On $q$-Isomonodromic Deformations and $q$-Nekrasov Functions

On $\boldsymbol{q}$-Isomonodromic Deformations
and $\boldsymbol{q}$-Nekrasov Functions

Hajime NAGOYA

H. Nagoya

School of Mathematics and Physics, Kanazawa University, Kanazawa, Ishikawa
920-1192, Japan<EMAIL_ADDRESS>
Received June 02, 2020, in final form May 04, 2021; Published online May 13,
2021

We construct a fundamental system of a $q$-difference Lax pair of rank $N$ in
terms of 5d Nekrasov functions with $q=t$. Our fundamental system degenerates
by the limit $q\to 1$ to a fundamental system of a differential Lax pair,
which yields the Fuji–Suzuki–Tsuda system. We introduce tau functions of our
system as Fourier transforms of 5d Nekrasov functions. Using asymptotic
expansions of the fundamental system at $0$ and $\infty$, we obtain several
determinantal identities of the tau functions.

isomonodromic deformations; Nekrasov functions; Painlevé equations;
determinantal identities

39A13; 33E17; 05A30

##  1 Introduction

Isomonodromic deformations of linear differential equations [18] have been
studied intensively and admit many applications both in mathematics and
physics. The recent paper [12] revealed a connection between isomonodromic
deformation and the Liouville conformal field theory that the tau function of
the sixth Painlevé equation $\mathrm{P_{VI}}$ is a Fourier transform of the
Virasoro conformal block with central charge $c=1$. Later, the tau functions
of other Painlevé equations $\mathrm{P_{V}}$, $\mathrm{P_{IV}}$,
$\mathrm{P_{III}}$, $\mathrm{P_{II}}$ were also recognized as Fourier
transforms of irregular conformal blocks [13, 22, 27, 28]. Furthermore, the
tau functions of isomonodromic deformation of particular Fuchsian systems of
rank $N$ was constructed by semi-degenerate conformal blocks of
$W_{N}$-algebra with central charge $c=N-1$ [15].

By AGT correspondence [1], Virasoro conformal blocks correspond to Nekrasov
partition functions of 4d supersymmetric gauge theories. Hence, the tau
functions of the Painlevé equations have explicit series expansions at regular
singular points. Correspondence between partition functions of gauge theories
and tau functions of Painlevé equations was investigated in [7]. In [3], using
Nakajima–Yoshioka blow-up equations [29] for the pure Nekrasov partition
functions, it was proved that the tau function of the Painlevé
$\mathrm{III}_{3}$ ($\mathrm{P_{III_{3}}}$) in terms of the pure Nekrasov
partition function satisfies the bilinear equation for it.

One of the reasons why conformal blocks appear in the construction of
isomonodromic tau functions is that when the central charge takes particular
values, monodromy invariant finite-systems can be constructed from conformal
blocks with the degenerate field [15, 16]. We note that by definition, local
monodromies of conformal blocks are independent of their singular points, and
connection problem of semi-degenerate $W_{N}$-conformal blocks with the
degenerate field is reduced to connection problem of the Clausen–Thomae
hypergeometric function ${}_{N}F_{N-1}$. Recall that $W_{2}$ is the Virasoro
algebra.

A $q$-analog of isomonodromic deformation of linear differential equations can
be considered. It corresponds to study connection preserved deformation of
linear $q$-difference equations, that is, $q$-difference Lax pairs with
rational coefficients of the independent variable. In [21], the $q$-Painlevé
VI ($q$-$\mathrm{P_{VI}}$) was obtained from the compatibility condition of a
Lax pair of rank 2. In [20], fundamental solutions of the Lax pair were
constructed in terms of the $q$-conformal blocks given as expectation values
of the intertwining operators of the Ding–Iohara–Miki algebra [2], and
expressed in terms of the $q$-Nekrasov functions. As a result, a general
solution to the $q$-Painlevé VI is expressed explicitly, and moreover, it was
conjectured that tau functions of $q$-$\mathrm{P_{VI}}$ defined as Fourier
transforms of the $q$-conformal blocks satisfy bilinear equations. Later, by
taking degeneration of $q$-$\mathrm{P_{VI}}$, explicit series representations
for the tau functions of the $q$-Painlevé V, $\mathrm{III_{1}}$,
$\mathrm{III_{2}}$, and $\mathrm{III_{3}}$ equations were obtained [23]. The
series representations of the tau function for $q$-$\mathrm{P_{{III}_{3}}}$
expressed by the pure 5d Nekrasov function had been proposed in [5] for
surface type $A_{7}^{(1)\prime}$ and [4] for surface type $A_{7}^{(1)}$. The
tau functions obtained in [23] are equivalent to those proposed in [4, 5]. A
Fredholm determinant representation for the tau functions of
$q$-$\mathrm{P_{III_{3}}}$ was presented in [6].

In this paper, we construct a fundamental system of a Lax pair of rank $N$ in
terms of 5d Nekrasov functions with $q=t$. Our fundamental system degenerates
by the limit $q\to 1$ to a fundamental system of a Lax pair $L_{M,N}$
($M\in\mathbb{Z}_{\geq 1}$) describing isomonodromy deformation of a Fuchsian
system with $M+3$ regular singular points

$\displaystyle\frac{{\rm d}Y}{{\rm
d}z}=\sum_{i=0}^{M+1}\frac{A_{i}}{z-x_{i}}Y,$

where $Y=Y(z)$, $A_{i}$ are $N$ by $N$ matrices, and whose spectral type is
given by the ($M+3$)-tuple

$\displaystyle\left(1^{N}\right),\quad\left(1^{N}\right),\quad\left(N-1,1\right),\quad\dots,\quad\left(N-1,1\right).$

Here, the spectral type expresses multiplicities of the eigenvalues of the
coefficient matrices $A_{i}$. The corresponding isomonodromic system was
studied from the point of view of UC hierarchy [37]. When $M=1$, the
isomonodromic system is also derived from similarity condition of the
Drinfeld–Sokolov hierarchy [11] ($N=3$), [35] ($N\geq 2$). We call this
isomonodromic system the Fuji–Suzuki–Tsuda system. Since a fundamental system
of $L_{M,N}$ was constructed by $W_{N}$ conformal blocks [15], the uniqueness
of a fundamental system of a Fuchsian system gives another justification of
AGT correspondence. We note that to be rigorous, we have to prove convergence
of $W_{N}$ conformal blocks and 5d Nekrasov partition functions, and moreover,
Fourier transforms of them.

A key ingredient to construct a monodromy invariant fundamental system from
conformal blocks is the braiding relation interchanging places of the variable
$x$ of the degenerate field with other singular points. In differential cases,
the braiding relation is derived from $W_{N}$ conformal field theory. In
$q$-difference cases; however, we have not yet understood braiding relations
of $q$-conformal blocks representation theoretically. Instead, taking
advantage of explicit expressions of $q$-Nekrasov functions, we directly prove
contiguity relations of $q$-Nekrasov functions with special values by
generalizing a method used in [20]. Consequently, we solve the connection
problem of $q$-conformal blocks with the degenerate field.

The remainder of this paper is organized as follows. In Section 2, we
introduce $q$-conformal blocks and investigate the connection problem of
degenerate $q$-conformal blocks. In Section 3, we construct a fundamental
system of a particular Lax pair of rank $N$, using degenerate $q$-conformal
blocks. We introduce tau functions of our system as Fourier transforms of
$q$-conformal blocks. Finally, we present determinantal identities of the tau
functions.

Notation. Throughout the paper we fix $q\in{\mathbb{C}}^{\times}$ such that
$|q|<1$. We set

$\displaystyle[u]=\frac{1-q^{u}}{1-q},\qquad(a;q)_{n}=\prod_{j=0}^{n-1}\big{(}1-aq^{j}\big{)},$
$\displaystyle(a_{1},\dots,a_{k};q)_{\infty}=\prod_{j=1}^{k}(a_{j};q)_{\infty},\qquad(a;q,q)_{\infty}=\prod_{j,k=0}^{\infty}\big{(}1-aq^{j+k}\big{)}.$

We use the $q$-gamma function, $q$-Barnes function and the $q$-theta function
defined by

$\displaystyle\Gamma_{q}(u)=\frac{(q;q)_{\infty}}{(q^{u};q)_{\infty}}(1-q)^{1-u},\qquad
G_{q}(u)=\frac{(q^{u};q,q)_{\infty}}{(q;q,q)_{\infty}}(q;q)_{\infty}^{u-1}(1-q)^{-(u-1)(u-2)/2},$
$\displaystyle\vartheta(u)=q^{u(u-1)/2}\Theta_{q}(q^{u}),\qquad\Theta_{q}(x)=(x,q/x,q;q)_{\infty},$

which satisfy $\Gamma_{q}(1)=G_{q}(1)=1$ and

$\displaystyle\Gamma_{q}(u+1)=[u]\Gamma_{q}(u),\qquad
G_{q}(u+1)=\Gamma_{q}(u)G_{q}(u),\qquad\vartheta(u+1)=-\vartheta(u)=\vartheta(-u).$

A partition is a finite sequence of positive integers
$\lambda=(\lambda_{1},\dots,\lambda_{l})$ such that
$\lambda_{1}\geq\cdots\geq$ $\lambda_{l}>0$. Denote the length of the
partition by $\ell(\lambda)=l$. The conjugate partition
$\lambda^{\prime}=(\lambda^{\prime}_{1},\dots,\lambda^{\prime}_{l^{\prime}})$
is defined by $\lambda^{\prime}_{j}=\sharp\\{i\mid\lambda_{i}\geq j\\}$,
$l^{\prime}=\lambda_{1}$. We regard a partition as a Young diagram. Namely, we
regard a partition $\lambda$ also as the subset
$\big{\\{}(i,j)\in{\mathbb{Z}}^{2}\mid 1\leq j\leq\lambda_{i},\ i\geq
1\big{\\}}$ of ${\mathbb{Z}}^{2}$, and denote its cardinality by $|\lambda|$.
We denote the set of all partitions by $\mathbb{Y}$. For
$\square=(i,j)\in{\mathbb{Z}}_{>0}^{2}$ we set
$a_{\lambda}(\square)=\lambda_{i}-j$ (the arm length of $\square$) and
$\ell_{\lambda}(\square)=\lambda^{\prime}_{j}-i$ (the leg length of
$\square$). In the last formulas we set $\lambda_{i}=0$ if $i>\ell(\lambda)$
(resp. $\lambda^{\prime}_{j}=0$ if $j>\ell(\lambda^{\prime})$). For a pair of
partitions ($\lambda,\mu$) and $u\in{\mathbb{C}}$ we set

$\displaystyle
N_{\lambda,\mu}(u)=\prod_{\square\in\lambda}\big{(}1-q^{-\ell_{\lambda}(\square)-a_{\mu}(\square)-1}u\big{)}\prod_{\square\in\mu}\big{(}1-q^{\ell_{\mu}(\square)+a_{\lambda}(\square)+1}u\big{)},$

which we call a Nekrasov factor. We set
$|\boldsymbol{\lambda}|=\sum_{i=1}^{N}\big{|}\lambda^{(i)}\big{|}$ for an
element $\boldsymbol{\lambda}=\big{(}\lambda^{(1)},\dots,\lambda^{(N)}\big{)}$
of ${\mathbb{Y}}^{N}$.

