq L$, we have

$\displaystyle t_{s}\prod_{j=1}^{N}\left(1-\frac{a_{j}}{b_{L+1}\cdots
b_{M}}T\right)\cdot(1-b_{s}T_{s})\cdot\mathcal{F}_{N,M}^{L}\left(\begin{matrix}\\{a_{j}\\}_{1\leq
j\leq N},\\{b_{i}\\}_{1\leq i\leq M}\\\ \\{c_{j}\\}_{1\leq j\leq
N}\end{matrix};\\{t_{i}\\}_{1\leq i\leq M}\right)$
$\displaystyle=t_{s}\prod_{j=1}^{N}\left(1-\frac{a_{j}}{b_{L+1}\cdots
b_{M}}T\right)\cdot(1-b_{s}T_{s})$
$\displaystyle\qquad{}\times\sum_{m_{1},\dots,m_{M}\geq
0}\prod_{j=1}^{N}\frac{(a_{j}/b_{L+1}\cdots
b_{M})_{m(L)}}{(c_{j}/b_{L+1}\cdots
b_{M})_{m(L)}}\prod_{i=1}^{M}\frac{(b_{i})_{m_{i}}}{(q)_{m_{i}}}\prod_{i=1}^{L}t_{i}^{m_{i}}\prod_{i=L+1}^{M}\left(\frac{q}{b_{i}t_{i}}\right)^{m_{i}}$
$\displaystyle=\sum_{m_{1},\dots,m_{M}\geq
0}t_{s}\prod_{j=1}^{N}\left(1-\frac{a_{j}q^{m(L)}}{b_{L+1}\cdots
b_{M}}\right)\cdot\big{(}1-b_{s}q^{m_{s}}\big{)}$
$\displaystyle\qquad{}\times\prod_{j=1}^{N}\frac{(a_{j}/b_{L+1}\cdots
b_{M})_{m(L)}}{(c_{j}/b_{L+1}\cdots
b_{M})_{m(L)}}\prod_{i=1}^{M}\frac{(b_{i})_{m_{i}}}{(q)_{m_{i}}}\prod_{i=1}^{L}t_{i}^{m_{i}}\prod_{i=L+1}^{M}\left(\frac{q}{b_{i}t_{i}}\right)^{m_{i}}$
$\displaystyle=\sum_{m_{1},\dots,m_{M}\geq
0}\prod_{j=1}^{N}\frac{(a_{j}/b_{L+1}\cdots
b_{M})_{m(L)+1}}{(c_{j}/b_{L+1}\cdots
b_{M})_{m(L)}}\prod_{\begin{subarray}{c}1\leq i\leq M\\\ i\neq
s\end{subarray}}\frac{(b_{i})_{m_{i}}}{(q)_{m_{i}}}$
$\displaystyle\qquad{}\times\frac{(b_{s})_{m_{s}+1}}{(q)_{m_{s}}}\prod_{\begin{subarray}{c}1\leq
i\leq L\\\ i\neq s\end{subarray}}t_{i}^{m_{i}}\cdot
t_{s}^{m_{s}+1}\prod_{i=L+1}^{M}\left(\frac{q}{b_{i}t_{i}}\right)^{m_{i}}$
$\displaystyle=\sum_{\begin{subarray}{c}m_{1},\dots,m_{s-1},m_{s+1},\dots,m_{M}\geq
0\end{subarray}}\sum_{m_{s}\geq-1}\Biggl{\\{}\prod_{j=1}^{N}\frac{(a_{j}/b_{L+1}\cdots
b_{M})_{m(L)+1}}{(c_{j}/b_{L+1}\cdots
b_{M})_{m(L)+1}}\prod_{j=1}^{N}\left(1-\frac{c_{j}q^{-1}q^{m(L)+1}}{b_{L+1}\cdots
b_{M}}\right)$ $\displaystyle\qquad\times\prod_{\begin{subarray}{c}1\leq i\leq
M\\\ i\neq
s\end{subarray}}\frac{(b_{i})_{m_{i}}}{(q)_{m_{i}}}\cdot\frac{(b_{s})_{m_{s}+1}}{(q)_{m_{s}+1}}\big{(}1-q^{m_{s}+1}\big{)}\prod_{\begin{subarray}{c}1\leq
i\leq L\\\ i\neq s\end{subarray}}t_{i}^{m_{i}}\cdot
t_{s}^{m_{s}+1}\prod_{i=L+1}^{M}\left(\frac{q}{b_{i}t_{i}}\right)^{m_{i}}\Biggr{\\}}$
$\displaystyle=\sum_{m_{1},\dots,m_{M}\geq
0}\prod_{j=1}^{N}\left(1-\frac{c_{j}q^{-1}q^{m(L)}}{b_{L+1}\cdots
b_{M}}\right)\cdot\big{(}1-q^{m_{s}}\big{)}$
$\displaystyle\qquad{}\times\prod_{j=1}^{N}\frac{(a_{j}/b_{L+1}\cdots
b_{M})_{m(L)}}{(c_{j}/b_{L+1}\cdots
b_{M})_{m(L)}}\prod_{i=1}^{M}\frac{(b_{i})_{m_{i}}}{(q)_{m_{i}}}\prod_{i=1}^{L}t_{i}^{m_{i}}\prod_{i=L+1}^{M}\left(\frac{q}{b_{i}t_{i}}\right)^{m_{i}}$
$\displaystyle=\prod_{n=1}^{N}\left(1-\frac{c_{n}q^{-1}T}{b_{L+1}\cdots
b_{M}}\right)\cdot(1-T_{s})\cdot\mathcal{F}_{N,M}^{L}\left(\begin{matrix}\\{a_{j}\\}_{1\leq
j\leq N},\\{b_{i}\\}_{1\leq i\leq M}\\\ \\{c_{j}\\}_{1\leq j\leq
N}\end{matrix};\\{t_{i}\\}_{1\leq i\leq M}\right).$

For $L+1\leq s\leq M$, we have

$\displaystyle t_{s}\prod_{j=1}^{N}\left(1-\frac{a_{j}}{b_{L+1}\cdots
b_{M}}T\right)\cdot(1-T_{s})\cdot\mathcal{F}_{N,M}^{L}\left(\begin{matrix}\\{a_{j}\\}_{1\leq
j\leq N},\\{b_{i}\\}_{1\leq i\leq M}\\\ \\{c_{j}\\}_{1\leq j\leq
N}\end{matrix};\\{t_{i}\\}_{1\leq i\leq M}\right)$
$\displaystyle=\sum_{m_{1},\dots,m_{M}\geq
0}t_{s}\prod_{j=1}^{N}\left(1-\frac{a_{j}q^{m(L)}}{b_{L+1}\cdots
b_{M}}\right)\cdot\big{(}1-q^{-m_{s}}\big{)}$
$\displaystyle\qquad{}\times\prod_{j=1}^{N}\frac{(a_{j}/b_{L+1}\cdots
b_{M})_{m(L)}}{(c_{j}/b_{L+1}\cdots
b_{M})_{m(L)}}\prod_{i=1}^{M}\frac{(b_{i})_{m_{i}}}{(q)_{m_{i}}}\prod_{i=1}^{L}t_{i}^{m_{i}}\prod_{i=L+1}^{M}\left(\frac{q}{b_{i}t_{i}}\right)^{m_{i}}$
$\displaystyle=\sum_{{m_{1},\dots,m_{s-1},m_{s+1},\dots,m_{M}\geq
0}}\sum_{m_{s}\geq 1}\Biggl{\\{}\prod_{j=1}^{N}\frac{(a_{j}/b_{L+1}\cdots
b_{M})_{m(L)+1}}{(c_{j}/b_{L+1}\cdots
b_{M})_{m(L)+1}}\prod_{j=1}^{N}\left(1-\frac{c_{j}q^{-1}q^{m(L)+1}}{b_{L+1}\cdots
b_{M}}\right)$ $\displaystyle\qquad{}\times\prod_{\begin{subarray}{c}1\leq
i\leq M\\\ i\neq
s\end{subarray}}\\!\frac{(b_{i})_{m_{i}}}{(q)_{m_{i}}}\cdot\frac{(b_{s})_{m_{s}-1}}{(q)_{m_{s}-1}}\cdot\left(1-\frac{q^{-m_{s}+1}}{b_{s}}\right)\prod_{i=1}^{L}t_{i}^{m_{i}}\\!\prod_{\begin{subarray}{c}L+1\leq
i\leq M\\\ i\neq
s\end{subarray}}\\!\left(\frac{q}{b_{i}t_{i}}\right)^{m_{i}}\cdot\left(\frac{q}{b_{s}t_{s}}\right)^{m_{s}-1}\Biggr{\\}}$
$\displaystyle=\sum_{m_{1},\dots,m_{M}\geq
0}\prod_{j=1}^{N}\left(1-\frac{c_{j}q^{-1}q^{m(L)}}{b_{L+1}\cdots
b_{M}}\right)\cdot\left(1-\frac{q^{-m_{s}}}{b_{s}}\right)$
$\displaystyle\qquad{}\times\prod_{j=1}^{N}\frac{(a_{j}/b_{L+1}\cdots
b_{M})_{m(L)}}{(c_{j}/b_{L+1}\cdots
b_{M})_{m(L)}}\prod_{i=1}^{M}\frac{(b_{i})_{m_{i}}}{(q)_{m_{i}}}\prod_{i=1}^{L}t_{i}^{m_{i}}\prod_{i=L+1}^{M}\left(\frac{q}{b_{i}t_{i}}\right)^{m_{i}}$
$\displaystyle=\prod_{j=1}^{N}\left(1-\frac{c_{j}q^{-1}}{b_{L+1}\cdots
b_{M}}T\right)\cdot\left(1-\frac{1}{b_{s}}T_{s}\right)\cdot\mathcal{F}_{N,M}^{L}\left(\begin{matrix}\\{a_{j}\\}_{1\leq
j\leq N},\\{b_{i}\\}_{1\leq i\leq M}\\\ \\{c_{j}\\}_{1\leq j\leq
N}\end{matrix};\\{t_{i}\\}_{1\leq i\leq M}\right).$

Similar to these calculations, we can check Proposition 3.2 directly. ∎

######  Remark 3.3.

If $N=1$, the solutions are $q$-analogs of the solutions of the differential
equation related with Lauricella function $F_{D}$, which was obtained by
Gelfand–Kapranov–Zelevinsky [9]. More precisely, the function (3.2), (3.3),
(3.4) are $q$-analogs of the function

$\displaystyle
F_{D,j}\left(\begin{matrix}\tilde{\alpha};\\{\tilde{\beta}_{i}\\}_{1\leq i\leq
M}\\\ \tilde{\gamma}\end{matrix};\\{x_{i}\\}_{1\leq i\leq
M}\right)=\sum_{m_{1},\dots,m_{M}\geq
0}\frac{(\tilde{\alpha})_{-m(j-1)}}{(\tilde{\gamma})_{-m(j-1)}}\prod_{i=1}^{M}\frac{(\tilde{\beta}_{i})_{m_{i}}}{(1)_{m_{i}}}\prod_{i=1}^{M}x_{i}^{m_{i}},$

by replacing some parameters and transforming variables. Only here,
$(\tilde{\alpha})_{n}=\Gamma(\tilde{\alpha}+n)/\Gamma(\tilde{\alpha})$ for
$n\in\mathbb{Z}$. The solutions of the equation related with $F_{D}$ are given
by using the function $F_{D,j}$.