##  2 Contiguity relations

###  2.1 $\boldsymbol{q}$-conformal blocks

For $\sigma\in{\mathbb{C}}^{N}$, set
$\Delta_{\sigma}=\sum_{i=1}^{N}\big{(}\sigma^{(i)}\big{)}^{2}/2$. We define
$h_{i}\in{\mathbb{C}}^{N}$, $i=1,\dots,N$, as

$\displaystyle h_{i}^{(j)}=\delta_{ij}-\frac{1}{N},$

which may be regarded as the weights of the vector representation of
$\mathfrak{sl}_{N}$. We choose the normalization factor as

$\displaystyle{\mathcal{N}}\Bigl{(}{\kern-1.0pt\theta\atop{\sigma_{2}\phantom{h_{2}}\sigma_{1}}}\Bigr{)}=\frac{\prod_{k,k^{\prime}=1}^{N}G_{q}\big{(}1+\sigma_{2}^{(k)}-\theta-\sigma_{1}^{(k^{\prime})}\big{)}}{\prod_{1\leq
k<k^{\prime}\leq
N}G_{q}\big{(}1+\sigma_{2}^{(k)}-\sigma_{2}^{(k^{\prime})}\big{)}G_{q}\big{(}1-\sigma_{1}^{(k)}+\sigma_{1}^{(k^{\prime})}\big{)}}$

for $\theta\in{\mathbb{C}}$, $\sigma_{i}\in{\mathbb{C}}^{N}$, $i=1,2$.

######  Definition 2.1.

Let $m\in{\mathbb{Z}}_{\geq 2}$. For $\theta_{k}\in{\mathbb{C}}$ ($1\leq k\leq
m$), and
$\sigma_{k}=\big{(}\sigma_{k}^{(1)},\dots,\sigma_{k}^{(N)}\big{)}\in{\mathbb{C}}^{N}$,
$k=0,\dots,m$, such that $\sum_{j=1}^{N}\sigma_{k}^{(j)}=0$, we define the
$m+2$ point $q$-conformal block by

$\displaystyle\mathcal{F}\left({{\phantom{000}\theta_{m}\phantom{h000}\theta_{m-1}\phantom{h0}\phantom{h0}\cdots\phantom{h0}\phantom{h0}\theta_{1}\phantom{h0}}\atop{\sigma_{m}\phantom{h0}\sigma_{m-1}\phantom{h0}\sigma_{m-2}\phantom{h0}\cdots\phantom{h_{0}}\sigma_{1}\phantom{h0}\sigma_{0}}};\,x_{m},\dots,x_{1}\right)$
$\displaystyle\qquad=\prod_{p=1}^{m}{\mathcal{N}}\Bigl{(}{\kern-1.0pt\theta_{p}\atop{\sigma_{p}\phantom{h_{2}}\sigma_{p-1}}}\Bigr{)}q^{N\theta_{p}\Delta_{\sigma_{p}}}\prod_{p=1}^{m}x_{p}^{\Delta_{\sigma_{p}}-\Delta_{N\theta_{p}h_{1}}-\Delta_{\sigma_{p-1}}}$
$\displaystyle\qquad\phantom{=}{}\times\sum_{\boldsymbol{\lambda}_{1},\dots,\boldsymbol{\lambda}_{m-1}\in{\mathbb{Y}}^{N}}\prod_{p=1}^{m-1}\bigg{(}\frac{q^{N\theta_{p}}x_{p}}{x_{p+1}}\bigg{)}^{\left|\boldsymbol{\lambda}_{p}\right|}\frac{\prod_{p=1}^{m}\prod_{k,k^{\prime}=1}^{N}N_{\lambda_{p}^{(k)},\lambda_{p-1}^{(k^{\prime})}}\big{(}q^{\sigma_{p}^{(k)}-\theta_{p}-\sigma_{p-1}^{(k^{\prime})}}\big{)}}{\prod_{p=1}^{m-1}\prod_{k,k^{\prime}=1}^{N}N_{\lambda_{p}^{(k)},\lambda_{p}^{(k^{\prime})}}\big{(}q^{\sigma_{p}^{(k)}-\sigma_{p}^{(k^{\prime})}}\big{)}},$
(2.1)

where
$\boldsymbol{\lambda}_{p}=\big{(}\lambda_{p}^{(1)},\dots,\lambda_{p}^{(N)}\big{)}$,
and
$\boldsymbol{\lambda}_{0}=\boldsymbol{\lambda}_{m}=(\varnothing,\dots,\varnothing)$.

Our $q$-conformal blocks can be obtained as expectation values of intertwining
operators of the Ding–Iohara–Miki algebra [2]. We omit a derivation and refer
to [20, Section 2].111If we replace $\mathcal{V}_{\theta,w}$ on p. 6 in [20]
by the tensor product of the $N$ Fock modules of the Ding–Iohara–Miki algebra,
then we will get (2.1), provided that the normalization is suitably chosen.
The series part of a $q$-conformal block is the 5d Nekrasov instanton
partition function.

First, we relate a $4$-point conformal block to the $q$-hypergeometric series,
which is defined as follows.

######  Definition 2.2.

Let
$\alpha_{1},\dots,\alpha_{N},\,\beta_{1},\dots,\beta_{N-1}\in{\mathbb{C}}$. A
series

${}_{N}\phi_{N-1}\left(\begin{matrix}\alpha_{1},\dots,\alpha_{N}\\\
\beta_{1},\dots,\beta_{N-1}\\\
\end{matrix};\,q,z\right)=\sum_{k=0}^{\infty}\frac{\big{(}q^{\alpha_{1}};q\big{)}_{k}\cdots\big{(}q^{\alpha_{N}};q\big{)}_{k}}{\big{(}q^{\beta_{1}};q\big{)}_{k}\cdots\big{(}q^{\beta_{N-1}};q\big{)}_{k}(q;q)_{k}}z^{k}$

is called the $q$-hypergeometric series.

Let us recall some elementally facts about the Nekrasov factor.

######  Lemma 2.3.

For $\lambda,\mu\in{\mathbb{Y}},w\in{\mathbb{C}}$, we have

  * $(1)$

$N_{\lambda,\varnothing}\big{(}q^{-1}\big{)}\neq 0$ if and only if
$\lambda=(1^{n})$, $n=0,1,2,\dots$,

  * $(2)$

$N_{(1^{n}),\varnothing}(w)=(q^{-n+1}w;q)_{n}$,

  * $(3)$

$N_{\varnothing,(1^{n})}(w)=(w;q)_{n}$,

  * $(4)$

$N_{(1^{n}),(1^{n})}(w)=\big{(}q^{-n}w;q\big{)}_{n}(qw;q)_{n}$.

We need to substitute a special value into the normalization factor as

$\displaystyle{\mathcal{N}}\left(\frac{1}{N}\atop{\sigma_{1}-h_{i}\phantom{h_{2}}\sigma_{1}}\right).$

But then it has a zero factor. We redefine the normalization factor in this
case as

$\displaystyle{\mathcal{N}}^{\prime}\left(\frac{1}{N}\atop{\sigma_{2}\phantom{h_{2}}\sigma_{2}+h_{i}}\right)=\lim_{\epsilon\to
0}\frac{1}{G_{q}(\epsilon)}{\mathcal{N}}\left(\frac{1}{N}-\epsilon\atop{\sigma_{2}\phantom{h_{2}}\sigma_{2}+h_{i}}\right).$

The $q$-conformal block with the special value above is called degenerate
$q$-conformal block. The reduction of the 5d Nekrasov partition function to
the $q$-hypergeometric series was explained in [8, 9], which is stated in our
notation as follows.

######  Proposition 2.4.

For $i=1,\dots,N$, we have

$\displaystyle\mathcal{F}\left({\frac{1}{N}\phantom{0000}\theta_{1}\atop\sigma_{2}\phantom{00}\sigma_{2}+h_{i}\phantom{00}\sigma_{0}};\,x_{1},x_{2}\right)$
$\displaystyle\qquad{}=\mathcal{N}q^{\Delta_{\sigma_{2}}}\bigg{(}\frac{q^{N\theta_{1}}x_{2}}{x_{1}}\bigg{)}^{\sigma_{2}^{(i)}+(1-1/N)/2}\frac{\prod_{k=1}^{N}\Gamma_{q}\big{(}1-\frac{1}{N}+\sigma_{2}^{(i)}-\theta_{1}-\sigma_{0}^{(k)}\big{)}}{\prod_{k=1}^{N}\Gamma_{q}\big{(}1+\sigma_{2}^{(i)}-\sigma_{2}^{(k)}\big{)}}$
$\displaystyle\qquad\phantom{=}{}\times{}_{N}\phi_{N-1}\left(\begin{matrix}\big{\\{}1-\tfrac{1}{N}+\sigma_{2}^{(i)}-\theta_{1}-\sigma_{0}^{(k)}\big{\\}}_{k=1}^{N}\\\\[2.15277pt]
\big{\\{}1+\sigma_{2}^{(i)}-\sigma_{2}^{(k)}\big{\\}}_{k=1,\,k\neq
i}^{N}\end{matrix};\,q,\frac{q^{N\theta_{1}}x_{2}}{x_{1}}\right)\\!,$
$\displaystyle\mathcal{F}\left({\theta_{1}\phantom{0000}\frac{1}{N}\atop\sigma_{2}\phantom{00}\sigma_{0}-h_{i}\phantom{00}\sigma_{0}};\,x_{2},x_{1}\right)$
$\displaystyle\qquad{}=\mathcal{N}q^{\Delta_{\sigma_{0}}}\bigg{(}\frac{qx_{1}}{x_{2}}\bigg{)}^{-\sigma_{0}^{(i)}+(1-1/N)/2}\frac{\prod_{k=1}^{N}\Gamma_{q}\big{(}1-\frac{1}{N}+\sigma_{2}^{(k)}-\theta_{1}-\sigma_{0}^{(i)}\big{)}}{\prod_{k=1}^{N}\Gamma_{q}\big{(}1+\sigma_{0}^{(k)}-\sigma_{0}^{(i)}\big{)}}$
$\displaystyle\qquad\phantom{=}{}\times{}_{N}\phi_{N-1}\left(\begin{matrix}\big{\\{}1-\frac{1}{N}+\sigma_{2}^{(k)}-\theta_{2}-\sigma_{0}^{(i)}\big{\\}}_{k=1}^{N}\\\\[2.15277pt]
\big{\\{}1+\sigma_{0}^{(k)}-\sigma_{0}^{(i)}\big{\\}}_{k=1,\,k\neq
i}^{N}\end{matrix};\,q,\frac{qx_{1}}{x_{2}}\right)\\!,$

where

$\displaystyle\mathcal{N}=q^{N\theta_{1}\Delta_{\sigma_{2}}}x_{1}^{-(1-1/N)/2}x_{2}^{\Delta_{\sigma_{2}}-\Delta_{N\theta_{1}h_{1}}-\Delta_{\sigma_{0}}}$
$\displaystyle\hphantom{\mathcal{N}=}{}\times\frac{\prod_{k,k^{\prime}=1}^{N}G_{q}\big{(}1-\frac{1}{N}+\sigma_{2}^{(k)}-\theta_{1}-\sigma_{0}^{(k^{\prime})}\big{)}}{\prod_{1\leq
k<k^{\prime}\leq
N}G_{q}\big{(}1+\sigma_{2}^{(k)}-\sigma_{2}^{(k^{\prime})}\big{)}G_{q}\big{(}1-\sigma_{0}^{(k)}+\sigma_{0}^{(k^{\prime})}\big{)}}\\!.$

###### Proof.

By the definition of $q$-conformal blocks and Lemma 2.3, we obtain the desired
results. ∎

###  2.2 Connection problem

The connection formula of the $q$-hypergeometric series is well known. See the
equation (4.5.2) on p. 121 in [14] for example. From Proposition 2.4, the
connection formula of the degenerate $4$-point $q$-conformal block reads as

$\displaystyle\mathcal{F}\left({\frac{1}{N}\phantom{0000}\theta_{1}\atop\sigma_{2}\phantom{00}\sigma_{2}+h_{i}\phantom{00}\sigma_{0}};\,x_{1},x_{2}\right)$
$\displaystyle\qquad{}=\sum_{j=1}^{N}\mathcal{F}\left({\theta_{1}\phantom{0000}\frac{1}{N}\atop\sigma_{2}\phantom{00}\sigma_{0}-h_{j}\phantom{00}\sigma_{0}};\,x_{2},x_{1}\right){\mathcal{B}}_{j,i}\left[\begin{matrix}\theta_{1}&\frac{1}{N}\\\
\sigma_{2}&\sigma_{0}\end{matrix}\Bigl{|}\frac{x_{2}}{x_{1}}\right]q^{N\theta_{1}^{2}/2-\theta_{1}/2}\bigg{(}\frac{x_{2}}{x_{1}}\bigg{)}^{\theta_{1}},$
(2.2)

where ${\mathcal{B}}\left[\begin{smallmatrix}\theta_{1}&\frac{1}{N}\\\
\sigma_{2}&\sigma_{0}\end{smallmatrix}\Bigl{|}x\right]$ is the $N$ by $N$
matrix given by

$\displaystyle{\mathcal{B}}_{j,i}\left[\begin{matrix}\theta_{1}&\frac{1}{N}\\\
\sigma_{2}&\sigma_{0}\end{matrix}\Bigl{|}x\right]=\frac{\vartheta\big{(}1\\!-\frac{1}{N}+\sigma_{2}^{(i)}\\!+(N\\!-1)\theta_{1}\\!-\sigma_{0}^{(j)}+u\big{)}}{\vartheta(N\theta_{1}+u)}\frac{\prod_{k=1,\,k\neq
i}^{N}\vartheta\big{(}\frac{1}{N}\\!-\sigma_{2}^{(k)}\\!+\theta_{1}+\sigma_{0}^{(j)}\big{)}}{\prod_{k=1,k\neq
j}^{N}\vartheta\big{(}\sigma_{0}^{(j)}\\!-\sigma_{0}^{(k)}\big{)}}$

with $x=q^{u}$. We mention that the connection problem of the
$q$-hypergeometric series was used to derive the three-point function for the
$q$-deformed correlators where conformal blocks are controlled by the
$q$-Virasoro algebra [30].