######  Remark 3.4.

For $0\leq L\leq M$, the functions (3.5), (3.6), (3.7) converge in the region

$\displaystyle D=\left\\{\\!|t_{i}|<1,\,1\leq i\leq
L,\,\left|\prod_{j=1}^{N}\frac{c_{j}}{a_{j}}\cdot\frac{q}{b_{i}t_{i}}\right|<1,\,L+1\leq
i\leq M,\,\left|\frac{qt_{i}}{b_{j}t_{j}}\right|<1,\,1\leq i<j\leq
M\\!\right\\},$

simultaneously. We put $x_{i}=t_{i}/t_{i+1}$, $1\leq i<L$, $x_{L}=t_{L}$,
$x_{L+1}=1/t_{L+1}$, $x_{i}=t_{i-1}/t_{i}$, $L+1<i\leq M$, then we have
$t_{i}=x_{i}\cdots x_{L}$, $1\leq i\leq L$, $1/t_{i}=x_{L+1}\cdots x_{i}$,
$L+1\leq i\leq M$ and

$\displaystyle\frac{t_{i}}{t_{j}}=\begin{cases}x_{i}\cdots x_{j-1},&1\leq
i<j\leq L,\\\ x_{i}\cdots x_{j},&1\leq i\leq L<j\leq M,\\\ x_{i+1}\cdots
x_{j},&L<i<j\leq M.\end{cases}$

Therefore the functions (3.5), (3.6), (3.7) are solutions of the system
$E_{N,M}$ in the region $\\{|x_{i}|\ll 1,\,1\leq i\leq M\\}$, and have the
following asymptotic behavior at $x=(0,\dots,0)$:

$\displaystyle\prod_{i=L+1}^{M}t_{i}^{-\beta_{i}}\cdot\mathcal{F}_{N,M}^{L}\left(\begin{matrix}\\{a_{j}\\}_{1\leq
j\leq N},\\{b_{i}\\}_{1\leq i\leq M}\\\ \\{c_{j}\\}_{1\leq j\leq
N}\end{matrix};\\{t_{i}\\}_{1\leq i\leq
M}\right)=\prod_{i=L+1}^{M}t_{i}^{-\beta_{i}}\cdot(1+O(||x||)),$
$\displaystyle
t_{l}^{1+\sum_{i=l+1}^{M}\beta_{i}-\gamma_{k}}\prod_{i=l+1}^{M}t_{i}^{-\beta_{i}}\cdot\mathcal{G}_{N,M}^{L;k,l}\left(\begin{matrix}\\{a_{j}\\}_{1\leq
j\leq N},\\{b_{i}\\}_{1\leq i\leq M}\\\ \\{c_{j}\\}_{1\leq j\leq
N}\end{matrix};\\{t_{i}\\}_{1\leq i\leq M}\right)$
$\displaystyle\qquad{}=t_{l}^{1+\sum_{i=l+1}^{M}\beta_{i}-\gamma_{k}}\prod_{i=l+1}^{M}t_{i}^{-\beta_{i}}\cdot(1+O(||x||)),$
$\displaystyle
t_{l}^{-\alpha_{k}+\sum_{i=l+1}^{M}\beta_{i}}\prod_{i=l+1}^{M}t_{i}^{-\beta_{i}}\cdot\mathcal{F}_{N,M}^{L;k,l}\left(\begin{matrix}\\{a_{j}\\}_{1\leq
j\leq N},\\{b_{i}\\}_{1\leq i\leq M}\\\ \\{c_{j}\\}_{1\leq j\leq
N}\end{matrix};\\{t_{i}\\}_{1\leq i\leq M}\right)$
$\displaystyle\qquad{}=t_{l}^{-\alpha_{k}+\sum_{i=l+1}^{M}\beta_{i}}\prod_{i=l+1}^{M}t_{i}^{-\beta_{i}}\cdot(1+O(||x||)),$

where $||x||=\sum_{i=1}^{M}|x_{i}|$. The system $E_{N,M}$ is rewritten as
follows:

$\displaystyle\begin{cases}\displaystyle\left\\{\frac{t_{s}}{t_{s+1}}\cdots\frac{t_{L-1}}{t_{L}}t_{L}\prod_{j=1}^{N}(1-a_{j}T)\cdot(1-b_{s}T_{s})-\prod_{j=1}^{N}\big{(}1-c_{j}q^{-1}T\big{)}\cdot(1-T_{s})\right\\}\mathcal{F}=0,\\\
\hskip 312.9803pt1\leq s\leq L,\\\
\displaystyle\left\\{\prod_{j=1}^{N}(1-a_{j}T)\cdot(1-b_{s}T_{s})-\frac{1}{t_{L+1}}\frac{t_{L+1}}{t_{L+2}}\cdots\frac{t_{s-1}}{t_{s}}\prod_{j=1}^{N}\big{(}1-c_{j}q^{-1}T\big{)}\cdot(1-T_{s})\right\\}\mathcal{F}=0,\\\
\hskip 312.9803ptL+1\leq s\leq M,\end{cases}$
$\displaystyle\left\\{\frac{t_{r}}{t_{s}}(1-b_{r}T_{r})(1-T_{s})-(1-b_{s}T_{s})(1-T_{r})\right\\}\mathcal{F}=0,\qquad
1\leq r<s\leq M.$

If a function $f(t)$, which is defined in the region $D$, is a solution of
$E_{N,M}$ and has asymptotic behavior

$\displaystyle f(t)=t_{1}^{\delta_{1}}\cdots t_{M}^{\delta_{M}}(1+O(||x||))$

at $x=(0,\dots,0)$. Then, by checking the coefficient of the lowest power of
$x$, we find that $\delta=(\delta_{1},\dots,\delta_{M})$ must satisfy

$\displaystyle\begin{cases}\displaystyle\prod_{j=1}^{N}\big{(}1-c_{j}q^{-1}q^{\delta_{1}+\cdots+\delta_{M}}\big{)}\big{(}1-q^{\delta_{s}}\big{)}=0,&1\leq
s\leq L,\\\
\displaystyle\prod_{j=1}^{N}\big{(}1-a_{j}q^{\delta_{1}+\cdots+\delta_{M}}\big{)}\big{(}1-b_{s}q^{\delta_{s}}\big{)}=0,&L<s\leq
M,\end{cases}$
$\displaystyle\big{(}1-b_{s}q^{\delta_{s}}\big{)}\big{(}1-q^{\delta_{r}}\big{)}=0,\qquad
1\leq r<s\leq M.$

In this sense, the solution $(\delta_{1},\dots,\delta_{M})$ of the above
equations should be called “the characteristic exponent of the system
$E_{N,M}$ at $x=(0,\dots,0)$”, like a characteristic exponent in one variable
case. Solving those equations, we have

$\displaystyle(\delta_{1},\dots,\delta_{M})=(0,\dots,0,-\beta_{L+1},\dots,-\beta_{M}),$
(3.8)
$\displaystyle\begin{cases}\displaystyle\left(1+\sum_{i=2}^{M}\beta_{i}-\gamma_{k},-\beta_{2},\dots,-\beta_{M}\right),\\\
\displaystyle\left(0,1+\sum_{i=3}^{M}\beta_{i}-\gamma_{k},-\beta_{3},\dots,-\beta_{M}\right),\\\
\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\\\
\displaystyle\left(0,\dots,0,1+\sum_{i=L+1}^{M}\beta_{i}-\gamma_{k},-\beta_{L+1},\dots,-\beta_{M}\right),\end{cases}$
(3.9)
$\displaystyle\begin{cases}\displaystyle\left(0,\dots,0,-\alpha_{k}+\sum_{i=L+2}^{M}\beta_{i},-\beta_{L+2},\dots,-\beta_{M}\right),\\\
\displaystyle\left(0,\dots,0,-\alpha_{k}+\sum_{i=L+3}^{M}\beta_{i},-\beta_{L+3},\dots,-\beta_{M}\right),\\\
\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\\\
(0,\dots,0,-\alpha_{k}),\end{cases}$ (3.10)

where $1\leq k\leq N$. The solutions (3.5), (3.6), (3.7) correspond to “the
characteristic exponents” (3.8), (3.9), (3.10), respectively. Also the
solutions (3.5), (3.6), (3.7) are characterized by the asymptotic behavior

$\displaystyle\prod_{i=L+1}^{M}t_{i}^{-\beta_{i}}(1+O(||x||)),\qquad
t_{l}^{1+\sum_{i=l+1}^{M}\beta_{i}-\gamma_{k}}\prod_{i=l+1}^{M}t_{i}^{-\beta_{i}}(1+O(||x||)),$
$\displaystyle
t_{l}^{-\alpha_{k}+\sum_{i=l+1}^{M}\beta_{i}}\prod_{i=l+1}^{M}t_{i}^{-\beta_{i}}(1+O(||x||)),$

respectively at
$x=(t_{1}/t_{2},\dots,t_{L-1}/t_{L},t_{L},1/t_{L+1},t_{L+1}/t_{L+2},\dots,t_{M-1}/t_{M})=(0,\dots,0)$.

We should check the linearly independence of the solutions (3.5), (3.6),
(3.7). The following lemma is useful to check the linearly independence.

######  Lemma 3.5.

For any $i\neq j$, we assume $\delta_{i}\neq\delta_{j}$. Here,
$\delta_{i}=(\delta_{1,i},\dots,\delta_{M,i})$ and $\delta_{k,i}\in\mathbb{C}$
$1\leq i\leq n$, $1\leq k\leq M$. Then the functions

$\displaystyle f_{i}(t_{1},\dots,t_{M})=t^{\delta_{i}}(1+O(||t||))$

are linearly independent on $K=\\{C(t);\,\mbox{for any
}i,\,T_{i}C(t)=C(t)\\}$. Here, $t^{\delta_{i}}=t_{1}^{\delta_{1,i}}\cdots
t_{M}^{\delta_{M,i}}$.

###### Proof.

We assume that $C_{1}(t)f_{1}(t)+\cdots+C_{n}(t)f_{n}(t)=0$, where
$C_{i}(t)\in K$, $1\leq i\leq n$. We can take
$m=(m_{1},\dots,m_{M})\in\mathbb{Z}^{M}$ such that
$(m,\delta_{i})\neq(m,\delta_{j})$ for any $i\neq j$, where
$(m,\delta_{i})=m_{1}\delta_{1,i}+\cdots+m_{M}\delta_{M,i}$. We define the
operator $R=T_{1}^{m_{1}}\cdots T_{M}^{m_{M}}$ for such $m$ and consider the
following determinant

$\displaystyle
D=\begin{vmatrix}C_{1}(t)f_{1}(t)&C_{2}(t)f_{2}(t)&C_{3}(t)f_{3}(t)&\cdots&C_{n}f_{n}(t)\\\
R(C_{1}(t)f_{1}(t))&R(C_{2}(t)f_{2}(t))&R(C_{3}(t)f_{3}(t))&\cdots&R(C_{n}(t)f_{n}(t))\\\
R^{2}(C_{1}(t)f_{1}(t))&R^{2}(C_{2}(t)f_{2}(t))&R^{2}(C_{3}(t)f_{3}(t))&\cdots&R^{2}(C_{n}(t)f_{n}(t))\\\
\vdots&\vdots&&&\vdots\\\
R^{n-1}(C_{1}(t)f_{1}(t))&R^{n-1}(C_{2}(t)f_{2}(t))&R^{n-1}(C_{3}(t)f_{3}(t))&\cdots&R^{n-1}(C_{n}(t)f_{n}(t))\end{vmatrix}.$

By definition of $R$ and $C_{i}(t)$, we have
$R^{k}(C_{i}(t)f_{i}(t))=C_{i}(t)R^{k}(f_{i}(t))$. Also, we have
$R^{k}(f_{i}(t))=q^{(m,\delta_{i})k}t^{\delta_{i}}(1+O(||t||))$ by definition
of $f_{i}(t)$. Thus we have