The connection matrix
${\mathcal{B}}\left[\begin{smallmatrix}\theta_{1}&\frac{1}{N}\\\
\sigma_{2}&\sigma_{0}\end{smallmatrix}\Bigl{|}x\right]$ enjoys the following
properties.

######  Lemma 2.5.

We have

$\displaystyle{\mathcal{B}}\left[\begin{matrix}\theta_{1}&\frac{1}{N}\\\
\sigma_{2}&\sigma_{0}\end{matrix}\Bigl{|}qx\right]={\mathcal{B}}\left[\begin{matrix}\theta_{1}&\frac{1}{N}\\\
\sigma_{2}&\sigma_{0}\end{matrix}\Bigl{|}x\right],$ (2.3)
$\displaystyle{\mathcal{B}}\left[\begin{matrix}\theta_{1}+1&\frac{1}{N}\\\
\sigma_{2}&\sigma_{0}\end{matrix}\Bigl{|}x\right]=(-1)^{N}{\mathcal{B}}\left[\begin{matrix}\theta_{1}&\frac{1}{N}\\\
\sigma_{2}&\sigma_{0}\end{matrix}\Bigl{|}x\right],$ (2.4)
$\displaystyle{\mathcal{B}}\left[\begin{matrix}\theta_{1}&\frac{1}{N}\\\
\sigma_{2}+h_{i}&\sigma_{0}+h_{j}\end{matrix}\Bigl{|}x\right]={\mathcal{B}}\left[\begin{matrix}\theta_{1}&\frac{1}{N}\\\
\sigma_{2}&\sigma_{0}\end{matrix}\Bigl{|}x\right],\qquad i,j=1,\dots,N,$ (2.5)
$\displaystyle{\mathcal{B}}\left[\begin{matrix}\theta_{1}+1/N&\frac{1}{N}\\\
\sigma_{2}&\sigma_{0}+h_{i}\end{matrix}\Bigl{|}x\right]=-{\mathcal{B}}\left[\begin{matrix}\theta_{1}&\frac{1}{N}\\\
\sigma_{2}&\sigma_{0}\end{matrix}\Bigl{|}x\right],\qquad i=1,\dots,N,$ (2.6)
$\displaystyle\det{\mathcal{B}}\left[\begin{matrix}\theta_{1}&\frac{1}{N}\\\
\sigma_{2}&\sigma_{0}\end{matrix}\Bigl{|}x\right]=(-1)^{N-1}\frac{\vartheta(u)}{\vartheta(N\theta_{1}+u)}\prod_{1\leq
i<j\leq
N}\frac{\vartheta\big{(}\sigma_{2}^{(i)}-\sigma_{2}^{(j)}\big{)}}{\vartheta\big{(}\sigma_{0}^{(i)}-\sigma_{0}^{(j)}\big{)}}.$
(2.7)

###### Proof.

The identities (2.3), (2.4), (2.5), and (2.6) are immediate consequences of
the definition of the connection matrix. The expression of the determinant
(2.7) of the connection matrix can be verified by straightforward calculations
as follows.

By definition, we have

$\displaystyle\det{\mathcal{B}}\left[\begin{matrix}\theta_{1}&\frac{1}{N}\\\
\sigma_{2}&\sigma_{0}\end{matrix}\Bigl{|}x\right]=\frac{1}{\vartheta(N\theta_{1}+u)^{N}\prod_{i=1}^{N}\prod_{k\neq
i}\vartheta\big{(}\sigma_{0}^{(i)}-\sigma_{0}^{(k)}\big{)}}\det(a_{ij}),$

where

$\displaystyle
a_{ij}=\vartheta\big{(}1-\tfrac{1}{N}+\sigma_{2}^{(j)}+(N-1)\theta_{1}-\sigma_{0}^{(i)}+u\big{)}\prod_{k\neq
j}^{N}\vartheta\big{(}\tfrac{1}{N}-\sigma_{2}^{(k)}+\theta_{1}+\sigma_{0}^{(i)}\big{)}.$

Put
$x_{ij}=1-\tfrac{1}{N}+\sigma_{2}^{(j)}+(N-1)\theta_{1}-\sigma_{0}^{(i)}+u$
and $y_{ij}=\tfrac{1}{N}-\sigma_{2}^{(j)}+\theta_{1}+\sigma_{0}^{(i)}$. Using
the relation on the theta function:

$\displaystyle\vartheta(x+y)\vartheta(x-y)\vartheta(u+v)\vartheta(u-v)-\vartheta(x+v)\vartheta(x-v)\vartheta(u+y)\vartheta(u-y)$
$\displaystyle\qquad{}=\vartheta(x+u)\vartheta(x-u)\vartheta(y+v)\vartheta(y-v),$

we obtain for $i,j\geq 2$

$\displaystyle
a_{11}a_{ij}-a_{i1}a_{1j}=\left(\vartheta(x_{11})\vartheta(y_{1j})\vartheta(x_{ij})\vartheta(y_{i1})-\vartheta(x_{i1})\vartheta(y_{ij})\vartheta(x_{1j})\vartheta(y_{11})\right)\prod_{k\neq
1,j}\vartheta(y_{1k})\vartheta(y_{ik})$
$\displaystyle\hphantom{a_{11}a_{ij}-a_{i1}a_{1j}}{}=\vartheta(1+N\theta_{1}+u)\vartheta\big{(}1-\tfrac{2}{N}+(N-2)\theta_{1}+u+\sigma_{2}^{(1)}+\sigma_{2}^{(j)}-\sigma_{0}^{(1)}-\sigma_{0}^{(i)}\big{)}$
$\displaystyle\hphantom{a_{11}a_{ij}-a_{i1}a_{1j}=}{}\times\vartheta\big{(}\sigma_{2}^{(1)}-\sigma_{2}^{(j)}\big{)}\vartheta\big{(}{-}\sigma_{0}^{(1)}+\sigma_{0}^{(i)}\big{)}\prod_{k\neq
1,j}\vartheta(y_{1k})\vartheta(y_{ik}).$

Hence, the determinant becomes

$\displaystyle\det{\mathcal{B}}\left[\begin{matrix}\theta_{1}&\frac{1}{N}\\\
\sigma_{2}&\sigma_{0}\end{matrix}\Bigl{|}x\right]=\frac{\prod_{k=2}^{N}\vartheta\big{(}\sigma_{2}^{(1)}-\sigma_{2}^{(k)}\big{)}}{\vartheta(N\theta_{1}+u)\vartheta(x_{11})^{N-2}\prod_{i=2}^{N}\prod_{k\neq
i}\vartheta\big{(}\sigma_{0}^{(i)}-\sigma_{0}^{(k)}\big{)}}\det(b_{ij}),$

where

$\displaystyle
b_{ij}=\vartheta\big{(}1-\tfrac{2}{N}+(N-2)\theta_{1}+u+\sigma_{2}^{(1)}+\sigma_{2}^{(j)}-\sigma_{0}^{(1)}-\sigma_{0}^{(i)}\big{)}\\!\prod_{k\neq
1,j}\\!\vartheta(y_{ik}),\qquad 2\leq i,j\leq N.$

In the same way, by computing $b_{22}b_{ij}-b_{i2}b_{2j}$, the determinant
$\det(b_{ij})$ reduces to a determinant of size $N-2$ whose entries are
products of the theta functions. Repeating the computation, we obtain the
desired result (2.7). ∎

We note that the connection matrix has periodicity on the variable $x$ and
parameters $\sigma_{0}$, $\sigma_{2}$, and $\theta_{1}$. Furthermore, the
determinant has a simple form in terms of the theta function.

We want to prove the following connection formula for $6$-point degenerate
$q$-conformal block

$\displaystyle\mathcal{F}\left({\theta_{4}\phantom{000}\frac{1}{N}\phantom{00000}\theta_{2}\phantom{000}\theta_{1}\phantom{}\atop\sigma_{4}\phantom{00}\sigma_{3}\phantom{00}\sigma_{3}+h_{i}\phantom{00}\sigma_{1}\phantom{00}\sigma_{0}};\,x_{4},x_{2},x_{3},x_{1}\right)$
$\displaystyle\qquad{}=\sum_{j=1}^{N}\mathcal{F}\left({\theta_{4}\phantom{000}\theta_{2}\phantom{00000}\frac{1}{N}\phantom{000}\theta_{1}\phantom{}\atop\sigma_{4}\phantom{00}\sigma_{3}\phantom{00}\sigma_{1}-h_{j}\phantom{00}\sigma_{1}\phantom{00}\sigma_{0}};\,x_{4},x_{3},x_{2},x_{1}\right){\mathcal{B}}_{j,i}\left[\begin{matrix}\theta_{2}&\frac{1}{N}\\\
\sigma_{3}&\sigma_{1}\end{matrix}\Bigl{|}\frac{x_{3}}{x_{2}}\right]$
$\displaystyle\qquad\phantom{=}{}\times
q^{N\theta_{2}^{2}/2-\theta_{2}/2}\bigg{(}\frac{x_{3}}{x_{2}}\bigg{)}^{\theta_{2}}.$
(2.8)

The constant with respect to the variables $x_{1}$, $x_{4}$ in the $6$-point
degenerate $q$-conformal block is $q$-hypergeometric series. In this case, the
connection problem is solved. We solve the connection problem (2.8) by solving
connection problem of each coefficient of $x_{1}$, $x_{4}$ in the $6$-point
degenerate $q$-conformal block. We consider the following series obtained by
taking the coefficient of
$x_{1}^{|\boldsymbol{\lambda}|}x_{4}^{-|\boldsymbol{\nu}|}$ in the $6$-point
$q$-conformal block:

$\displaystyle
S_{\boldsymbol{\lambda},\boldsymbol{\nu}}\left({\theta_{3}\phantom{000}\theta_{2}\atop\sigma_{3}\phantom{00}\sigma_{2}\phantom{00}\sigma_{1}};\,x_{3},x_{2}\right)=\sum_{\boldsymbol{\mu}\in{\mathbb{Y}}^{N}}\bigg{(}\frac{q^{N\theta_{2}}x_{2}}{x_{3}}\bigg{)}^{|\boldsymbol{\mu}|}\prod_{k,k^{\prime}=1}^{N}\frac{N_{\lambda_{k},\mu_{k^{\prime}}}(w_{k,k^{\prime}})N_{{\mu_{k}},\nu_{k^{\prime}}}(z_{k,k^{\prime}})}{N_{\mu_{k},\mu_{k^{\prime}}}(u_{k,k^{\prime}})}.$
(2.9)

Here we put

$\displaystyle
w_{k,k^{\prime}}=q^{\sigma_{k}^{(3)}-\theta_{3}-\sigma_{k^{\prime}}^{(2)}},\qquad
z_{k,k^{\prime}}=q^{\sigma_{k}^{(2)}-\theta_{2}-\sigma_{k^{\prime}}^{(1)}},\qquad
u_{k,k^{\prime}}=q^{\sigma_{k}^{(2)}-\sigma_{k^{\prime}}^{(2)}}.$

We note that for
$\boldsymbol{\lambda}=\boldsymbol{\nu}=(\varnothing,\dots,\varnothing)$,

$\displaystyle\mathcal{F}\left({\theta_{3}\phantom{000}\theta_{2}\atop\sigma_{3}\phantom{00}\sigma_{2}\phantom{00}\sigma_{1}};\,x_{3},x_{2}\right)$
$\displaystyle\qquad{}=\prod_{p=2}^{3}{\mathcal{N}}\Bigl{(}{\kern-1.0pt\theta_{p}\atop{\sigma_{p}\phantom{h_{2}}\sigma_{p-1}}}\Bigr{)}q^{N\theta_{p}\Delta_{\sigma_{p}}}x_{p}^{\Delta_{\sigma_{p}}-\Delta_{N\theta_{p}h_{1}}-\Delta_{\sigma_{p-1}}}\,S_{\varnothing,\varnothing}\left({\theta_{3}\phantom{000}\theta_{2}\atop\sigma_{3}\phantom{00}\sigma_{2}\phantom{00}\sigma_{1}};\,x_{3},x_{2}\right)\\!.$

From Proposition 2.4 if we set $\theta_{3}=1/N$, $\sigma_{2}=\sigma_{3}+h_{i}$
or $\theta_{2}=1/N$, $\sigma_{2}=\sigma_{1}-h_{i}$ in (2.9), then
$S_{\varnothing,\varnothing}$ becomes the $q$-hypergeomeric series
${}_{N}\phi_{N-1}$. In that case, $\boldsymbol{\mu}$ in the right hand side of
(2.9) restricts to $\mu_{i}=(1^{n})$ ($n=0,1,2,\dots$), $\mu_{k}=\varnothing$
($k\neq i$). For the general $\boldsymbol{\lambda}$, $\boldsymbol{\nu}$, a
tuple $\boldsymbol{\mu}$ of partitions also takes the following particular
forms.

Let us introduce the following operations on partitions
$\lambda\in{\mathbb{Y}}$:

$\displaystyle\bar{\lambda}=(\lambda_{1}-1,\dots,\lambda_{\ell(\lambda)}-1)\,,$
(2.10) $\displaystyle
r_{n}(\lambda)=(\lambda_{1}+1,\dots,\lambda_{n}+1,\lambda_{n+2},\dots),\qquad
n\in{\mathbb{Z}}_{\geq 0}.$ (2.11)

Here we identify a partition $\lambda$ with a sequence
$\big{(}\lambda_{1},\dots,\lambda_{\ell(\lambda)},0,0,\dots\big{)}$.

Let us recall the following lemma.

######  Lemma 2.6.

For $\lambda,\mu\in{\mathbb{Y}}$, we have

  * $(1)$

$N_{\lambda,\mu}(1)\neq 0$ if and only if $\lambda=\mu$,

  * $(2)$

$N_{\lambda,\mu}(q^{-1})\neq 0$ if and only if $\lambda=r_{n}(\mu)$,
$n=0,1,2,\dots$,

  * $(3)$

$N_{\lambda,\mu}(u)=N_{\mu^{\prime},\lambda^{\prime}}(u)$.

Lemma 2.6(1) is easily obtained from the definition of the Nekrasov factor and
Lemma 2.6(2), (3) were proved in [20] as Lemmas A.2 and A.3, respectively.

Suppose that $\theta_{3}=1/N$, $\sigma_{2}=\sigma_{3}+h_{i}$ in (2.9), then
the following holds:

$\displaystyle w_{i,i}=q^{\sigma_{3}^{(i)}-1/N-\sigma_{2}^{(i)}}=q^{-1},\qquad
w_{k,k}=q^{\sigma_{3}^{(k)}-1/N-\sigma_{2}^{(k)}}=1,\qquad k\neq i.$

Hence by Lemma 2.6(1), (2), we obtain

$\displaystyle\mu_{i}=r_{n}(\nu_{i}),\qquad\mu_{k}=\nu_{k},\qquad k\neq i.$

Similarly, suppose that $\theta_{2}=1/N$, $\sigma_{2}=\sigma_{1}-h_{i}$ in
(2.9), then the following holds:

$\displaystyle z_{i,i}=q^{\sigma_{2}^{(i)}-1/N-\sigma_{1}^{(i)}}=q^{-1},\qquad
z_{k,k}=q^{\sigma_{2}^{(k)}-1/N-\sigma_{1}^{(k)}}=1,\qquad k\neq i.$

Hence by Lemma 2.6(1), (2), we obtain

$\displaystyle\mu_{i}=r_{n}(\lambda_{i}),\qquad\mu_{k}=\lambda_{k},\qquad
k\neq i.$

We also recall the following lemma.

######  Lemma 2.7 ([20, Lemma A.4]).

Let $\lambda,\mu\in{\mathbb{Y}}$ and $n\in{\mathbb{Z}}_{\geq 0}$. Using the
notation (2.10) and (2.11), we set

$\displaystyle\ell=\ell(\lambda),\qquad
k=\ell(\mu),\qquad\eta=r_{n}(\lambda),\qquad\gamma=(r_{n}(\mu^{\prime}))^{\prime},$
$\displaystyle\tilde{\eta}=\begin{cases}\bar{\eta},&n\leq\ell-1,\\\
\big{(}\lambda,1^{n-\ell+1}\big{)},&n\geq\ell.\end{cases}$

Then we have

$\displaystyle\frac{N_{\eta,\lambda}(q^{-1})}{N_{\eta,\eta}(1)}=\frac{N_{\tilde{\eta},\bar{\lambda}}(q^{-1})}{N_{\tilde{\eta},\tilde{\eta}}(1)}\big{(}1-q^{|\tilde{\eta}|-|\lambda|}\big{)},$
$\displaystyle
N_{\mu,\eta}(u)=N_{\mu,\tilde{\eta}}(q^{-1}u)\prod_{j=1}^{\mu_{\ell}}\frac{1-q^{j-1}u}{1-q^{-\ell_{\mu}(\ell,j)+j-2}u}\prod_{i=1}^{\ell-1}\big{(}1-q^{\ell-i+a_{\mu}(i,1)}u\big{)},$
$\displaystyle
N_{\mu,\lambda}(u)=N_{\mu,\bar{\lambda}}(q^{-1}u)\prod_{j=1}^{\mu_{\ell+1}}\frac{1-q^{j-1}u}{1-q^{-\ell_{\mu}(\ell+1,j)+j-2}u}\prod_{i=1}^{\ell}\big{(}1-q^{\ell-i+a_{\mu}(i,1)+1}u\big{)},$
$\displaystyle\frac{N_{\mu,\lambda}(u)}{N_{\mu,\eta}(qu)}=\frac{N_{\mu,\bar{\lambda}}(q^{-1}u)}{N_{\mu,\tilde{\eta}}(u)}(1-u),$
$\displaystyle\frac{N_{\mu,\lambda}(u)}{N_{\mu,\eta}(qu)}=\frac{N_{\bar{\mu},\lambda}(qu)}{N_{\bar{\mu},\eta}(q^{2}u)}\frac{1-q^{|\eta|-|\lambda|+1-k}u}{1-qu},$
$\displaystyle
N_{\gamma,\lambda}(u)=N_{\gamma,\bar{\lambda}}(q^{-1}u)\big{(}1-q^{\ell+|\gamma|-|\mu|-1}u\big{)}\prod_{j=1}^{\mu_{\ell}}\frac{1-q^{j-1}u}{1-q^{-\ell_{\mu}(\ell,j)+j-2}u}\prod_{i=1}^{\ell-1}\big{(}1-q^{\ell-i+a_{\mu}(i,1)}u\big{)}.$

We denote by $T_{q,z}$, the $q$-shift operator with respect to the variable
$z$, namely, $T_{q,z}(f(z))=f(qz)$ for a function $f(z)$. For a partition
$\lambda=(\lambda_{1},\lambda_{2},\dots)$, we define

$\displaystyle\lambda+1^{m}=(\lambda_{1}+1,\lambda_{2}+1,\dots,\lambda_{m}+1,\lambda_{m+1},\dots).$

######  Theorem 2.8.

For $i,j$ $(1\leq i,j\leq N)$, let $m\geq\ell(\lambda_{j})$, and

$\displaystyle\hat{\boldsymbol{\lambda}}=\big{(}\lambda_{1},\dots,\lambda_{j}+1^{m},\dots,\lambda_{N}\big{)},$
$\displaystyle
A_{\boldsymbol{\lambda},\boldsymbol{\nu}}^{(j)}(\theta_{2},\sigma_{1},\sigma_{3})=\prod_{k=1}^{N}\frac{N_{\nu_{k},\hat{\lambda}_{j}}\big{(}q^{\sigma_{3}^{(k)}-\theta_{2}-\sigma_{1}^{(j)}+1-1/N}\big{)}}{N_{\nu_{k},\lambda_{j}}\big{(}q^{\sigma_{3}^{(k)}-\theta_{2}-\sigma_{1}^{(j)}-1/N}\big{)}},$

we have

$\displaystyle
S_{\hat{\boldsymbol{\lambda}},\boldsymbol{\nu}}\left({\frac{1}{N}\phantom{000}\theta_{2}-\frac{1}{N}\atop\sigma_{3}\phantom{00}\sigma_{3}+h_{i}\phantom{00}\sigma_{1}-h_{j}};\,x_{2},x_{3}\right)$
$\displaystyle\qquad{}=q^{m-|\boldsymbol{\nu}|+1-1/N+\sigma_{3}^{(i)}-\theta_{2}-\sigma_{1}^{(j)}}A_{\boldsymbol{\lambda},\boldsymbol{\nu}}^{(j)}(\theta_{2},\sigma_{1},\sigma_{3})\frac{1}{1-q^{\sigma_{3}^{(i)}-\theta_{2}-\sigma_{1}^{(j)}+1-1/N}}$
$\displaystyle\qquad\phantom{=}{}\times\big{(}q^{|\boldsymbol{\nu}|-m-1+1/N-\sigma_{3}^{(i)}+\theta_{2}+\sigma_{1}^{(j)}}T_{q,x_{2}}-1\big{)}S_{\boldsymbol{\lambda},\boldsymbol{\nu}}\left({\frac{1}{N}\phantom{000}\theta_{2}\atop\sigma_{3}\phantom{00}\sigma_{3}+h_{i}\phantom{00}\sigma_{1}};\,x_{2},x_{3}\right)\\!,$
(2.12) $\displaystyle
S_{\hat{\boldsymbol{\lambda}},\boldsymbol{\nu}}\left({\theta_{2}-\frac{1}{N}\phantom{000}\frac{1}{N}\atop\sigma_{3}\phantom{00}\sigma_{1}-h_{i}-h_{j}\phantom{00}\sigma_{1}-h_{j}};\,x_{3},x_{2}\right)=\left(\frac{qx_{2}}{x_{3}}\right)^{m}A_{\boldsymbol{\lambda},\boldsymbol{\nu}}^{(j)}(\theta_{2},\sigma_{1},\sigma_{3})$
$\displaystyle\qquad\phantom{=}{}\times\frac{1-q^{-m-|\boldsymbol{\lambda}|+\sigma_{1}^{(j)}-\sigma_{1}^{(i)}}T_{q,x_{2}}}{1-q^{\sigma_{1}^{(j)}-\sigma_{1}^{(i)}}}S_{\boldsymbol{\lambda},\boldsymbol{\nu}}\left({\theta_{2}\phantom{000}\frac{1}{N}\atop\sigma_{3}\phantom{00}\sigma_{1}-h_{i}\phantom{00}\sigma_{1}};\,x_{3},x_{2}\right),\qquad
i\neq j,$ (2.13) $\displaystyle
S_{\hat{\boldsymbol{\lambda}},\boldsymbol{\nu}}\left({\theta_{2}-\frac{1}{N}\phantom{000}\frac{1}{N}\atop\sigma_{3}\phantom{00}\sigma_{1}-2h_{i}\phantom{00}\sigma_{1}-h_{i}};\,x_{3},x_{2}\right)$
$\displaystyle\qquad{}=\bigg{(}\frac{qx_{2}}{x_{3}}\bigg{)}^{m-1}A_{\boldsymbol{\lambda},\boldsymbol{\nu}}^{(j)}(\theta_{2},\sigma_{1},\sigma_{3})\prod_{k=1}^{N}\frac{1-q^{\sigma_{1}^{(k)}-\sigma_{1}^{(i)}+1}}{1-q^{\sigma_{3}^{(k)}-\theta_{2}-\sigma_{1}^{(j)}+1-1/N}}\frac{1-q^{-m-|\boldsymbol{\lambda}|}T_{q,x_{2}}}{1-q}$
$\displaystyle\qquad\phantom{=}{}\times
S_{\boldsymbol{\lambda},\boldsymbol{\nu}}\left({\theta_{2}\phantom{000}\frac{1}{N}\atop\sigma_{3}\phantom{00}\sigma_{1}-h_{i}\phantom{00}\sigma_{1}};\,x_{3},x_{2}\right)\\!.$
(2.14)