$\displaystyle D=C_{1}(t)\cdots C_{M}(t)t^{\delta_{1}}\cdots t^{\delta_{n}}$
$\displaystyle\hphantom{D=}{}\times\begin{vmatrix}1&1&1&\cdots&1\\\
q^{(m,\delta_{1})}&q^{(m,\delta_{2})}&q^{(m,\delta_{3})}&\cdots&q^{(m,\delta_{n})}\\\
\big{(}q^{(m,\delta_{1})}\big{)}^{2}&\big{(}q^{(m,\delta_{2})}\big{)}^{2}&\big{(}q^{(m,\delta_{3})}\big{)}^{2}&\cdots&\big{(}q^{(m,\delta_{n})}\big{)}^{2}\\\
\vdots&\vdots&\vdots&&\vdots\\\
\big{(}q^{(m,\delta_{1})}\big{)}^{n-1}&\big{(}q^{(m,\delta_{2})}\big{)}^{n-1}&\big{(}q^{(m,\delta_{3})}\big{)}^{n-1}&\cdots&\big{(}q^{(m,\delta_{n})}\big{)}^{n-1}\end{vmatrix}(1+O(||t||)).$

By the assumption $C_{1}(t)f_{1}(t)+\cdots+C_{n}(t)f_{n}(t)=0$, we have $D=0$.
Also, by the condition $(m,\delta_{i})\neq(m,\delta_{j})$ $(i\neq j)$, the
Vandermonde determinant is not $0$. Therefore we have that there exists $i$
such that $C_{i}(t)=0$. By induction on $n$, we have
$C_{1}(t)=\cdots=C_{n}(t)=0$. ∎

By Lemma 3.5, we have that solutions (3.5), (3.6) (3.7) are linearly
independent on $K$ if parameters $\\{a_{j}\\}_{1\leq j\leq N}$,
$\\{b_{i}\\}_{1\leq i\leq M}$, $\\{c_{j}\\}_{1\leq j\leq N}$ satisfy the
condition (3.1). Also, if a function $f\left(\begin{matrix}\\{a_{j}\\}_{1\leq
j\leq N},\\{b_{i}\\}_{1\leq i\leq M}\\\ \\{c_{j}\\}_{1\leq j\leq
N}\end{matrix};\\{t_{i}\\}_{1\leq i\leq M}\right)$ satisfies the
$q$-difference system $E_{N,M}$, then the function
$f\left(\begin{matrix}\\{a_{j}\\}_{1\leq j\leq N},\\{b_{\sigma(i)}\\}_{1\leq
i\leq M}\\\ \\{c_{j}\\}_{1\leq j\leq N}\end{matrix};\\{t_{\sigma(i)}\\}_{1\leq
i\leq M}\right)$ satisfies the same system for $\sigma\in\mathfrak{S}_{M}$.
Therefore we have the following proposition.

######  Proposition 3.6.

For $0\leq L\leq M$ and $\sigma\in\mathfrak{S}_{M}$, we set

$\displaystyle
u_{0}^{L,\sigma}=\prod_{i=L+1}^{M}t_{\sigma_{i}}^{-\beta_{\sigma_{i}}}\cdot\mathcal{F}_{N,M}^{L}\left(\begin{matrix}\\{a_{j}\\}_{1\leq
j\leq N},\\{b_{\sigma(i)}\\}_{1\leq i\leq M}\\\ \\{c_{j}\\}_{1\leq j\leq
N}\end{matrix};\\{t_{\sigma(i)}\\}_{1\leq i\leq M}\right),$ $\displaystyle
u_{k,l}^{L,\sigma}=\begin{cases}\displaystyle
t_{\sigma_{l}}^{1+\sum_{i=l+1}^{M}\beta_{\sigma_{i}}-\gamma_{k}}\prod_{i=l+1}^{M}t_{\sigma_{i}}^{-\beta_{\sigma_{i}}}\cdot\mathcal{G}_{N,M}^{L;k,l}\left(\begin{matrix}\\{a_{j}\\}_{1\leq
j\leq N},\\{b_{\sigma(i)}\\}_{1\leq i\leq M}\\\ \\{c_{j}\\}_{1\leq j\leq
N}\end{matrix};\\{t_{\sigma(i)}\\}_{1\leq i\leq M}\right),\\\ \hskip
227.62204pt1\leq k\leq N,\ 1\leq l\leq L,\\\ \displaystyle
t_{\sigma_{l}}^{-\alpha_{k}+\sum_{i=l+1}^{M}\beta_{\sigma_{i}}}\prod_{i=l+1}^{M}t_{\sigma_{i}}^{-\beta_{\sigma_{i}}}\cdot\mathcal{F}_{N,M}^{L;k,l}\left(\begin{matrix}\\{a_{j}\\}_{1\leq
j\leq N},\\{b_{\sigma(i)}\\}_{1\leq i\leq M}\\\ \\{c_{j}\\}_{1\leq j\leq
N}\end{matrix};\\{t_{\sigma(i)}\\}_{1\leq i\leq M}\right),\\\ \hskip
227.62204pt1\leq k\leq N,\ L+1\leq l\leq M.\end{cases}$

We set

$\displaystyle{\bm{u}}^{L,\sigma}={}^{\rm
T}\big{(}u_{0}^{L,\sigma},u_{1,1}^{L,\sigma},\dots,u_{1,M}^{L,\sigma},u_{2,1}^{L,\sigma},\dots,u_{N,M}^{L,\sigma}\big{)}.$

Then ${\bm{u}}^{L,\sigma}$ is a fundamental solution of the $q$-difference
system $E_{N,M}$ in the region $D^{L,\sigma}$ if parameters
$\\{a_{j}\\}_{1\leq j\leq N}$, $\\{b_{i}\\}_{1\leq i\leq M}$,
$\\{c_{j}\\}_{1\leq j\leq N}$ satisfy the condition

$\displaystyle a_{j}/a_{k},c_{j}/c_{k},a_{j}/b_{\sigma(i)}\cdots
b_{\sigma(M)},c_{k}/b_{\sigma(i)}\cdots b_{\sigma(M)}\notin q^{\mathbb{Z}},$
(3.11)

for $1\leq i\leq M+1$, $1\leq j\neq k\leq N$. Here,

$\displaystyle D^{L,\sigma}=\Biggl{\\{}|t_{\sigma(i)}|<1,\,1\leq i\leq
L,\,\Bigg{|}\prod_{j=1}^{N}\frac{c_{j}}{a_{j}}\cdot\frac{q}{b_{\sigma(i)}t_{\sigma(i)}}\Bigg{|}<1,\,L+1\leq
i\leq M,$
$\displaystyle\hphantom{D^{L,\sigma}=\Biggl{\\{}}{}\left|\frac{qt_{\sigma(i)}}{b_{\sigma(j)}t_{\sigma(j)}}\right|<1,\,1\leq
i<j\leq M\Biggr{\\}}.$

###  3.2 Connection matrices

In this subsection, we take a fundamental solution $\bm{u}^{L,\sigma}$ on
$D^{L,\sigma}$ and we consider the connection formula between
$\bm{u}^{L_{1},\sigma_{1}}$ and $\bm{u}^{L_{2},\sigma_{2}}$, where $0\leq
L_{1},L_{2}\leq M$ and $\sigma_{1},\sigma_{2}\in\mathfrak{S}_{M}$. We assume
the condition (3.11) for $1\leq i\leq M+1$, $1\leq j\neq k\leq N$ and for any
$\sigma\in\mathfrak{S}_{M}$. We can solve this problem in principle by
calculating the following matrices:

  * •

the matrix which connects $\bm{u}^{L,\mathrm{id}}$ with
$\bm{u}^{L+1,\mathrm{id}}$, $0\leq L\leq M-1$,

  * •

the matrix which connects $\bm{u}^{L,\mathrm{id}}$ with
$\bm{u}^{L-1,\mathrm{id}}$, $1\leq L\leq M$,

  * •

the matrix which connects $\bm{u}^{M,\mathrm{id}}$ with $\bm{u}^{M,s_{r}}$,
$0\leq r\leq M-1$,

where $s_{r}=(r,r+1)\in\mathfrak{S}_{M}$. These matrices can be calculated by
the Thomae–Watson’s formula [23, 25].

######  Lemma 3.7 ([23, 25]).

The connection formula of ${}_{N+1}\varphi_{N}$ is given as follows:

$\displaystyle{}_{N+1}\varphi_{N}\left(\begin{cases}a_{1},\dots,a_{N+1}\\\
b_{1},\dots,b_{N}\end{cases};t\right)=\sum_{k=1}^{N+1}\Biggl{\\{}\prod_{j=1}^{N}\frac{(b_{j}/a_{k})_{\infty}}{(b_{j})_{\infty}}\prod_{\begin{subarray}{c}1\leq
j\leq N+1\\\ j\neq
k\end{subarray}}\frac{(a_{j})_{\infty}}{(a_{j}/a_{k})_{\infty}}$
$\displaystyle\qquad{}\times\frac{\theta(ta_{k})}{\theta(t)}{}_{N+1}\varphi_{N}\left(\begin{cases}\\{qa_{k}/b_{j}\\}_{1\leq
j\leq N},\,a_{k}\\\ \\{qa_{k}/a_{j}\\}_{1\leq j\leq N+1,\,j\neq
k}\end{cases};\frac{b_{1}\cdots b_{N}q}{a_{1}\cdots
a_{N+1}t}\right)\Biggr{\\}}.$ (3.12)

This formula can be derived by applying the Cauchy’s residue theorem to the
following integral:

$\displaystyle\int_{C}\frac{(b_{1}x,\dots,b_{N}x,qx/t,t/x)_{\infty}}{(a_{1}x,\dots,a_{N+1}x,1/x)_{\infty}}\frac{{\rm
d}x}{x}.$

Here the contour $C$ is a deformation of the positively oriented unit circle
so that the poles of $1/(a_{1}x,\dots,a_{N+1}x)_{\infty}$ lie outside $C$, and
the poles of $1/(1/x)_{\infty}$ and 0 lie inside $C$. For more details, see
[7, Section 4.10].

First, we consider the matrix which connects $\bm{u}^{L,\mathrm{id}}$ with
$\bm{u}^{L+1,\mathrm{id}}$. If $l\neq L+1$, then we have

$\displaystyle u_{k,l}^{L,\mathrm{id}}=u_{k,l}^{L+1,\mathrm{id}},$

easily. Hence, we should calculate the connection formula of
$u_{0}^{L,\mathrm{id}}$, $u_{k,L+1}^{L,\mathrm{id}}$. By rewriting the
definition (3.2), we have

$\displaystyle\mathcal{F}_{N,M}^{L}\left(\begin{matrix}\\{a_{j}\\},\\{b_{i}\\}\\\
\\{c_{j}\\}\end{matrix};\\{t_{i}\\}\right)$
$\displaystyle=\sum_{m_{1},\dots,m_{M}\geq
0}\prod_{j=1}^{N}\frac{(a_{j}/b_{L+1}\cdots
b_{M})_{m(L)}}{(c_{j}/b_{L+1}\cdots
b_{M})_{m(L)}}\prod_{i=1}^{M}\frac{(b_{i})_{m_{i}}}{(q)_{m_{i}}}\prod_{i=1}^{L}t_{i}^{m_{i}}\prod_{i=L+1}^{M}\left(\frac{q}{b_{i}t_{i}}\right)^{m_{i}}$
$\displaystyle=\sum_{m_{1},\dots,m_{L},m_{L+2},\dots,m_{M}\geq
0}\Biggl{\\{}\prod_{j=1}^{N}\frac{(a_{j}/b_{L+1}\cdots
b_{M})_{m(L)^{\prime}}}{(c_{j}/b_{L+1}\cdots
b_{M})_{m(L)^{\prime}}}\prod_{\begin{subarray}{c}1\leq i\leq M\\\ i\neq
L+1\end{subarray}}\frac{(b_{i})_{m_{i}}}{(q)_{m_{i}}}\prod_{i=1}^{L}t_{i}^{m_{i}}\prod_{i=L+2}^{M}\left(\frac{q}{b_{i}t_{i}}\right)^{m_{i}}$
$\displaystyle\qquad\times{}_{N+1}\varphi_{N}\left(\begin{matrix}\\{qb_{L+1}\cdots
b_{M}/c_{j}q^{m(L)^{\prime}}\\}_{1\leq j\leq N},\,b_{L+1}\\\ \\{qb_{L+1}\cdots
b_{M}/a_{j}q^{m(L)^{\prime}}\\}_{1\leq j\leq
N}\end{matrix};\prod_{j=1}^{N}\frac{c_{j}}{a_{j}}\cdot\frac{q}{b_{L+1}t_{L+1}}\right)\Biggr{\\}}.$
(3.13)