###### Proof.

We can verify the equations (2.12), (2.13), and (2.14) by direct computations
using Lemma 2.7. We omit the calculation. ∎

We note that if
$\boldsymbol{\lambda}=\boldsymbol{\nu}=(\varnothing,\dots,\varnothing)$, then
we can take $m=0$ in Theorem 2.8 and then
$\hat{\boldsymbol{\lambda}}=(\varnothing,\dots,\varnothing)$. In that case,
the equations (2.12), (2.13), and (2.14) are equal to the contiguity relations
for the $q$-hypergeometric series. Hence, we may regard the contiguity
relations (2.12), (2.13), and (2.14) as a generalization of the contiguity
relation of the $q$-hypergeometric series.

Let us denote the coefficient of
$x_{1}^{|\boldsymbol{\lambda}|}x_{4}^{-|\boldsymbol{\nu}|}$ in the $6$-point
$q$-conformal block by

$\displaystyle{\mathcal{F}}_{\boldsymbol{\lambda},\boldsymbol{\nu}}\left({\theta_{3}\phantom{000}\theta_{2}\atop\sigma_{3}\phantom{00}\sigma_{2}\phantom{00}\sigma_{1}};\,x_{3},x_{2}\right)$
$\displaystyle\qquad{}=\prod_{p=2}^{3}{\mathcal{N}}\Bigl{(}{\kern-1.0pt\theta_{p}\atop{\sigma_{p}\phantom{h_{2}}\sigma_{p-1}}}\Bigr{)}q^{N\theta_{p}\Delta_{\sigma_{p}}}x_{p}^{\Delta_{\sigma_{p}}-\Delta_{N\theta_{p}h_{1}}-\Delta_{\sigma_{p-1}}}x_{3}^{|\boldsymbol{\nu}|}x_{2}^{-|\boldsymbol{\lambda}|}S_{\boldsymbol{\lambda},\boldsymbol{\nu}}\left({\theta_{3}\phantom{000}\theta_{2}\atop\sigma_{3}\phantom{00}\sigma_{2}\phantom{00}\sigma_{1}};\,x_{3},x_{2}\right)\\!.$

We note that if
$\boldsymbol{\lambda}=\boldsymbol{\nu}=(\varnothing,\dots,\varnothing)$, then
it is a 4-point $q$-conformal block, namely

$\displaystyle{\mathcal{F}}_{\varnothing,\varnothing}\left({\theta_{3}\phantom{000}\theta_{2}\atop\sigma_{3}\phantom{00}\sigma_{2}\phantom{00}\sigma_{1}};\,x_{3},x_{2}\right)={\mathcal{F}}\left({\theta_{3}\phantom{000}\theta_{2}\atop\sigma_{3}\phantom{00}\sigma_{2}\phantom{00}\sigma_{1}};\,x_{3},x_{2}\right)\\!.$

######  Corollary 2.9.

We have

$\displaystyle{\mathcal{F}}_{\hat{\boldsymbol{\lambda}},\boldsymbol{\nu}}\left({\frac{1}{N}\phantom{000}\theta_{2}-\frac{1}{N}\atop\sigma_{3}\phantom{00}\sigma_{3}+h_{i}\phantom{00}\sigma_{1}-h_{j}};\,x_{2},x_{3}\right)$
$\displaystyle\qquad{}=C\big{(}q^{-m+\theta_{2}+\sigma_{1}^{(j)}}T_{q,x_{2}}-1\big{)}{\mathcal{F}}_{\boldsymbol{\lambda},\boldsymbol{\nu}}\left({\frac{1}{N}\phantom{000}\theta_{2}\atop\sigma_{3}\phantom{00}\sigma_{3}+h_{i}\phantom{00}\sigma_{1}};\,x_{2},x_{3}\right)\\!,$
$\displaystyle{\mathcal{F}}_{\hat{\boldsymbol{\lambda}},\boldsymbol{\nu}}\left({\theta_{2}-\frac{1}{N}\phantom{000}\frac{1}{N}\atop\sigma_{3}\phantom{00}\sigma_{1}-h_{i}-h_{j}\phantom{00}\sigma_{1}-h_{j}};\,x_{3},x_{2}\right)$
$\displaystyle\qquad{}=Cq^{\theta_{2}}\left(\frac{x_{3}}{qx_{2}}\right)^{1/N}\big{(}1-q^{-m+\sigma_{1}^{(j)}}T_{q,x_{2}}\big{)}{\mathcal{F}}_{\boldsymbol{\lambda},\boldsymbol{\nu}}\left({\theta_{2}\phantom{000}\frac{1}{N}\atop\sigma_{3}\phantom{00}\sigma_{1}-h_{i}\phantom{00}\sigma_{1}};\,x_{3},x_{2}\right)\\!,$

where

$\displaystyle
C=q^{m-|\boldsymbol{\nu}|-\Delta_{\sigma_{3}}+(1-1/N)/2-\theta_{2}-\sigma_{1}^{(j)}}x_{3}^{-m+(N\theta_{2}-1)(1-1/N)+\sigma_{1}^{(j)}}A_{\boldsymbol{\lambda},\boldsymbol{\nu}}^{(j)}(\theta_{2},\sigma_{1},\sigma_{3}).$

Corollary 2.9 enable us to reduce the connection problem of degenerate
${\mathcal{F}}_{\hat{\boldsymbol{\lambda}},\boldsymbol{\nu}}(x_{2})$ to the
connection problem of degenerate
${\mathcal{F}}_{\boldsymbol{\lambda},\boldsymbol{\nu}}(x_{2})$. Firstly, we
let $C\big{(}q^{-m+\theta_{2}+\sigma_{1}^{(1)}}T_{q,x_{2}}-1\big{)}$ act on
the connection formula (2.2) of the degenerate 4-point $q$-conformal block.
Then we have the connection formula

$\displaystyle{\mathcal{F}}_{\hat{\boldsymbol{\lambda}},\varnothing}\left({\frac{1}{N}\phantom{000}\theta_{2}-\frac{1}{N}\atop\sigma_{3}\phantom{00}\sigma_{3}+h_{i}\phantom{00}\sigma_{1}-h_{1}};\,x_{2},x_{3}\right)=-\sum_{j=1}^{N}\mathcal{F}_{\hat{\boldsymbol{\lambda}},\varnothing}\left({\theta_{2}-\frac{1}{N}\phantom{0000}\frac{1}{N}\atop\sigma_{3}\phantom{00}\sigma_{1}-h_{j}-h_{1}\phantom{00}\sigma_{1}-h_{1}};\,x_{2},x_{1}\right)$
$\displaystyle\hphantom{{\mathcal{F}}_{\hat{\boldsymbol{\lambda}},\varnothing}\left({\frac{1}{N}\phantom{000}\theta_{2}-\frac{1}{N}\atop\sigma_{3}\phantom{00}\sigma_{3}+h_{i}\phantom{00}\sigma_{1}-h_{1}};\,x_{2},x_{3}\right)=}{}\times{\mathcal{B}}_{j,i}\left[\begin{matrix}\theta_{2}&\frac{1}{N}\\\
\sigma_{3}&\sigma_{1}\end{matrix}\Bigl{|}\frac{x_{3}}{x_{2}}\right]q^{N\theta_{2}^{2}/2-3\theta_{2}/2+1/N}\bigg{(}\frac{x_{3}}{x_{2}}\bigg{)}^{\theta_{2}-1/N}\\!\\!,$

where $\hat{\boldsymbol{\lambda}}=(1^{m},\varnothing,\dots,\varnothing)$.
Substituting $\theta_{2}+1/N$, $\sigma_{1}-h_{1}$ into $\theta_{2}$,
$\sigma_{1}$, respectively, we obtain

$\displaystyle{\mathcal{F}}_{\hat{\boldsymbol{\lambda}},\varnothing}\left({\frac{1}{N}\phantom{000}\theta_{2}\atop\sigma_{3}\phantom{00}\sigma_{3}+h_{i}\phantom{00}\sigma_{1}};\,x_{2},x_{3}\right)=\sum_{j=1}^{N}\mathcal{F}_{\hat{\boldsymbol{\lambda}},\varnothing}\left({\theta_{2}\phantom{0000}\frac{1}{N}\atop\sigma_{3}\phantom{00}\sigma_{1}-h_{j}\phantom{00}\sigma_{1}};\,x_{2},x_{1}\right)$
$\displaystyle\hphantom{{\mathcal{F}}_{\hat{\boldsymbol{\lambda}},\varnothing}\left({\frac{1}{N}\phantom{000}\theta_{2}\atop\sigma_{3}\phantom{00}\sigma_{3}+h_{i}\phantom{00}\sigma_{1}};\,x_{2},x_{3}\right)=}{}\times{\mathcal{B}}_{j,i}\left[\begin{matrix}\theta_{2}&\frac{1}{N}\\\
\sigma_{3}&\sigma_{1}\end{matrix}\Bigl{|}\frac{x_{3}}{x_{2}}\right]q^{N\theta_{2}^{2}/2-\theta_{2}/2}\left(\frac{x_{3}}{x_{2}}\right)^{\theta_{2}}$

due to the periodicity (2.6) of the connection matrix
${\mathcal{B}}\left[\begin{smallmatrix}\theta_{2}&\frac{1}{N}\\\
\sigma_{3}&\sigma_{1}\end{smallmatrix}\bigl{|}x\right]$. In this way, by using
Corollary 2.9 repeatedly and by the symmetry of
$\mathcal{F}_{\boldsymbol{\lambda},\boldsymbol{\nu}}$ with respect to
$\boldsymbol{\lambda},\boldsymbol{\nu}$, we obtain the connection formulas for
any degenerate $\mathcal{F}_{\boldsymbol{\lambda},\boldsymbol{\nu}}$ whose
connection matrix is the same of degenerate
$\mathcal{F}_{\varnothing,\varnothing}$. Therefore, we obtain