Here and in the following, we use the notation

$\displaystyle m(l)^{\prime}=\sum_{i=1}^{l}m_{i}-\sum_{i=l+2}^{M}m_{i},$

for $m=(m_{1},\dots,m_{M})$ and $0\leq l\leq M$. By applying the formula
(3.12) to ${}_{N+1}\varphi_{N}$, we obtain

$\displaystyle\mathcal{F}_{N,M}^{L}\left(\begin{matrix}\\{a_{j}\\}_{1\leq
j\leq N},\\{b_{i}\\}_{1\leq i\leq M}\\\ \\{c_{j}\\}_{1\leq j\leq
N}\end{matrix};\\{t_{i}\\}_{1\leq i\leq M}\right)$
$\displaystyle=\prod_{j=1}^{N}\frac{(qb_{L+2}\cdots b_{M}/a_{j},qb_{L+1}\cdots
b_{M}/c_{j})_{\infty}}{(qb_{L+1}\cdots b_{M}/a_{j},qb_{L+2}\cdots
b_{M}/c_{j})_{\infty}}\cdot\frac{\theta(t_{L+1}a_{1}\cdots a_{N}/c_{1}\cdots
c_{N})}{\theta(t_{L+1}b_{L+1}a_{1}\cdots a_{N}/c_{1}\cdots c_{N})}$
$\displaystyle\qquad{}\times\mathcal{F}_{N,M}^{L+1}\left(\begin{matrix}\\{a_{j}\\}_{1\leq
j\leq N},\\{b_{i}\\}_{1\leq i\leq M}\\\ \\{c_{j}\\}_{1\leq j\leq
N}\end{matrix};\\{t_{i}\\}_{1\leq i\leq M}\right)$
$\displaystyle\qquad{}+\sum_{d=1}^{N}\Biggl{\\{}\prod_{j=1}^{N}\frac{(c_{d}/a_{j})_{\infty}}{(qb_{L+1}\cdots
b_{M}/a_{j})_{\infty}}\prod_{\begin{subarray}{c}1\leq j\leq N\\\ j\neq
d\end{subarray}}\frac{(qb_{L+1}\cdots
b_{M}/c_{j})_{\infty}}{(c_{d}/c_{j})_{\infty}}\cdot\frac{(b_{L+1})_{\infty}}{(c_{d}/qb_{L+2}\cdots
b_{M})_{\infty}}$ $\displaystyle\qquad{}\times\frac{\theta(t_{L+1}a_{1}\cdots
a_{N}c_{d}/qb_{L+2}\cdots b_{M}c_{1}\cdots
c_{N})}{\theta(t_{L+1}b_{L+1}a_{1}\cdots a_{N}/c_{1}\cdots c_{N})}$
$\displaystyle\qquad{}\times\mathcal{F}_{N,M}^{L+1;d,L+1}\left(\begin{matrix}\\{a_{j}\\}_{1\leq
j\leq N},\\{b_{i}\\}_{1\leq i\leq M}\\\ \\{c_{j}\\}_{1\leq j\leq
N}\end{matrix};\\{t_{i}\\}_{1\leq i\leq M}\right)\Biggr{\\}}.$

By similar calculation of (3.13), we have

$\displaystyle\mathcal{F}_{N,M}^{L;k,L+1}\left(\begin{matrix}\\{a_{j}\\}_{1\leq
j\leq N},\\{b_{i}\\}_{1\leq i\leq M}\\\ \\{c_{j}\\}_{1\leq j\leq
N}\end{matrix};\\{t_{i}\\}_{1\leq i\leq M}\right)$
$\displaystyle=\prod_{\begin{subarray}{c}1\leq j\leq N\\\ j\neq
k\end{subarray}}\frac{(qb_{L+2}\cdots
b_{M}/a_{j})_{\infty}}{(qa_{k}/a_{j})_{\infty}}\cdot\frac{(q/b_{L+1})_{\infty}}{(qa_{k}/b_{L+1}\cdots
b_{M})_{\infty}}\prod_{j=1}^{N}\frac{(qa_{k}/c_{j})_{\infty}}{(qb_{L+2}\cdots
b_{M}/c_{j})_{\infty}}$
$\displaystyle{}\times\frac{\theta(t_{L+1}b_{L+1}\cdots b_{M}a_{1}\cdots
a_{N}/a_{k}c_{1}\cdots c_{N})}{\theta(t_{L+1}b_{L+1}a_{1}\cdots
a_{N}/c_{1}\cdots
c_{N})}\mathcal{F}_{N,M}^{L+1}\left(\begin{matrix}\\{a_{j}\\}_{1\leq j\leq
N},\\{b_{i}\\}_{1\leq i\leq M}\\\ \\{c_{j}\\}_{1\leq j\leq
N}\end{matrix};\\{t_{i}\\}_{1\leq i\leq M}\right)$
$\displaystyle{}+\sum_{d=1}^{N}\Biggl{\\{}\prod_{\begin{subarray}{c}1\leq
j\leq N\\\ j\neq
k\end{subarray}}\frac{(c_{d}/a_{j})_{\infty}}{(qa_{k}/a_{j})_{\infty}}\cdot\frac{(c_{d}/b_{L+1}\cdots
b_{M})_{\infty}}{(qa_{k}/b_{L+1}\cdots
b_{M})_{\infty}}\prod_{\begin{subarray}{c}1\leq j\leq N\\\ j\neq
d\end{subarray}}\frac{(qa_{k}/c_{j})_{\infty}}{(c_{d}/c_{j})_{\infty}}\cdot\frac{(a_{k}/b_{L+2}\cdots
b_{M})_{\infty}}{(c_{d}/qb_{L+2}\cdots b_{M})_{\infty}}$
$\displaystyle{}\times\frac{\theta(t_{L+1}b_{L+1}a_{1}\cdots
a_{N}c_{d}/qc_{1}\cdots c_{N}a_{k})}{\theta(t_{L+1}b_{L+1}a_{1}\cdots
a_{N}/c_{1}\cdots
c_{N})}\mathcal{G}_{N,M}^{L+1;d,L+1}\left(\begin{matrix}\\{a_{j}\\}_{1\leq
j\leq N},\\{b_{i}\\}_{1\leq i\leq M}\\\ \\{c_{j}\\}_{1\leq j\leq
N}\end{matrix};\\{t_{i}\\}_{1\leq i\leq M}\right)\Biggr{\\}}.$

Therefore, we set

$\displaystyle
A^{L,\mathrm{id}}=A^{L,\mathrm{id}}\left(\begin{matrix}\\{a_{j}\\}_{1\leq
j\leq N},\\{b_{i}\\}_{1\leq i\leq M}\\\ \\{c_{j}\\}_{1\leq j\leq
N}\end{matrix};t_{L+1}\right)=\begin{pmatrix}A_{0,0}&A_{0,1}&\cdots&A_{0,N}\\\
A_{1,0}&A_{1,1}&\cdots&A_{1,N}\\\ \vdots&\vdots&&\vdots\\\
A_{N,0}&A_{N,1}&\cdots&A_{N,N}\\\ \end{pmatrix},$ $\displaystyle
A_{0,0}=\prod_{j=1}^{N}\frac{(qb_{L+2}\cdots b_{M}/a_{j},qb_{L+1}\cdots
b_{M}/c_{j})_{\infty}}{(qb_{L+1}\cdots b_{M}/a_{j},qb_{L+2}\cdots
b_{M}/c_{j})_{\infty}}\cdot\frac{\theta(t_{L+1}a_{1}\cdots a_{N}/c_{1}\cdots
c_{N})}{\theta(t_{L+1}b_{L+1}a_{1}\cdots a_{N}/c_{1}\cdots
c_{N})}t_{L+1}^{-\beta_{L+1}},$ $\displaystyle
A_{0,d}=(A_{0,(1,d)},A_{0,(2,d)},\dots,A_{0,(M,d)})=(0,\dots,0,A_{0,(L+1,d)},0,\dots,0),$
$\displaystyle
A_{0,(L+1,d)}=\prod_{j=1}^{N}\frac{(c_{d}/a_{j})_{\infty}}{(qb_{L+1}\cdots
b_{M}/a_{j})_{\infty}}\prod_{\begin{subarray}{c}1\leq j\leq N\\\ j\neq
d\end{subarray}}\frac{(qb_{L+1}\cdots
b_{M}/c_{j})_{\infty}}{(c_{d}/c_{j})_{\infty}}\cdot\frac{(b_{L+1})_{\infty}}{(c_{d}/qb_{L+2}\cdots
b_{M})_{\infty}}$
$\displaystyle\hphantom{A_{0,(L+1,d)}=}{}\times\frac{\theta(t_{L+1}a_{1}\cdots
a_{N}c_{d}/qb_{L+2}\cdots b_{M}c_{1}\cdots
c_{N})}{\theta(t_{L+1}b_{L+1}a_{1}\cdots a_{N}/c_{1}\cdots
c_{N})}t_{L+1}^{-1-\sum_{i=L+1}^{M}\beta_{i}+\gamma_{d}},$ $\displaystyle
A_{k,0}={}^{\rm T}(A_{(1,k),0},A_{(2,k),0},\dots,A_{(N,k),0})={}^{\rm
T}(0,\dots,0,A_{(L+1,k),0},0,\dots,0),$ $\displaystyle
A_{(L+1,k),0}=\prod_{\begin{subarray}{c}1\leq j\leq N\\\ j\neq
k\end{subarray}}\frac{(qb_{L+2}\cdots
b_{M}/a_{j})_{\infty}}{(qa_{k}/a_{j})_{\infty}}\cdot\frac{(q/b_{L+1})_{\infty}}{(qa_{k}/b_{L+1}\cdots
b_{M})_{\infty}}\prod_{j=1}^{N}\frac{(qa_{k}/c_{j})_{\infty}}{(qb_{L+2}\cdots
b_{M}/c_{j})_{\infty}}$
$\displaystyle\hphantom{A_{(L+1,k),0}=}{}\times\frac{\theta(t_{L+1}b_{L+1}\cdots
b_{M}a_{1}\cdots a_{N}/a_{k}c_{1}\cdots
c_{N})}{\theta(t_{L+1}b_{L+1}a_{1}\cdots a_{N}/c_{1}\cdots
c_{N})}t_{L+1}^{-\alpha_{k}+\sum_{i=L+2}^{M}\beta_{i}},$ $\displaystyle
A_{k,d}=\begin{pmatrix}I_{L}&O&O\\\ O&A_{(L+1,k),(L+1,d)}&O\\\
O&O&I_{M-L-1}\end{pmatrix},$ $\displaystyle
A_{(L+1,k),(L+1,d)}=\prod_{\begin{subarray}{c}1\leq j\leq N\\\ j\neq
k\end{subarray}}\frac{(c_{d}/a_{j})_{\infty}}{(qa_{k}/a_{j})_{\infty}}\cdot\frac{(c_{d}/b_{L+1}\cdots
b_{M})_{\infty}}{(qa_{k}/b_{L+1}\cdots b_{M})_{\infty}}$
$\displaystyle\hphantom{A_{(L+1,k),(L+1,d)}=}{}\times\prod_{\begin{subarray}{c}1\leq
j\leq N\\\ j\neq
d\end{subarray}}\frac{(qa_{k}/c_{j})_{\infty}}{(c_{d}/c_{j})_{\infty}}\cdot\frac{(a_{k}/b_{L+2}\cdots
b_{M})_{\infty}}{(c_{d}/qb_{L+2}\cdots b_{M})_{\infty}}$
$\displaystyle\hphantom{A_{(L+1,k),(L+1,d)}=}{}\times\frac{\theta(t_{L+1}b_{L+1}a_{1}\cdots
a_{N}c_{d}/qc_{1}\cdots c_{N}a_{k})}{\theta(t_{L+1}b_{L+1}a_{1}\cdots
a_{N}/c_{1}\cdots c_{N})}t_{L+1}^{-1-\alpha_{k}+\gamma_{d}},$

where $1\leq k,d\leq N$, $I_{n}$ is the unit matrix of degree $n$ and $O$ is
the null matrix. Then we have

$\displaystyle\bm{u}^{L,\mathrm{id}}=A^{L,\mathrm{id}}\bm{u}^{L+1,\mathrm{id}}.$

Secondly, we consider the matrix which connects $\bm{u}^{L,\mathrm{id}}$ with
$\bm{u}^{L-1,\mathrm{id}}$. This can be calculated by a similar method for
deriving the matrix $A^{L,\mathrm{id}}$. We set

$\displaystyle
B^{L,\mathrm{id}}=B^{L,\mathrm{id}}\left(\begin{matrix}\\{a_{j}\\}_{1\leq
j\leq N},\\{b_{i}\\}_{1\leq i\leq M}\\\ \\{c_{j}\\}_{1\leq j\leq
N}\end{matrix};t_{L}\right)=\begin{pmatrix}B_{0,0}&B_{0,1}&\cdots&B_{0,N}\\\
B_{1,0}&B_{1,1}&\cdots&B_{1,N}\\\ \vdots&\vdots&&\vdots\\\
B_{N,0}&B_{N,1}&\cdots&B_{N,N}\\\ \end{pmatrix},$ $\displaystyle
B_{0,0}=\prod_{j=1}^{N}\frac{(a_{j}/b_{L+1}\cdots b_{M},c_{j}/b_{L}\cdots
b_{M})_{\infty}}{(a_{j}/b_{L}\cdots b_{M},c_{j}/b_{L+1}\cdots
b_{M})_{\infty}}\cdot\frac{\theta(t_{L}b_{L})}{\theta(t_{L})}t_{L}^{\beta_{L}},$
$\displaystyle
B_{0,d}=(B_{0,(1,d)},B_{0,(2,d)},\dots,B_{0,(M,d)})=(0,\dots,0,B_{0,(L,d)},0,\dots,0),$
$\displaystyle
B_{0,(L,d)}=\prod_{j=1}^{N}\frac{(c_{j}/a_{d})_{\infty}}{(c_{j}/b_{L+1}\cdots
b_{M})_{\infty}}\prod_{\begin{subarray}{c}1\leq j\leq N\\\ j\neq
d\end{subarray}}\frac{(a_{j}/b_{L+1}\cdots
b_{M})_{\infty}}{(a_{j}/a_{d})_{\infty}}\cdot\frac{(b_{L})_{\infty}}{(b_{L}\cdots
b_{M}/a_{d})_{\infty}}$
$\displaystyle\hphantom{B_{0,(L,d)}=}{}\times\frac{\theta(t_{L}a_{d}/b_{L+1}\cdots
b_{M})}{\theta(t_{L})}t_{L}^{\alpha_{d}-\sum_{i=L+1}^{M}\beta_{i}},$
$\displaystyle B_{k,0}={}^{\rm
T}(B_{(1,k),0},B_{(2,k),0},\dots,B_{(M,k),0})={}^{\rm
T}(0,\dots,0,B_{(L,k),0},0,\dots,0),$ $\displaystyle
B_{(L,k),0}=\prod_{\begin{subarray}{c}1\leq j\leq N\\\ j\neq
k\end{subarray}}\frac{(c_{j}/b_{L}\cdots
b_{M})_{\infty}}{(qc_{j}/c_{k})_{\infty}}\cdot\frac{(q/b_{L})_{\infty}}{(q^{2}b_{L+1}\cdots
b_{M}/c_{k})_{\infty}}\prod_{j=1}^{N}\frac{(qa_{j}/c_{k})_{\infty}}{(a_{j}/b_{L}\cdots
b_{M})_{\infty}}$
$\displaystyle\hphantom{B_{(L,k),0}=}{}\times\frac{\theta(t_{L}qb_{L}\cdots
b_{M}/c_{k})}{\theta(t_{L})}t_{L}^{1+\sum_{i=L}^{M}\beta_{i}-\gamma_{k}},$
$\displaystyle B_{k,d}=\begin{pmatrix}I_{L-1}&O&O\\\ O&B_{(L,k),(L,d)}&O\\\
O&O&I_{M-L}\end{pmatrix},$ $\displaystyle
B_{(L,k),(L,d)}=\prod_{\begin{subarray}{c}1\leq j\leq N\\\ j\neq
k\end{subarray}}\frac{(c_{j}/a_{d})_{\infty}}{(qc_{j}/c_{k})_{\infty}}\cdot\frac{(qb_{L+1}\cdots
b_{M}/a_{d})_{\infty}}{(q^{2}b_{L+1}\cdots
b_{M}/c_{k})_{\infty}}\prod_{\begin{subarray}{c}1\leq j\leq N\\\ j\neq
d\end{subarray}}\frac{(qa_{j}/c_{k})_{\infty}}{(a_{j}/a_{d})_{\infty}}\cdot\frac{(qb_{L}\cdots
b_{M}/c_{k})_{\infty}}{(b_{L}\cdots b_{M}/a_{d})_{\infty}}$
$\displaystyle\hphantom{B_{(L,k),(L,d)}=}{}\times\frac{\theta(t_{L}qa_{d}/c_{k})}{\theta(t_{L})}t_{L}^{1+\alpha_{d}-\gamma_{k}},$

where $1\leq k,d\leq N$. Then we have

$\displaystyle\bm{u}^{L,\mathrm{id}}=B^{L,\mathrm{id}}\bm{u}^{L-1,\mathrm{id}}.$

Finally, we consider the matrix which connects $\bm{u}^{M,\mathrm{id}}$ with
$\bm{u}^{M,s_{r}}$. We have

$\displaystyle u_{0}^{M,s_{r}}=u_{0}^{M,\mathrm{id}},\qquad
u_{k,l}^{M,s_{r}}=u_{k,l}^{M,\mathrm{id}},$

easily if $l\neq r,r+1$. Thus we should calculate the connection formula of
$u_{k,r}^{M,s_{r}}$, $u_{k,r+1}^{M,s_{r}}$. We have

$\displaystyle\mathcal{G}_{N,M}^{M;k,r}\left(\begin{matrix}\\{a_{j}\\}_{1\leq
j\leq N},\\{b_{s_{r}(i)}\\}_{1\leq i\leq M}\\\ \\{c_{j}\\}_{1\leq j\leq
N}\end{matrix};\\{t_{s_{r}(i)}\\}_{1\leq i\leq M}\right)$
$\displaystyle=\sum_{m_{1},\dots,m_{M}\geq
0}\Biggl{\\{}\prod_{k=1}^{N}\frac{(qa_{k}/c_{j})_{m_{M}}}{(qc_{k}/c_{j})_{m_{M}}}\prod_{i=1}^{r-1}\frac{(b_{i})_{m_{i}}}{(q)_{m_{i}}}\prod_{i=r+2}^{M}\frac{(b_{i})_{m_{i-1}}}{(q)_{m_{i-1}}}\cdot\frac{(b_{r})_{m_{r}}}{(q)_{m_{r}}}\frac{(c_{k}/qb_{r}b_{r+2}\cdots
b_{M})_{m(r-1)}}{(c_{k}/b_{r}\cdots b_{M})_{m(r-1)}}$
$\displaystyle\qquad{}\times\prod_{i=1}^{r-1}\left(\frac{qt_{i}}{b_{r+1}t_{r+1}}\right)^{m_{i}}\prod_{i=r+2}^{M}\left(\frac{b_{r+1}t_{r+1}}{b_{i}t_{i}}\right)^{m_{i-1}}\cdot\left(\frac{b_{r+1}t_{r+1}}{b_{r}t_{r}}\right)^{m_{r}}\left(\frac{b_{r+1}t_{r+1}}{q}\right)^{m_{M}}\Biggr{\\}}$
$\displaystyle=\sum_{m_{1},\dots,m_{r-1},m_{r+1},\dots,m_{M}\geq
0}\Biggl{\\{}\prod_{k=1}^{N}\frac{(qa_{k}/c_{j})_{m_{M}}}{(qc_{k}/c_{j})_{m_{M}}}\prod_{i=1}^{r-1}\frac{(b_{i})_{m_{i}}}{(q)_{m_{i}}}\prod_{i=r+2}^{M}\frac{(b_{i})_{m_{i-1}}}{(q)_{m_{i-1}}}$
$\displaystyle\qquad{}\times\frac{(c_{k}/qb_{r}b_{r+2}\cdots
b_{M})_{m(r-1)^{\prime}}}{(c_{k}/b_{r}\cdots
b_{M})_{m(r-1)^{\prime}}}\\!\prod_{i=1}^{r-1}\\!\left(\frac{qt_{i}}{b_{r+1}t_{r+1}}\right)^{m_{i}}\\!\prod_{i=r+2}^{M}\\!\left(\frac{b_{r+1}t_{r+1}}{b_{i}t_{i}}\right)^{m_{i-1}}\\!\cdot\left(\frac{b_{r+1}t_{r+1}}{q}\right)^{m_{M}}$
$\displaystyle\qquad{}\times{}_{2}\varphi_{1}\left(\begin{matrix}b_{r},qb_{r}\cdots
b_{M}/c_{k}q^{m(r-1)^{\prime}}\\\ q^{2}b_{r}b_{r+2}\cdots
b_{M}/c_{k}q^{m(r-1)^{\prime}}\end{matrix};\frac{qt_{r+1}}{b_{r}t_{r}}\right)\Biggr{\\}},$

and by applying the formula (3.12) to ${}_{2}\varphi_{1}$, we obtain

$\displaystyle\mathcal{G}_{N,M}^{M;k,r}\left(\begin{matrix}\\{a_{j}\\}_{1\leq
j\leq N},\\{b_{s_{r}(i)}\\}_{1\leq i\leq M}\\\ \\{c_{j}\\}_{1\leq j\leq
N}\end{matrix};\\{t_{s_{r}(i)}\\}_{1\leq i\leq M}\right)$
$\displaystyle=\frac{(q/b_{r+1},b_{r})_{\infty}}{(q^{2}b_{r}b_{r+2}\cdots
b_{M}/c_{k},c_{k}/qb_{r+1}\cdots
b_{M})_{\infty}}\frac{\theta(t_{r}c_{k}/t_{r+1}qb_{r+1}\cdots
b_{M})}{\theta(t_{r}b_{r}/t_{r+1})}$
$\displaystyle\qquad{}\times\mathcal{G}_{N,M}^{M;k,r}\left(\begin{matrix}\\{a_{j}\\}_{1\leq
j\leq N},\\{b_{i}\\}_{1\leq i\leq M}\\\ \\{c_{j}\\}_{1\leq j\leq
N}\end{matrix};\\{t_{i}\\}_{1\leq i\leq M}\right)$
$\displaystyle\qquad{}+\frac{(q^{2}b_{r+2}\cdots b_{M}/c_{k},qb_{r}\cdots
b_{M}/c_{k})_{\infty}}{(q^{2}b_{r}b_{r+2}\cdots b_{M}/c_{k},q_{r+1}\cdots
b_{M}/c_{k})_{\infty}}\frac{\theta(t_{r}/t_{r+1})}{\theta(t_{r}b_{r}/t_{r+1})}$
$\displaystyle\qquad{}\times\mathcal{G}_{N,M}^{M;j,r+1}\left(\begin{matrix}\\{a_{j}\\}_{1\leq
j\leq N},\\{b_{i}\\}_{1\leq i\leq M}\\\ \\{c_{j}\\}_{1\leq j\leq
N}\end{matrix};\\{t_{i}\\}_{1\leq i\leq M}\right).$