######  Theorem 2.10.

We have

$\displaystyle\mathcal{F}\left({\theta_{4}\phantom{000}\frac{1}{N}\phantom{00000}\theta_{2}\phantom{000}\theta_{1}\phantom{}\atop\sigma_{4}\phantom{00}\sigma_{3}\phantom{00}\sigma_{3}+h_{i}\phantom{00}\sigma_{1}\phantom{00}\sigma_{0}};\,x_{4},x_{2},x_{3},x_{1}\right)$
$\displaystyle\qquad{}=\sum_{j=1}^{N}\mathcal{F}\left({\theta_{4}\phantom{000}\theta_{2}\phantom{00000}\frac{1}{N}\phantom{000}\theta_{1}\phantom{}\atop\sigma_{4}\phantom{00}\sigma_{3}\phantom{00}\sigma_{1}-h_{j}\phantom{00}\sigma_{1}\phantom{00}\sigma_{0}};\,x_{4},x_{3},x_{2},x_{1}\right){\mathcal{B}}_{j,i}\left[\begin{matrix}\theta_{2}&\frac{1}{N}\\\
\sigma_{3}&\sigma_{1}\end{matrix}\Bigl{|}\frac{x_{3}}{x_{2}}\right]$
$\displaystyle\qquad\phantom{=}{}\times
q^{N\theta_{2}^{2}/2-\theta_{2}/2}\bigg{(}\frac{x_{3}}{x_{2}}\bigg{)}^{\theta_{2}}.$

##  3 CFT construction

In this section, we construct a monodromy invariant fundamental solution of a
linear $q$-difference system as a Fourier transform of $q$-Nekrasov functions
multiplied by some factors. Our fundamental system degenerates by the limit
$q\to 1$ to a monodromy invariant fundamental system of a Fuchsian system of
rank $N$ with spectral type

$\displaystyle\left(1^{N}\right),\quad\left(1^{N}\right),\quad\left(N-1,1\right),\quad\dots,\quad\left(N-1,1\right).$

Another monodromy invariant fundamental solution of this Fuchsian system was
constructed in [15] by using the $W$ conformal field theory. The corresponding
tau function was expressed as a Fourier transform of the semi-degenerate
$W_{N}$ conformal block with the central charge $c=N-1$. Hence, by the
uniqueness of a fundamental system of a Fuchsian system, $W$ conformal blocks
are equal to 4d Nekrasov functions. We note that in what follows, we assume
convergence of 5d Nekrasov partition functions, and moreover, Fourier
transforms of them. Analyticity of 5d Nekrasov partition functions in cases
different from our case was discussed in [10].

###  3.1 Lax form

Let $R$ be defined by

$\displaystyle
R=\left\\{n\in{\mathbb{Z}}^{N}\,\middle|\,\sum_{i=1}^{N}n^{(i)}=0\right\\}\\!,$

which may be regarded as the root lattice of $\mathfrak{sl}_{N}$. We set

$\displaystyle\tilde{x}=q^{N\sum_{i=1}^{m+1}\theta_{i}}x,\qquad\tilde{t}_{i}=q^{N\sum_{j=i+1}^{m+1}\theta_{j}}t_{i},\qquad
i=1,\dots,m+1,$ $\displaystyle s=\prod_{i=1}^{m}\prod_{j=1}^{N}s_{i,j},\qquad
s_{i}=\prod_{j=1}^{N}s_{i,j},\qquad i=1,\dots,m$

for variables $x$, $t_{i}$, and $s_{i,j}$, $i=1,\dots,m$, $j=1,\dots,N$. We
denote $s_{i}^{n}=\prod_{j=1}^{N}s_{i,j}^{n^{(j)}}$ for $n\in R$ and
$s^{n}=\prod_{i=1}^{m}s_{i}^{n_{i}}$ for $n=(n_{1},\dots,n_{m})\in R^{m}$.

We define a $N$ by $N$ matrix for
$\theta_{1},\dots,\theta_{m+1}\in{\mathbb{C}}$ and
$\theta_{0},\theta_{\infty},\sigma_{1},\dots,\sigma_{m+1}\in{\mathbb{C}}^{N}$
by

$\displaystyle Y^{(0,1)}_{i,j}(x,t)=\frac{1}{\tau_{i}^{(0,1)}(t)}$
$\displaystyle\hphantom{Y^{(0,1)}_{i,j}(x,t)=}{}\times\\!\\!\\!\sum_{n=(n_{1},\dots,n_{m})\in
R^{m}}\\!\\!\\!\\!\\!s^{n}{\mathcal{F}}\\!\left({\phantom{00}\frac{1}{N}\phantom{000000000}(\theta_{\ell})_{\ell=1}^{m+1}\atop\theta_{\infty}-h_{i}\phantom{00}\theta_{\infty}-h_{i}+h_{j}\phantom{00}(\sigma_{\ell}+n_{\ell})_{\ell=1}^{m}\phantom{00}\theta_{0}};\,\tilde{x},(\tilde{t}_{\ell})_{\ell=1}^{m+1}\right)\\!.$

Here, the function $\tau_{i}^{(0,1)}(t)$ is the normalization factor so that

$\displaystyle
Y^{(0,1)}(x,t)=\big{(}I+Y_{1}(t)x^{-1}+O(x^{-2})\big{)}\mathop{\rm
diag}\big{(}x^{-\theta_{\infty}^{(1)}},\dots,x^{-\theta_{\infty}^{(N)}}\big{)},$

where $I$ is the unit matrix of rank $N$. The matrix $Y^{(0,1)}(x,t)$ is an
expansion around $x=\infty$. We define the $N$ by $N$ matrices by

$\displaystyle
Y^{(k,k+1)}_{i,j}(x,t)=\frac{x^{-\sum_{\ell=1}^{k}\theta_{\ell}}}{\tau_{i}^{(k,k+1)}(t)}\sum_{n\in
R^{m}}s^{n}s_{k}^{h_{j}-h_{N}}$
$\displaystyle\qquad{}\times{\mathcal{F}}\Bigg{(}{\phantom{000}(\theta_{\ell})_{\ell=1}^{k}\phantom{00000000099}\frac{1}{N}\phantom{00}(\theta_{\ell})_{\ell=k+1}^{m+1}\phantom{}\atop\theta_{\infty}-h_{i}\phantom{00}(\sigma_{\ell}-h_{N}+n_{\ell})_{\ell=1}^{k}\phantom{00}\sigma_{k}+h_{j}-h_{N}+n_{k}\phantom{00}(\sigma_{\ell}+n_{\ell})_{\ell=k+1}^{m}\phantom{00}\theta_{0}};$
$\displaystyle\qquad\hphantom{\times{\mathcal{F}}\Bigg{(}0}(\tilde{t}_{\ell})_{\ell=1}^{k},\tilde{x},(\tilde{t}_{\ell})_{\ell=k+1}^{m+1}\Bigg{)},$

for $k=1,\dots,m$, and

$\displaystyle Y^{(m+1,m+2)}_{i,j}(x,t)$
$\displaystyle\qquad{}=\frac{x^{-\sum_{\ell=1}^{m+1}\theta_{\ell}}}{\tau_{i}^{(m+1,m+2)}(t)}\sum_{n\in
R^{m}}s^{n}{\mathcal{F}}\left({\phantom{000}(\theta_{\ell})_{\ell=1}^{m+1}\phantom{00000000099}\frac{1}{N}\phantom{}\atop\theta_{\infty}-h_{i}\phantom{00}(\sigma_{\ell}-h_{N}+n_{\ell})_{\ell=1}^{m}\phantom{00}\theta_{0}-h_{j}\phantom{00}\theta_{0}};\,(\tilde{t}_{\ell})_{\ell=1}^{m+1},\tilde{x}\right)\\!,$

where the functions $\tau_{i}^{(k,k+1)}(t)$ are given recursively by

$\displaystyle\tau_{i}^{(k,k+1)}(t)=\tau_{i}^{(k-1,k)}(t)\,q^{N\theta_{k}^{2}/2+\theta_{k}/2+N\theta_{k}\sum_{i=1}^{k-1}\theta_{i}}t_{k}^{-\theta_{k}}.$
(3.1)

By the definition of $q$-conformal blocks, asymptotic behaviours of the
matrices of $Y^{(0,1)}(x,t)$ and $Y^{(m+1,m+2)}(x,t)$ are given by

$\displaystyle Y^{(0,1)}(x,t)=\widehat{Y}^{(0,1)}(x,t)\mathop{\rm
diag}\big{(}x^{-\theta_{\infty}^{(1)}},\dots,x^{-\theta_{\infty}^{(N)}}\big{)},$
(3.2)
$\displaystyle\widehat{Y}^{(0,1)}(x,t)=I+Y_{1}(t)x^{-1}+O\big{(}x^{-2}\big{)},$
$\displaystyle Y^{(m+1,m+2)}(x,t)=\widehat{Y}^{(m+1,m+2)}(x,t)\mathop{\rm
diag}\big{(}x^{-\theta_{0}^{(1)}},\dots,x^{-\theta_{0}^{(N)}}\big{)}x^{-\sum_{i=1}^{m+1}\theta_{i}},$
(3.3) $\displaystyle\widehat{Y}^{(m+1,m+2)}(x,t)=G(t)(I+O(x)),$

where $I$ is the unit matrix of rank $N$, $Y_{1}(t)$ and $G(t)$ are $N$ by $N$
matrices. The off diagonal elements of $Y_{1}(t)$ and all elements of the
coefficient matrix $G(t)$ will be expressed as ratios of tau functions.
Furthermore, asymptotic behaviours of the matrices $Y^{(k,k+1)}(x,t)$,
$k=1,\dots,m$, are given by

$\displaystyle Y^{(k,k+1)}(x,t)=\widehat{Y}^{(k,k+1)}(x,t)\mathop{\rm
diag}\big{(}x^{-\sigma_{k}^{(1)}-{1}/{N}}\\!,\dots,x^{-\sigma_{k}^{(N-1)}-{1}/{N}},x^{-\sigma_{k}^{(N)}+1-{1}/{N}}\big{)}x^{-\sum_{i=1}^{k}\theta_{i}},$
$\displaystyle\widehat{Y}^{(k,k+1)}(x,t)=\sum_{n=-\infty}^{\infty}Y_{n}^{(k,k+1)}(t)x^{n}.$
(3.4)

We assume that each $\widehat{Y}^{(k,k+1)}(x,t)$ is holomorphic in the domain

$\displaystyle|t_{1}|>|t_{2}|>\cdots>|t_{k}|>\big{|}q^{1+N\sum_{i=1}^{k}\theta_{i}}x\big{|},\qquad\big{|}q^{N\sum_{i=1}^{k}\theta_{i}}x\big{|}>|t_{k+1}|>\cdots>|t_{m+1}|,$
(3.5)

which is a natural assumption from the point of view of conformal blocks.