Similarly, we have

$\displaystyle\mathcal{G}_{N,M}^{M;k,r+1}\left(\begin{matrix}\\{a_{j}\\}_{1\leq
j\leq N},\\{b_{s_{r}(i)}\\}_{1\leq i\leq M}\\\ \\{c_{j}\\}_{1\leq j\leq
N}\end{matrix};\\{t_{s_{r}(i)}\\}_{1\leq i\leq M}\right)$
$\displaystyle=\frac{(c_{k}/b_{r}\cdots b_{M},c_{k}/qb_{r+2}\cdots
b_{M})_{\infty}}{(c_{k}/b_{r}b_{r+2}\cdots b_{M},c_{k}/qb_{r+1}\cdots
b_{M})_{\infty}}\frac{\theta(t_{r}b_{r}/t_{r+1}b_{r+1})}{\theta(t_{r}b_{r}/t_{r+1})}$
$\displaystyle\qquad{}\times\mathcal{G}_{N,M}^{M;k,r}\left(\begin{matrix}\\{a_{j}\\}_{1\leq
j\leq N},\\{b_{i}\\}_{1\leq i\leq M}\\\ \\{c_{j}\\}_{1\leq j\leq
N}\end{matrix};\\{t_{i}\\}_{1\leq i\leq M}\right)$
$\displaystyle\qquad{}+\frac{(q/b_{r},b_{r+1})_{\infty}}{(c_{k}/b_{r}b_{r+2}\cdots
b_{M},qb_{r+1}\cdots
b_{M}/c_{k})_{\infty}}\frac{\theta(t_{r}qb_{r}b_{r+2}\cdots
b_{M}/t_{r+1}c_{k})}{\theta(t_{r}b_{r}/t_{r+1})}$
$\displaystyle\qquad{}\times\mathcal{G}_{N,M}^{M;k,r+1}\left(\begin{matrix}\\{a_{j}\\}_{1\leq
j\leq N},\\{b_{i}\\}_{1\leq i\leq M}\\\ \\{c_{j}\\}_{1\leq j\leq
N}\end{matrix};\\{t_{i}\\}_{1\leq i\leq M}\right).$

Therefore, we set

$\displaystyle
S_{s_{r}}^{M,\mathrm{id}}=S_{s_{r}}^{M,\mathrm{id}}\left(\begin{matrix}\\{b_{i}\\}_{1\leq
i\leq M}\\\ \\{c_{j}\\}_{1\leq j\leq
N}\end{matrix};\frac{t_{r}}{t_{r+1}}\right)=\begin{pmatrix}1&O&\cdots&\cdots&O\\\
O&S_{r}^{1}&O&\cdots&O\\\ \vdots&O&S_{r}^{2}&&\vdots\\\
\vdots&\vdots&&\ddots&O\\\ O&O&\cdots&O&S_{r}^{N}\end{pmatrix},$ (3.14)
$\displaystyle S_{r}^{k}=\begin{pmatrix}I_{r-1}&O&O&O\\\
O&S_{r,r}^{k}&S_{r,r+1}^{k}&O\\\ O&S_{r+1,r}^{k}&S_{r+1,r+1}^{k}&O\\\
O&O&O&I_{M-r-1}\end{pmatrix},$ $\displaystyle
S_{r,r}^{k}=\frac{(q/b_{r+1},b_{r})_{\infty}}{(q^{2}b_{r}b_{r+2}\cdots
b_{M}/c_{k},c_{k}/qb_{r+1}\cdots b_{M})_{\infty}}$
$\displaystyle\hphantom{S_{r,r}^{k}=}{}\times\frac{\theta(t_{r}c_{k}/t_{r+1}qb_{r+1}\cdots
b_{M})}{\theta(t_{r}b_{r}/t_{r+1})}\left(\frac{t_{r}}{t_{r+1}}\right)^{-1-\sum_{i=r}^{M}\beta_{i}+\gamma_{k}},$
$\displaystyle S_{r,r+1}^{k}=\frac{(q^{2}b_{r+2}\cdots
b_{M}/c_{k},qb_{r}\cdots b_{M}/c_{k})_{\infty}}{(q^{2}b_{r}b_{r+2}\cdots
b_{M}/c_{k},qb_{r+1}\cdots
b_{M}/c_{k})_{\infty}}\frac{\theta(t_{r}/t_{r+1})}{\theta(t_{r}b_{r}/t_{r+1})}\left(\frac{t_{r}}{t_{r+1}}\right)^{-\beta_{r}},$
$\displaystyle S_{r+1,r}^{k}=\frac{(c_{k}/b_{r}\cdots
b_{M},c_{k}/qb_{r+2}\cdots b_{M})_{\infty}}{(c_{k}/b_{r}b_{r+2}\cdots
b_{M},c_{k}/qb_{r+1}\cdots
b_{M})_{\infty}}\frac{\theta(t_{r}b_{r}/t_{r+1}b_{r+1})}{\theta(t_{r}b_{r}/t_{r+1})}\left(\frac{t_{r}}{t_{r+1}}\right)^{-\beta_{r+1}},$
$\displaystyle
S_{r+1,r+1}^{k}=\frac{(q/b_{r},b_{r+1})_{\infty}}{(c_{k}/b_{r}b_{r+2}\cdots
b_{M},qb_{r+1}\cdots b_{M}/c_{k})_{\infty}}$
$\displaystyle\hphantom{S_{r+1,r+1}^{k}=}{}\times\frac{\theta(t_{r}qb_{r}b_{r+2}\cdots
b_{M}/t_{r+1}c_{k})}{\theta(t_{r}b_{r}/t_{r+1})}\left(\frac{t_{r}}{t_{r+1}}\right)^{1+\sum_{i=r+2}^{M}\beta_{i}-\gamma_{k}},$

where $1\leq k\leq N$. Then we have

$\displaystyle\bm{u}^{M,s_{r}}=S_{s_{r}}^{M,\mathrm{id}}\bm{u}^{M,\mathrm{id}}.$

Moreover, for $\sigma\in\mathfrak{S}_{M}$, we set

$\displaystyle
A^{L,\sigma}=A^{L,\mathrm{id}}\left(\begin{matrix}\\{a_{j}\\}_{1\leq j\leq
N},\\{b_{\sigma(i)}\\}_{1\leq i\leq M}\\\ \\{c_{j}\\}_{1\leq j\leq
N}\end{matrix};t_{\sigma(L+1)}\right),$ $\displaystyle
B^{L,\sigma}=B^{L,\mathrm{id}}\left(\begin{matrix}\\{a_{j}\\}_{1\leq j\leq
N},\\{b_{\sigma(i)}\\}_{1\leq i\leq M}\\\ \\{c_{j}\\}_{1\leq j\leq
N}\end{matrix};t_{\sigma(L)}\right),$ $\displaystyle
S_{s_{r}}^{M,\sigma}=S_{s_{r}}^{M,\mathrm{id}}\left(\begin{matrix}\\{b_{\sigma(i)}\\}_{1\leq
i\leq M}\\\ \\{c_{j}\\}_{1\leq j\leq
N}\end{matrix};\frac{t_{\sigma(r)}}{t_{\sigma(r+1)}}\right).$

Then we have

$\displaystyle\bm{u}^{L,\sigma}=A^{L,\sigma}\bm{u}^{L+1,\sigma},\qquad\bm{u}^{L,\sigma}=B^{L,\sigma}\bm{u}^{L-1,\sigma},\qquad\bm{u}^{M,s_{r}\sigma}=S_{s_{r}}^{M,\sigma}\bm{u}^{M,\sigma}.$

Therefore, we obtain the following theorem:

######  Theorem 3.8.

We assume the condition (3.11) for $1\leq i\leq M+1$, $1\leq j\neq k\leq N$
and for any $\sigma\in\mathfrak{S}_{M}$. For $0\leq L_{1},L_{2}\leq M$ and
$\sigma_{1},\sigma_{2}\in\mathfrak{S}_{M}$, we have

$\displaystyle\bm{u}^{L_{2},\sigma_{2}}=A^{L_{2},\sigma_{2}}A^{L_{2}+1,\sigma_{2}}\cdots
A^{M-1,\sigma_{2}}S_{s_{r_{1}}}^{M,s_{r_{2}}\cdots
s_{r_{I}}\sigma_{1}}S_{s_{r_{2}}}^{M,s_{r_{3}}\cdots
s_{r_{I}}\sigma_{1}}\cdots S_{s_{r_{I}}}^{M,\sigma_{1}}$
$\displaystyle\hphantom{\bm{u}^{L_{2},\sigma_{2}}=}{}\times
B^{M,\sigma_{1}}B^{M-1,\sigma_{1}}\cdots
B^{L_{1}+1,\sigma_{1}}\bm{u}^{L_{1},\sigma_{1}},$

if $\sigma_{2}=s_{r_{1}}\cdots s_{r_{I}}\sigma_{1}$, where
$s_{r}=(r,r+1)\in\mathfrak{S}_{M}$.

######  Remark 3.9.

Each element of the matrices $A^{L,\sigma}$, $B^{L,\sigma}$,
$S_{s_{r}}^{M,\sigma}$ is a pseudo constant, i.e.,

$\displaystyle T_{s}A^{L,\sigma}=A^{L,\sigma},\qquad
T_{s}B^{L,\sigma}=B^{L,\sigma},\qquad
T_{s}S_{s_{r}}^{M,\sigma}=S_{s_{r}}^{M,\sigma}.$

##  4 A solution of the Yang–Baxter equation

In this section, we assume $b_{i}=q^{\beta_{i}}$, $1\leq i\leq M$, and
$c_{j}=q^{\gamma_{j}}$, $1\leq j\leq N$. We obtain an elliptic solution of the
Yang–Baxter equation as an application of Theorem 3.8. The functions

$\displaystyle
v_{0}^{\sigma}=\mathcal{F}_{N,M}\left(\begin{matrix}\\{a_{j}\\}_{1\leq j\leq
N},\\{b_{i}\\}_{1\leq i\leq M}\\\ \\{c_{j}\\}_{1\leq j\leq
N}\end{matrix};\\{t_{i}\\}_{1\leq i\leq M}\right),$ $\displaystyle
v_{k,l}^{\sigma}=t_{\sigma(l)}^{1+\sum_{i=l}^{M}\beta_{\sigma(i)}-\gamma_{k}}\prod_{m=l+1}^{M}t_{\sigma(m)}^{-\beta_{\sigma(m)}}\cdot\mathcal{F}_{N,M}^{M;k,l}\left(\begin{matrix}\\{a_{j}\\}_{1\leq
j\leq N},\\{b_{\sigma(i)}\\}_{1\leq i\leq M}\\\ \\{c_{j}\\}_{1\leq j\leq
N}\end{matrix};\\{t_{\sigma(i)}\\}_{1\leq i\leq M}\right),$

where $1\leq k\leq N$, $1\leq l\leq M$ and $\sigma\in\mathfrak{S}_{M}$, are
solutions of the $q$-difference system $E_{N,M}$ in the region

$\displaystyle D^{M,\sigma}=\left\\{|t_{i}|<1,\,1\leq i\leq
M,\,\left|\frac{qt_{\sigma(i)}}{b_{\sigma(j)}t_{\sigma(j)}}\right|<1,\,1\leq
i<j\leq M\right\\}.$

Similar to the calculation of the matrix (3.14), we set

$\displaystyle\bm{v}^{\sigma}={}^{\rm
T}\big{(}v_{0}^{\sigma},v_{1,1}^{\sigma},\dots,v_{1,M}^{\sigma},v_{2,1}^{\sigma},\dots,v_{N,M}^{\sigma}\big{)},$

and we set

$\displaystyle\tilde{S}_{s_{r}}^{\mathrm{id}}=\tilde{S}_{s_{r}}^{\mathrm{id}}\left(\begin{matrix}\\{b_{i}\\}_{1\leq
i\leq M}\\\ \\{c_{j}\\}_{1\leq j\leq
N}\end{matrix};\frac{t_{r}}{t_{r+1}}\right)=\begin{pmatrix}1&O&\cdots&\cdots&O\\\
O&\tilde{S}_{r}^{1}&O&\cdots&O\\\ \vdots&O&\tilde{S}_{r}^{2}&&\vdots\\\
\vdots&\vdots&&\ddots&O\\\ O&O&\cdots&O&\tilde{S}_{r}^{N}\end{pmatrix},$
$\displaystyle\tilde{S}_{r}^{k}=\tilde{S}_{r}\left(\begin{matrix}\\{b_{i}\\}_{1\leq
i\leq M}\\\
c_{k}\end{matrix};\frac{t_{r}}{t_{r+1}}\right)=\begin{pmatrix}I_{r-1}&O&O&O\\\
O&\tilde{S}_{r,r}^{k}&\tilde{S}_{r,r+1}^{k}&O\\\
O&\tilde{S}_{r+1,r}^{k}&\tilde{S}_{r+1,r+1}^{k}&O\\\
O&O&O&I_{M-r-1}\end{pmatrix},$ (4.1)
$\displaystyle\tilde{S}_{r,r}^{k}=\frac{(q/b_{r+1},b_{r})_{\infty}}{(q^{2}b_{r}b_{r+2}\cdots
b_{M}/c_{k},c_{k}/qb_{r+1}\cdots b_{M})_{\infty}}$
$\displaystyle\hphantom{\tilde{S}_{r,r}^{k}=}{}\times\frac{\theta(t_{r}c_{k}/t_{r+1}qb_{r+1}\cdots
b_{M})}{\theta(t_{r}b_{r}/t_{r+1})}\left(\frac{t_{r}}{t_{r+1}}\right)^{-1-\sum_{i=r}^{M}\beta_{i}+\gamma_{k}},$
$\displaystyle\tilde{S}_{r,r+1}^{k}=\frac{(q^{2}b_{r+2}\cdots
b_{M}/c_{k},qb_{r}\cdots b_{M}/c_{k})_{\infty}}{(q^{2}b_{r}b_{r+2}\cdots
b_{M}/c_{k},qb_{r+1}\cdots
b_{M}/c_{k})_{\infty}}\frac{\theta(t_{r}/t_{r+1})}{\theta(t_{r}b_{r}/t_{r+1})}\left(\frac{t_{r}}{t_{r+1}}\right)^{-\beta_{r}},$
$\displaystyle\tilde{S}_{r+1,r}^{k}=\frac{(c_{k}/b_{r}\cdots
b_{M},c_{k}/qb_{r+2}\cdots b_{M})_{\infty}}{(c_{k}/b_{r}b_{r+2}\cdots
b_{M},c_{k}/qb_{r+1}\cdots
b_{M})_{\infty}}\frac{\theta(t_{r}b_{r}/t_{r+1}b_{r+1})}{\theta(t_{r}b_{r}/t_{r+1})}\left(\frac{t_{r}}{t_{r+1}}\right)^{-\beta_{r+1}},$
$\displaystyle\tilde{S}_{r+1,r+1}^{k}=\frac{(q/b_{r},b_{r+1})_{\infty}}{(c_{k}/b_{r}b_{r+2}\cdots
b_{M},qb_{r+1}\cdots b_{M}/c_{k})_{\infty}}$
$\displaystyle\hphantom{\tilde{S}_{r+1,r+1}^{k}=}{}\times\frac{\theta(t_{r}qb_{r}b_{r+2}\cdots
b_{M}/t_{r+1}c_{k})}{\theta(t_{r}b_{r}/t_{r+1})}\left(\frac{t_{r}}{t_{r+1}}\right)^{1+\sum_{i=r+2}^{M}\beta_{i}-\gamma_{k}},$

where $1\leq k\leq N$ and $s_{r}=(r,r+1)\in\mathfrak{S}_{M}$. Then we have

$\displaystyle\bm{v}^{s_{r}}=\tilde{S}_{s_{r}}^{\mathrm{id}}\bm{v}^{\mathrm{id}}.$

In addition, we set

$\displaystyle\tilde{S}_{s_{r}}^{\sigma}=\tilde{S}_{s_{r}}^{\mathrm{id}}\left(\begin{matrix}\\{b_{\sigma(i)}\\}_{1\leq
i\leq M}\\\ \\{c_{j}\\}_{1\leq j\leq
N}\end{matrix};\frac{t_{\sigma(r)}}{t_{\sigma(r+1)}}\right),$

for $\sigma\in\mathfrak{S}_{M}$, and then we have

$\displaystyle\bm{v}^{s_{r}\sigma}=\tilde{S}_{s_{r}}^{\sigma}\bm{v}^{\sigma}.$

######  Remark 4.1.

The matrices $\tilde{S}_{s_{r}}^{\mathrm{id}}$, $1\leq r\leq M-1$, depend only
on $t_{r}/t_{r+1}$ and the parameters $\\{b_{i}\\}$, $\\{c_{j}\\}$.

By the braid relation $(r,r+2)=s_{r}s_{r+1}s_{r}=s_{r+1}s_{r}s_{r+1}$, we have

$\displaystyle\bm{v}^{(r,r+2)}=\tilde{S}_{s_{r}}^{s_{r+1}s_{r}}\tilde{S}_{s_{r+1}}^{s_{r}}\tilde{S}_{s_{r}}^{\mathrm{id}}\bm{v}^{\mathrm{id}}=\tilde{S}_{s_{r+1}}^{s_{r}s_{r+1}}\tilde{S}_{s_{r}}^{s_{r+1}}\tilde{S}_{s_{r+1}}^{\mathrm{id}}\bm{v}^{\mathrm{id}}.$

In particular, we find that the matrices $\tilde{S}_{r}$ satisfy the
Yang–Baxter equation

$\displaystyle\tilde{S}_{r}\left(\begin{matrix}\\{b_{s_{r+1}s_{r}(i)}\\}_{1\leq
i\leq M}\\\
c_{k}\end{matrix};u\right)\tilde{S}_{r+1}\left(\begin{matrix}\\{b_{s_{r}(i)}\\}_{1\leq
i\leq M}\\\
c_{k}\end{matrix};uv\right)\tilde{S}_{r}\left(\begin{matrix}\\{b_{i}\\}_{1\leq
i\leq M}\\\ c_{k}\end{matrix};v\right)$
$\displaystyle\qquad{}=\tilde{S}_{r+1}\left(\begin{matrix}\\{b_{s_{r}s_{r+1}(i)}\\}_{1\leq
i\leq M}\\\
c_{k}\end{matrix};v\right)\tilde{S}_{r}\left(\begin{matrix}\\{b_{s_{r+1}(i)}\\}_{1\leq
i\leq M}\\\
c_{k}\end{matrix};uv\right)\tilde{S}_{r+1}\left(\begin{matrix}\\{b_{i}\\}_{1\leq
i\leq M}\\\ c_{k}\end{matrix};u\right),$

where $u=t_{r+1}/t_{r+2}$, $v=t_{r}/t_{r+1}$.

######  Remark 4.2.

For the details of the Yang–Baxter equation, see Jimbo’s text [11].

######  Remark 4.3.

Aomoto, Kato and Mimachi [4] obtained an elliptic solution of the Yang–Baxter
equation by considering the connection matrices of a holonomic $q$-difference
system which was studied in [16]. They obtained that the matrices

$\displaystyle P_{i}(u)=\begin{pmatrix}I_{i-1}&O&O\\\
O&W(\alpha^{\prime}+(i-1)\beta^{\prime},\beta^{\prime};u)&O\\\
O&O&I_{n-i-1}\end{pmatrix},$ $\displaystyle
W(\alpha,\beta;u)=\begin{pmatrix}\displaystyle
u^{\alpha+3\beta+1}\frac{\theta\big{(}q^{-\beta}\big{)}\theta\big{(}uq^{\alpha+2\beta+1}\big{)}}{\theta\big{(}q^{-\alpha-2\beta}\big{)}\theta\big{(}uq^{-\beta}\big{)}}&\displaystyle
q^{\beta+1}u^{\beta}\frac{\theta(u)\theta\big{(}q^{-\alpha-\beta+1}\big{)}\theta\big{(}q^{\alpha+3\beta+2}\big{)}}{\theta\big{(}q^{-\alpha-2\beta}\big{)}^{2}\theta\big{(}uq^{-\beta}\big{)}}\vspace{1mm}\\\
\displaystyle\displaystyle
u^{\beta}\frac{\theta(u)}{\theta\big{(}uq^{-\beta}\big{)}}&\displaystyle
u^{-\alpha-\beta}\frac{\theta\big{(}q^{-\beta}\big{)}\theta\big{(}uq^{-\alpha-2\beta}\big{)}}{\theta\big{(}q^{-\alpha-2\beta}\big{)}\theta\big{(}uq^{-\beta}\big{)}},\end{pmatrix},$

where $1\leq i\leq n-1$, satisfy the Yang–Baxter equation

$\displaystyle P_{i}(u)P_{i+1}(uv)P_{i}(v)=P_{i+1}(v)P_{i}(uv)P_{i+1}(u).$

They also found that the matrix $W$ is identified as the matrix $W^{\prime}$
with entires of Boltzmann weight $\begin{matrix}\kappa\vspace{-1mm}\\\
\mu\,\Box\,\sigma\vspace{-2mm}\\\ \nu\end{matrix}$ of the $A_{1}^{(1)}$ face
model discussed by Jimbo, Miwa and Okado [12]. The matrix $W^{\prime}$ is
expressed as

$\displaystyle
W^{\prime}=\begin{pmatrix}\displaystyle\frac{[a-u]}{[a]}&\displaystyle\frac{[u][a+1][a-1]}{[1][a]^{2}}\vspace{1mm}\\\
\displaystyle\frac{[u]}{[1]}&\displaystyle\frac{[a+u]}{[a]}\end{pmatrix},$
(4.2)

where $[u]=\theta_{1}(\pi u/L,q)$, $L\neq 0$ is an arbitrary complex parameter
and

$\displaystyle\theta_{1}(u,q)=2q^{1/8}\sin
u\prod_{k=1}^{\infty}\big{(}1-2q^{k}\cos
2u+q^{2k}\big{)}\big{(}1-q^{k}\big{)}$
$\displaystyle\hphantom{\theta_{1}(u,q)}{}=2q^{1/8}\sin u\big{(}{\rm
e}^{2\sqrt{-1}u}q,{\rm e}^{-2\sqrt{-1}u}q,q\big{)}_{\infty}.$

The matrix $W$ is equivalent to $W^{\prime}$ as follows:

$\displaystyle x^{2g_{c}}\begin{pmatrix}x^{-g_{a+1}}&0\\\
0&x^{-g_{a-1}}\end{pmatrix}W^{\prime}\begin{pmatrix}x^{-g_{a+1}}&0\\\
0&x^{-g_{a-1}}\end{pmatrix}=\frac{\theta\big{(}xq^{-\beta}\big{)}}{\theta\big{(}q^{-\beta}\big{)}}W(\alpha,\beta;x),$

with ${\rm e}^{2\pi\sqrt{-1}u/L}=x$, ${\rm e}^{2\pi\sqrt{-1}/L}=q^{\beta+1}$,
${\rm e}^{2\pi\sqrt{-1}a/L}=q^{-\alpha-2\beta}$,
$g_{a-1}=\frac{\alpha+\beta}{2}$, $g_{a+1}=\frac{-\alpha-3\beta}{2}$,
$2g_{c}=\frac{1}{2}$. On the other hand, by specializing parameters $b_{i}$ as
$b_{1}=\cdots=b_{M}=q^{\beta}$, the matrices
$\tilde{S}_{r}(u)=\tilde{S}_{r}\left(\begin{matrix}q^{\beta}\\\
c_{k}\end{matrix};u\right)$ satisfy the Yang–Baxter equation