######  Theorem 3.1.

Let

$\displaystyle 1<\big{|}q^{-N\theta_{i}}\big{|}<\frac{1}{|q|},\qquad
i=1,\dots,m+1,$ (3.6)
$\displaystyle\big{|}q^{-N\sum_{j=i}^{k+1}\theta_{j}}t_{k+1}\big{|}<|t_{i}|,\qquad
i=1,\dots,m,\ k=i,\dots,m.$ (3.7)

Then, the matrices $Y^{(k,k+1)}(x,t)$, $k=0,1,\dots,m$, form a fundamental
solution of the $q$-linear difference system

$\displaystyle T_{q,x}(Y(x,t))=A(x,t)Y(x,t),\quad\
T_{q,t_{i}}(Y(x,t))=B_{i}(x,t)Y(x,t),\quad\ i=1,\dots,m+1,\\!\\!$ (3.8)

with $N$ by $N$ matrices $A(x,t)$, $B_{i}(x,t)$ satisfying

$\displaystyle
A(x,t)=\frac{A_{m+1}x^{m+1}+\sum_{k=0}^{m}A_{k}(t)x^{k}}{\prod_{k=1}^{m+1}\big{(}x-q^{-1-N\sum_{j=1}^{k-1}\theta_{j}}t_{k}\big{)}},\qquad\det
A(x,t)=\prod_{k=1}^{m+1}\frac{x-q^{-1-N\sum_{j=1}^{k}\theta_{j}}t_{k}}{x-q^{-1-N\sum_{j=1}^{k-1}\theta_{j}}t_{k}},$
(3.9) $\displaystyle A_{m+1}=\mathop{\rm
diag}(q^{-\theta_{\infty}^{(1)}},\dots,q^{-\theta_{\infty}^{(N)}}),$
$\displaystyle
A_{0}(t)=(-1)^{m+1}q^{-m-1-\sum_{i=1}^{m+1}\theta_{i}}\prod_{k=0}^{m}\big{(}q^{-N\sum_{j=1}^{k}\theta_{j}}t_{k+1}\big{)}G(t)\mathop{\rm
diag}(q^{-\theta_{0}^{(1)}},\dots,q^{-\theta_{0}^{(N)}})G(t)^{-1},$
$\displaystyle
B_{i}(x,t)=\frac{xI+B_{i,0}(t)}{x-q^{-N\sum_{j=1}^{i}\theta_{j}}t_{i}},\qquad\det
B_{i}(x,t)=\frac{x-q^{-N\sum_{j=1}^{i-1}\theta_{j}}t_{i}}{x-q^{-N\sum_{j=1}^{i}\theta_{j}}t_{i}}.$
(3.10)

###### Proof.

Thanks to Theorem 2.10, we have for $k=0,1,\dots,m+1$

$\displaystyle
Y^{(k,k+1)}_{i,j}(x,t)=\frac{x^{-\sum_{\ell=1}^{k}\theta_{\ell}}}{\tau_{i}^{(k,k+1)}(t)}\sum_{n\in
R^{m}}s^{n}s_{k}^{h_{j}-h_{N}}$
$\displaystyle\qquad{}\times\sum_{p=1}^{N}{\mathcal{F}}\Bigg{(}{\phantom{000}(\theta_{\ell})_{\ell=1}^{k+1}\phantom{00000000099}\frac{1}{N}\phantom{00}(\theta_{\ell})_{\ell=k+2}^{m+1}\phantom{}\atop\theta_{\infty}-h_{i}\phantom{00}(\sigma_{\ell}-h_{N}+n_{\ell})_{\ell=1}^{k}\phantom{00}\sigma_{k+1}+n_{k+1}-h_{p}\phantom{00}(\sigma_{\ell}+n_{\ell})_{\ell=k+1}^{m}\phantom{00}\theta_{0}};$
$\displaystyle\qquad\hphantom{\times\sum_{p=1}^{N}{\mathcal{F}}\Bigg{(}0}(\tilde{t}_{\ell})_{\ell=1}^{k+1},\tilde{x},(\tilde{t}_{\ell})_{\ell=k+2}^{m+1}\Bigg{)}{\mathcal{B}}_{p,j}\left[\begin{matrix}\theta_{k+1}&\frac{1}{N}\\\
\sigma_{k}-h_{N}+n_{k}&\sigma_{k+1}+n_{k+1}\end{matrix}\Bigl{|}\frac{t_{k+1}}{q^{N\sum_{i=1}^{k+1}\theta_{i}}x}\right]$
$\displaystyle\qquad{}\times
q^{N\theta_{k+1}^{2}/2-\theta_{k+1}/2}\bigg{(}\frac{t_{k+1}}{q^{N\sum_{i=1}^{k+1}\theta_{i}}x}\bigg{)}^{\theta_{k+1}}.$

By the periodicity (2.4), (2.6) of the connection matrix ${\mathcal{B}}$ and
the definition (3.1) of the function $\tau_{i}^{(k,k+1)}(t)$, we obtain

$\displaystyle
Y^{(k,k+1)}_{i,j}(x,t)=\frac{x^{-\sum_{\ell=1}^{k+1}\theta_{\ell}}}{\tau_{i}^{(k+1,k+2)}(t)}\sum_{p=1}^{N}{\mathcal{B}}_{p,j}\left[\begin{matrix}\theta_{k+1}&\frac{1}{N}\\\
\sigma_{k}-h_{N}&\sigma_{k+1}\end{matrix}\Bigl{|}\frac{t_{k+1}}{q^{N\sum_{i=1}^{k+1}\theta_{i}}x}\right]\sum_{n\in
R^{m}}s^{n}s_{k}^{h_{j}-h_{N}}$
$\displaystyle\qquad\times{\mathcal{F}}\Bigg{(}{\phantom{000}(\theta_{\ell})_{\ell=1}^{k+1}\phantom{00000000099}\frac{1}{N}\phantom{00}(\theta_{\ell})_{\ell=k+2}^{m+1}\phantom{}\atop\theta_{\infty}-h_{i}\phantom{00}(\sigma_{\ell}-h_{N}+n_{\ell})_{\ell=1}^{k}\phantom{00}\sigma_{k+1}+n_{k+1}-h_{p}\phantom{00}(\sigma_{\ell}+n_{\ell})_{\ell=k+1}^{m}\phantom{00}\theta_{0}};$
$\displaystyle\qquad\hphantom{\times{\mathcal{F}}\Bigg{(}0}(\tilde{t}_{\ell})_{\ell=1}^{k+1},\tilde{x},(\tilde{t}_{\ell})_{\ell=k+2}^{m+1}\Bigg{)}.$

Hence, we have

$\displaystyle
Y^{(k,k+1)}_{i,j}(x,t)=s_{k}^{h_{j}-h_{N}}\sum_{p=1}^{N}Y^{(k+1,k+2)}_{i,p}(x,t){\mathcal{B}}_{p,j}\left[\begin{matrix}\theta_{k+1}&\frac{1}{N}\\\
\sigma_{k}-h_{N}&\sigma_{k+1}\end{matrix}\Bigl{|}\frac{t_{k+1}}{q^{N\sum_{i=1}^{k+1}\theta_{i}}x}\right]\\!.$

Therefore we obtain for $k=0,1,\dots,m+1$

$\displaystyle
Y^{(k,k+1)}(x,t)=Y^{(k+1,k+2)}(x,t){\mathcal{B}}\left[\begin{matrix}\theta_{k+1}&\frac{1}{N}\\\
\sigma_{k}-h_{N}&\sigma_{k+1}\end{matrix}\Bigl{|}\frac{t_{k+1}}{q^{N\sum_{i=1}^{k+1}\theta_{i}}x}\right]$
$\displaystyle\hphantom{Y^{(k,k+1)}(x,t)=}{}\times\mathop{\rm
diag}\big{(}s_{k}^{h_{1}-h_{N}},\dots,s_{k}^{h_{N-1}-h_{N}},1\big{)},$ (3.11)

where $\sigma_{0}=\theta_{\infty}+h_{N}$, $\sigma_{m+1}=\theta_{0}$,
$t_{0}=1$, and $s_{0}=1$.

Put

$\displaystyle A(x,t)=T_{q,x}\big{(}Y^{(0,1)}(x,t)\big{)}Y^{(0,1)}(x,t)^{-1},$
(3.12) $\displaystyle
B_{i}(x,t)=T_{q,t_{i}}\big{(}Y^{(0,1)}(x,t)\big{)}Y^{(0,1)}(x,t)^{-1},\qquad
i=1,\dots,m+1.$ (3.13)

Since all connection matrices are independent of the $q$-shifts with respect
to the variables $x$ and $t_{i}$ ($i=1,\dots,m+1$) by (2.3), from (3.1) the
matrix functions $A(x,t)$, $B_{i}(x,t)$ of $x$ are analytically continued on
$\mathbb{P}^{1}$ as

$\displaystyle
A(x,t)\\!=T_{q,x}\big{(}Y^{(0,1)}(x,t)\big{)}Y^{(0,1)}(x,t)^{-1}\\!=\cdots\\!=T_{q,x}\big{(}Y^{(m+1,m+2)}(x,t)\big{)}Y^{(m+1,m+2)}(x,t)^{-1},$
$\displaystyle
B_{i}(x,t)\\!=T_{q,t_{i}}\big{(}Y^{(0,1)}(x,t)\big{)}Y^{(0,1)}(x,t)^{-1}\\!=\cdots\\!=T_{q,t_{i}}\big{(}Y^{(m+1,m+2)}(x,t)\big{)}Y^{(m+1,m+2)}(x,t)^{-1}\\!.$

Therefore by the assumption on holomorphicity of $\widehat{Y}^{k,k+1}(x,t)$,
the functions $A(x,t)$, $B_{i}(x,t)$ of $x$ are meromorphic functions of $x$
on $\mathbb{P}^{1}$, namely, rational functions.

As a result, the matrices $Y^{(k,k+1)}(x,t)$ satisfy the $q$-linear difference
system

$\displaystyle T_{q,x}(Y(x,t))=A(x,t)Y(x,t),\qquad
T_{q,t_{i}}(Y(x,t))=B_{i}(x,t)Y(x,t),\qquad i=1,\dots,m+1,$

with $N\times N$ matrix rational functions $A(x,t)$, $B_{i}(x,t)$ of $x$.

The forms of $A(x,t)$ and $B_{i}(x,t)$ can be deduced from $\det
Y^{(0,1)}(x,t)$. The connection formula (3.1) yields

$\displaystyle\prod_{i=0}^{m}\frac{\big{(}q^{-N\sum_{j=1}^{i}\theta_{j}}t_{i+1}/x;q\big{)}_{\infty}}{\big{(}q^{-N\sum_{j=1}^{i+1}\theta_{j}}t_{i+1}/x;q\big{)}_{\infty}}\,\det
Y^{(0,1)}(x,t)$
$\displaystyle\qquad{}=q^{N\sum_{i=1}^{k}\theta_{i}(N\sum_{i=1}^{k}\theta_{i}+1)/2}x^{N\sum_{i=1}^{k}\theta_{i}}\prod_{i=0}^{k-1}s_{i}^{\sum_{j=1}^{N-1}(h_{j}-h_{N})}t_{i+1}^{-N\theta_{i+1}}\prod_{1\leq
i<j\leq
N}\frac{\vartheta\big{(}\theta_{\infty}^{(i)}-\theta_{\infty}^{(j)}\big{)}}{\vartheta\big{(}\theta_{k}^{(i)}-\theta_{k}^{(j)}\big{)}}$
$\displaystyle\qquad\phantom{=}{}\times\prod_{i=k}^{m}\frac{\big{(}q^{-N\sum_{j=1}^{i}\theta_{j}}t_{i+1}/x;q\big{)}_{\infty}}{\big{(}q^{-N\sum_{j=1}^{i+1}\theta_{j}}t_{i+1}/x;q\big{)}_{\infty}}\\!\prod_{i=0}^{k-1}\frac{\big{(}q^{1+N\sum_{j=1}^{i+1}\theta_{j}}x/t_{i+1};q\big{)}_{\infty}}{\big{(}q^{1+N\sum_{j=1}^{i}\theta_{j}}x/t_{i+1};q\big{)}_{\infty}}\,\det
Y^{(k,k+1)}(x,t),\\!\\!$ (3.14)

where $k=1,\dots,m+1$. The asymptotic behaviours (3.2), (3.3), (3.4) imply
that (3.14) is a meromorphic function of $x$ on $\mathbb{P}^{1}$. Since
$\widehat{Y}^{(k,k+1)}(x,t)$ is holomorphic in the domain
$\big{|}q^{-N\sum_{i=1}^{k}\theta_{i}}t_{k+1}\big{|}<|x|<\big{|}q^{-1-N\sum_{i=1}^{k-1}\theta_{i}}t_{k}\big{|}$
from the assumption (3.5), the meromorphic function (3.14) of $x$ is
holomorphic on $\mathbb{P}^{1}$ due to the condition (3.7). Hence, it is a
constant with respect to $x$, and therefore, using (3.2) again, we obtain