$\displaystyle\tilde{S}_{r}(u)\tilde{S}_{r+1}(uv)\tilde{S}_{r}(v)=\tilde{S}_{r+1}(v)\tilde{S}_{r}(uv)\tilde{S}_{r+1}(u).$

By definition (4.1), we have

$\displaystyle\tilde{S}_{r}(u)=\begin{pmatrix}I_{r-1}&O&O\\\
O&\tilde{W}(\gamma_{k}-2-(M-r-2)\beta,-\beta;u)&O\\\
O&O&I_{M-r-1}\end{pmatrix},$ (4.3)
$\displaystyle\tilde{W}(\alpha,\beta;u)=\begin{pmatrix}\displaystyle
u^{\alpha+3\beta+1}\frac{\theta\big{(}q^{-\beta}\big{)}\theta\big{(}uq^{\alpha+2\beta+1}\big{)}}{\theta\big{(}q^{-\alpha-2\beta}\big{)}\theta\big{(}uq^{-\beta}\big{)}}&\displaystyle
u^{\beta}\frac{\theta(u)\big{(}q^{-\alpha-\beta},q^{-\alpha-3\beta-1}\big{)}_{\infty}}{\theta\big{(}uq^{-\beta}\big{)}\big{(}q^{-\alpha-2\beta},q^{-\alpha-2\beta-1}\big{)}_{\infty}}\vspace{1mm}\\\
\displaystyle
u^{\beta}\frac{\theta(u)\big{(}q^{\alpha+3\beta+2},q^{\alpha+\beta+1}\big{)}_{\infty}}{\theta\big{(}uq^{-\beta}\big{)}\big{(}q^{\alpha+2\beta+2},q^{\alpha+2\beta+1}\big{)}_{\infty}}&\displaystyle
u^{-\alpha-\beta-1}\frac{\theta\big{(}uq^{-\alpha-2\beta-1}\big{)}\theta\big{(}q^{-\beta}\big{)}}{\theta\big{(}uq^{-\beta}\big{)}\theta\big{(}q^{-\alpha-2\beta-1}\big{)}}\end{pmatrix},$

and by means of easy calculations, we find that the matrices $W$ and
$\tilde{W}$ are conjugate as follows:

$\displaystyle
W(\alpha,\beta;u)=A(\alpha,\beta)^{-1}\tilde{W}(\alpha,\beta;u)A(\alpha,\beta)=B(\alpha,\beta)\tilde{W}(\alpha,\beta;u)B(\alpha,\beta)^{-1},$

where

$\displaystyle A(\alpha,\beta)=\begin{pmatrix}1&0\\\
0&f(\alpha,\beta)\end{pmatrix},\qquad
B(\alpha,\beta)=\begin{pmatrix}f(\alpha,\beta)&0\\\ 0&1\end{pmatrix},$
$\displaystyle
f(\alpha,\beta)=\frac{\big{(}q^{\alpha+3\beta+2},q^{\alpha+\beta+1}\big{)}_{\infty}}{\big{(}q^{\alpha+2\beta+2},q^{\alpha+2\beta+1}\big{)}_{\infty}}.$

In conclusion, our matrix $\tilde{W}$ is identified as Jimbo, Miwa and Okado’s
matrix $W^{\prime}$ and with $n=M$, $\alpha^{\prime}=\gamma_{k}-2-(M-3)\beta$
and $\beta^{\prime}=-\beta$, our matrices $\tilde{S}_{r}(u)$ and Aomoto, Kato
and Mimachi’s matrices $P_{r}(u)$ are conjugate.

##  5 Summary and discussion

A summary of our results is as follows. The main result of this paper is
Theorem 3.8, which gives the connection matrices for fundamental solutions
$\bm{u}^{L,\sigma}$ of the $q$-difference system $E_{N,M}$ (2.5) and (2.6).
The fundamental solution $\bm{u}^{L,\sigma}$ converges in the region
$\\{|t_{\sigma(1)}|\ll\cdots\ll|t_{\sigma(L)}|\ll
1\ll|t_{\sigma(L+1)}|\ll\cdots\ll|t_{\sigma(M)}|\\}$. The component of the
fundamental solution $\bm{u}^{L,\mathrm{id}}$ has the asymptotic behavior of
the form $t_{1}^{\delta_{1}}\cdots t_{M}^{\delta_{M}}(1+O(||x||))$ at
$x=(t_{1}/t_{2},\dots,t_{L-1}/t_{L},t_{L},1/t_{L+1},t_{L+1}/t_{L+2},\dots,\allowbreak
t_{M-1}/t_{M})=(0,\dots,0)$ for some $\delta$. More precisely, see Remark 3.4.
The way to get the connection matrices is to calculate “easy” connection
matrices many times, i.e.,

$\displaystyle\bm{u}^{L_{1},\sigma_{1}}\to\bm{u}^{L_{1}+1,\sigma_{1}}\to\cdots\to\bm{u}^{M,\sigma_{1}}$
$\displaystyle\hphantom{\bm{u}^{L_{1},\sigma_{1}}}{}\to\bm{u}^{M,s_{r_{1}}\sigma_{1}}\to\bm{u}^{M,s_{r_{2}}s_{r_{1}}\sigma_{1}}\to\cdots\to\bm{u}^{M,s_{r_{I}}\cdots
s_{r_{1}}\sigma_{1}}=\bm{u}^{M,\sigma_{2}}$
$\displaystyle\hphantom{\bm{u}^{L_{1},\sigma_{1}}}{}\to\bm{u}^{M-1,\sigma_{2}}\to\bm{u}^{M-2,\sigma_{2}}\to\cdots\to\bm{u}^{L_{2},\sigma_{2}}.$

Here, $s_{r}=(r,r+1)\in\mathfrak{S}_{M}$. Each step can be calculated by using
Thomae–Watson’s formula [23, 25], which is a connection formula of the
function ${}_{N+1}\varphi_{N}$. In addition, as an application of Theorem 3.8,
we obtained a solution of the Yang–Baxter equation by considering the
connection matrix between $\bm{u}^{M,(r,r+2)}$ and $\bm{u}^{M,\mathrm{id}}$.
Also we showed that our matrix (4.3) is identified as Jimbo, Miwa, Okado’s
matrix (4.2), and our solution and Aomoto, Kato, Mimachi’s solution [4] are
conjugate.

There are many problems related to our results. We mention four of them here.

  * (i)

By taking the limit $q\to 1$ with $a_{j}=q^{\alpha_{j}}$,
$b_{i}=q^{\beta_{i}}$, $c_{j}=q^{\gamma_{j}}$, we obtain fundamental solutions
of Tsuda’s hypergeometric equations [24]

$\displaystyle\left\\{t_{s}(\beta_{s}+\mathcal{D}_{s})\prod_{j=1}^{N}(\alpha_{j}+\mathcal{D})-\mathcal{D}_{s}\prod_{j=1}^{N}(\gamma_{j}-1+\mathcal{D})\right\\}y=0,\qquad
1\leq s\leq M,$
$\displaystyle\\{t_{r}(\beta_{r}+\mathcal{D}_{r})\mathcal{D}_{s}-t_{s}(\beta_{s}+\mathcal{D}_{s})\mathcal{D}_{r}\\}y=0,\qquad
1\leq r<s\leq M,$

where $\mathcal{D}_{s}=t_{s}\frac{\partial}{\partial t_{s}}$ and
$\mathcal{D}=\sum_{s=1}^{M}\mathcal{D}_{s}$. Similar to the method of Theorem
3.8, it is expected that the connection problem of Tsuda’s equations will be
solved. In this case, it must be noted that solutions are multivalued
functions. The connection formula of Tsuda’s hypergeometric function
$F_{N+1,M}$, which is a solution of Tsuda’s equations, depends on the path on
$X=\big{\\{}(t_{1},\dots,t_{M})\in\mathbb{C}^{M};\,t_{i}\neq t_{j},\,i\neq
j,\,t_{i}\neq 0,1\big{\\}}$. In [15], a path of connection for solutions of GG
system [8] was discussed by Matsubara-Heo. Thus the connection problem of
Tsuda’s equations will also be solved by Matsubara’s method. Note that in the
case of $N=1$, some of connection formulas related with $F_{1}$ were obtained
by Olsson [18], and related with $F_{D}$ were obtained by Mimachi [17] from
the viewpoint of the Jordan–Pochhammer integral and the intersection theory.

  * (ii)

The function is a generalization of the $q$-Lauricella function $\varphi_{D}$.
In differential case and two variable case, i.e.,

$\displaystyle[t_{s}(\beta_{s}+\mathcal{D}_{s})(\alpha+\mathcal{D}_{s}+\mathcal{D}_{r})-\mathcal{D}_{s}(\gamma-1+\mathcal{D}_{s}+\mathcal{D}_{r})]F=0,\qquad(i,j)=(1,2),\,(2,1),$

there are power series solutions of the equation which converge near
$(t_{1},t_{2})=(0,0)$, $(0,1)$, $(0,\infty)$, $(1,0)$, $(1,1)$, $(1,\infty)$,
$(\infty,0)$, $(\infty,1)$, $(\infty,\infty)$. Here we show two solutions as
examples:

$\displaystyle F_{1}\left(\begin{matrix}\alpha;\beta_{1},\beta_{2}\\\
\alpha+\beta_{1}+\beta_{2}+1-\gamma\end{matrix};1-t_{1},1-t_{2}\right),$
$\displaystyle
t_{1}^{-\beta_{1}}t_{2}^{-\beta_{2}}F_{1}\left(\begin{matrix}\beta_{1}+\beta_{2}+1-\gamma;\beta_{1},\beta_{2}\\\
\alpha+\beta_{1}+\beta_{2}+1-\gamma\end{matrix};\frac{t_{1}-1}{t_{1}},\frac{t_{2}-1}{t_{2}}\right).$

60 solutions like these series were obtained by Appell [5], Le Vavasseur [14],
and other solutions which are given by the Horn’s $G_{2}$ function were
obtained by Erdélyi [6]. In this paper we get power series solutions for the
system $E_{N,M}$, however in the case $N=1$ and $M=2$, the series are
$q$-analogs of only a part of Appell, Le Vavasseur, Erdélyi’s solutions. In
case of one variable, Hahn [10] obtained $q$-analogs of Kummer’s 24 solutions.
In future we hope that other solutions for $E_{N,M}$, which contain
$q$-analogs of Appell, Le Vavasseur, Erdélyi’s solutions will be obtained. It
is expected that these solutions will be generalizations of Hahn’s solutions.

  * (iii)

Our fundamental solutions $\bm{u}^{L,\sigma}$ are given by series. On the
other hand, a solution $\mathcal{F}_{N,M}$ has the Euler-type Jackson integral
representation (2.4). It is also expected that other solutions of the
$q$-difference system $E_{N,M}$ have the Euler-type Jackson integral
representation with suitable domain of integration.

  * (iv)

In recent years, the theory of elliptic difference equations has progressed.
For example, considering discrete isomonodromic deformations of a linear
difference system, an elliptic Garnier system which is a generalization of
elliptic Painlevé equation defined by Sakai [22] was obtained by Ormerod,
Rains [19] and Yamada [26]. Also, by using representation theory of the
elliptic quantum group $U_{q,p}\big{(}\widehat{\mathfrak{sl}}_{N}\big{)}$, an
explicit formula for elliptic hypergeometric integral solutions of the face
type elliptic $q$-KZ equation was obtained by Konno [13]. We hope that an
elliptic analog of the hypergeometric function $\mathcal{F}_{N,M}$ and its
related isomonodromic system will be obtained and our result will be extended
to the elliptic hypergeometric function.

## Acknowledgements

The author would like to thank Professor Yasuhiko Yamada for useful
discussions, valuable suggestions and his encouragement. He also thanks
Professor Saiei-Jaeyeong Matsubara-Heo for useful comments. He is also
grateful to Professor Wayne Rossman for careful reading and corrections in the
manuscript. In addition, he thanks the referees for reading carefully the
original manuscript and giving valuable suggestions.

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