$\displaystyle\det
Y^{(0,1)}(x,t)=\prod_{k=0}^{m}\frac{\big{(}q^{-N\sum_{i=1}^{k+1}\theta_{i}}t_{k+1}/x;q\big{)}_{\infty}}{\big{(}q^{-N\sum_{i=1}^{k}\theta_{i}}t_{k+1}/x;q\big{)}_{\infty}}.$
(3.15)

Then by the definitions (3.12), (3.13) we obtain

$\displaystyle\det
A(x,t)=\prod_{k=1}^{m+1}\frac{x-q^{-1-N\sum_{i=1}^{k}\theta_{i}}t_{k}}{x-q^{-1-N\sum_{i=1}^{k-1}\theta_{i}}t_{k}},$
$\displaystyle\det
B_{i}(x,t)=\frac{x-q^{-N\sum_{j=1}^{i-1}\theta_{j}}t_{i}}{x-q^{-N\sum_{j=1}^{i}\theta_{j}}t_{i}},\qquad
i=1,\dots,m+1.$

We recall that for each $k=0,1,\dots,m+1$, we have

$\displaystyle
A(x,t)=T_{q,x}\big{(}Y^{(k,k+1)}(x,t)\big{)}Y^{(k,k+1)}(x,t)^{-1}.$

from the definitions (3.12), (3.13) and the connection relations (3.1). Hence,
in the domain

$\displaystyle\big{|}q^{-1-N\sum_{i=1}^{k}\theta_{i}}t_{k+1}\big{|}<|x|<\big{|}q^{-1-N\sum_{i=1}^{k}\theta_{i}}t_{k}\big{|},$

possible poles of $A(x,t)$ are the zeros of
$\det\left(Y^{(k,k+1)}(x,t)\right)$. From (3.14) we can read where the zeros
of $\det\left(Y^{(k,k+1)}(x,t)\right)$ is. The conditions (3.6) and (3.7)
imply that $A(x,t)$ only has simple poles at
$x=q^{-1-N\sum_{i=1}^{k-1}\theta_{i}t_{k}}$, $k=1,\dots,m+1$.

Therefore, together with the asymptotic behaviour (3.2), we have

$\displaystyle
A=\frac{A_{m+1}+\sum_{k=0}^{m}A_{k}(t)x^{k}}{\prod_{k=1}^{m+1}\big{(}x-q^{-1-N\sum_{j=1}^{k-1}\theta_{j}}t_{k}\big{)}}.$

The $q$-difference equation $Y(qx)=A(x)Y(x)$ entails that possible (simple)
poles of the analytic continuations of $Y^{(0,1)}(x,t)^{\pm 1}$,
$Y^{(m+1,m+2)}(x,t)^{\pm 1}$ are

$\displaystyle Y^{(0,1)}(x,t)\colon\ q^{\ell-N\sum_{j=1}^{k-1}}t_{k},\qquad
Y^{(0,1)}(x,t)^{-1}\colon\ q^{\ell-N\sum_{j=1}^{k-1}}t_{k},\
q^{\ell-N\sum_{j=1}^{k}}t_{k},$ $\displaystyle Y^{(m+1,m+2)}(x,t)\colon\
q^{-\ell-1-N\sum_{j=1}^{k-1}}t_{k},\ q^{-\ell-1-N\sum_{j=1}^{k}}t_{k},$
$\displaystyle Y^{(m+1,m+2)}(x,t)^{-1}\colon\
q^{-\ell-1-N\sum_{j=1}^{k-1}}t_{k},$

where $\ell\in{\mathbb{Z}}_{\geq 0}$, $k=1,\dots,m+1$. Hence, since

$\displaystyle
B_{i}(x,t)=T_{q,t_{i}}\big{(}Y^{(0,1)}(x,t)\big{)}Y^{(0,1)}(x,t)^{-1}=T_{q,t_{i}}\big{(}Y^{(m+1,m+2)}(x,t)\big{)}Y^{(m+1,m+2)}(x,t)^{-1},$

$B_{i}(x,t)$ only has a simple pole at
$x=q^{-N\sum_{j=1}^{i}\theta_{j}}t_{i}$. Therefore, we obtain (3.10). ∎

The compatibility conditions

$\displaystyle T_{q,t_{i}}(A(x,t))B_{i}(x,t)=B_{i}(qx,t)A(x,t),\qquad
i=1,\dots,m+1,$ (3.16)

of the Lax form (3.8) with rational functions $A(x,t)$ and $B_{i}(x,t)$
satisfying (3.9) and (3.10) yield nonlinear $q$-difference system satisfied by
the elements of $A(x,t)$, $B_{i}(x,t)$ as functions of the variables $t$’s
with the parameters $\theta$’s. When $m=1$ and $N=2$, the derived nonlinear
$q$-difference system is the $q$-difference VI equation [21]. Fortunately, the
dependent variables of $q$-$\mathrm{P_{VI}}$ are simply written by the
elements of $A(x,t)$ and $B_{2}(x,t)$ and moreover they are expressed in terms
of Fourier transforms of $q$-conformal blocks, namely, tau functions [20].
However, in the general case, nonlinear $q$-difference systems derived from
compatibility conditions are complicated and we have not found a way to nicely
write down equation satisfied by the elements of $A(x,t)$ and $B_{i}(x,t)$. We
refer to [32] where the author investigated the $N=2$ case and obtained a
$q$-analog of the Garnier system, see also [24, 25, 26, 31]. We also refer to
[36] where the authors verified that the Lax form of a generalization
$q$-$P_{(n,n)}$ of $q$-$\mathrm{P_{VI}}$ obtained from the Drinfeld–Sokolov
hierarchy is transformed to our Lax form (3.8) for $m=1$ by $q$-Laplace
transformations [36, Theorem 3.6].

The rational function $B_{i}(x,t)$ can be computed from the fundamental
solution $Y^{(k,k+1)}(x,t)$. In particular, $B_{i,0}(t)$ has two expressions

$\displaystyle
B_{i,0}(t)=-q^{-N\sum_{j=1}^{i}\theta_{j}}t_{i}I+T_{q,t_{i}}(Y_{1}(t))-Y_{1}(t)=-q^{-N\sum_{j=1}^{i}\theta_{j}}t_{i}T_{q,t_{i}}(G(t))G(t)^{-1},$
(3.17)

derived from the asymptotic behaviours of the fundamental solution around $0$
and $\infty$. Recall that $Y_{1}(t)$, $G(t)$ are defined by the asymtotoic
expansions (3.2) and (3.3). This produces non-trivial relations among elements
of $Y_{1}(t)$ and $G(t)$, which can be viewed as Fourier transforms of
$m+3$-point $q$-conformal blocks.

We define the tau function by

$\displaystyle\tau\left[\theta|s,\sigma,t\right]=q^{-N\Delta_{\theta_{\infty}}\sum_{k=1}^{m+1}\theta_{k}+N\sum_{k=1}^{m}\Delta_{N\theta_{k}h_{1}}\sum_{j=k+1}^{m+1}\theta_{j}}$
$\displaystyle\hphantom{\tau\left[\theta|s,\sigma,t\right]=}{}\times\prod_{1\leq
k<k^{\prime}\leq
N}G_{q}\big{(}1+\theta_{\infty}^{(k)}-\theta_{\infty}^{(k^{\prime})}\big{)}G_{q}\big{(}1-\theta_{0}^{(k)}+\theta_{0}^{(k^{\prime})}\big{)}$
$\displaystyle\hphantom{\tau\left[\theta|s,\sigma,t\right]=}{}\times\sum_{n\in
R^{m}}s^{n}{\mathcal{F}}\left({(\theta_{p})_{p=1}^{m+1}\atop\theta_{\infty}\phantom{00}(\sigma_{p}+n_{p})_{p=1}^{m}\phantom{00}\theta_{0}};\,(\tilde{t}_{p})_{p=1}^{m+1}\right)\\!.$

Then

$\displaystyle\tau\left[\theta|s,\sigma,t\right]=\sum_{n\in
R^{m}}s^{n}\prod_{p=1}^{m+1}t_{p}^{\Delta_{\sigma_{p-1}+n_{p-1}}-\Delta_{N\theta_{p}h_{1}}-\Delta_{\sigma_{p}+n_{p}}}C[\theta|\sigma+n]Z[\theta|\sigma+n,t]$

with the definition

$\displaystyle
C[\theta|\sigma]=\frac{\prod_{p=1}^{m+1}\prod_{k,k^{\prime}=1}^{N}G_{q}\big{(}1+\sigma_{p-1}^{(k)}-\theta_{p}-\sigma_{p}^{(k^{\prime})}\big{)}}{\prod_{p=1}^{m}\prod_{k,k^{\prime}=1}^{N}G_{q}\big{(}1+\sigma_{p}^{(k)}-\sigma_{p}^{(k^{\prime})}\big{)}},$
$\displaystyle
Z[\theta|\sigma,t]=\sum_{\boldsymbol{\lambda}_{1},\dots,\boldsymbol{\lambda}_{m}\in{\mathbb{Y}}^{N}}\prod_{p=1}^{m}\bigg{(}\frac{t_{p+1}}{t_{p}}\bigg{)}^{\left|\boldsymbol{\lambda}_{p}\right|}\frac{\prod_{p=1}^{m+1}\prod_{k,k^{\prime}=1}^{N}N_{\lambda_{p-1}^{(k)},\lambda_{p}^{(k^{\prime})}}\big{(}q^{\sigma_{p-1}^{(k)}-\theta_{p}-\sigma_{p}^{(k^{\prime})}}\big{)}}{\prod_{p=1}^{m}\prod_{k,k^{\prime}=1}^{N}N_{\lambda_{p}^{(k)},\lambda_{p}^{(k^{\prime})}}\big{(}q^{\sigma_{p}^{(k)}-\sigma_{p}^{(k^{\prime})}}\big{)}}.$

We note that the normalization factors $\tau_{i}^{(0,1)}(t)$, $i=1,\dots,N$,
are related to the tau function as

$\displaystyle\tau_{i}^{(0,1)}(t)=q^{N\Delta_{\theta_{\infty}}\sum_{k=1}^{m+1}\theta_{k}-N\sum_{k=1}^{m}\Delta_{N\theta_{k}h_{1}}\sum_{j=k+1}^{m+1}\theta_{j}+\Delta_{\theta_{\infty}-h_{i}}}$
$\displaystyle\hphantom{\tau_{i}^{(0,1)}(t)=}{}\times\bigg{(}\prod_{1\leq
k<k^{\prime}\leq
N}G_{q}\big{(}1+\theta_{\infty}^{(k)}-\theta_{\infty}^{(k^{\prime})}\big{)}G_{q}\big{(}1-\theta_{0}^{(k)}+\theta_{0}^{(k^{\prime})}\big{)}\bigg{)}^{-1}$
$\displaystyle\hphantom{\tau_{i}^{(0,1)}(t)=}{}\times{\mathcal{N}}^{\prime}\Bigl{(}{\kern-1.0pt\frac{1}{N}\atop{\theta_{\infty}-h_{i}\phantom{h_{2}}\theta_{\infty}}}\Bigr{)}\tau\left[\theta|s,\sigma,t\right].$

Let us introduce other tau functions by

$\displaystyle\tau=\tau\left[\theta|s,\sigma,t\right],\qquad\tau_{ij}=\tau\left[\big{(}\theta_{\infty}-h_{i}+h_{j},(\theta_{p})_{p=1}^{m+2}\big{)}|s,\sigma,t\right],$
$\displaystyle\tilde{\tau}_{ij}=\tau\left[\big{(}\theta_{\infty}-h_{i},(\theta_{p})_{p=1}^{m+1},\theta_{0}-h_{j}\big{)}|s,\sigma-
h_{N},t\right]$

for $i,j=1,\dots,N$.

######  Proposition 3.2.

For $p=1,\dots,m+1$ and $i,j=1,\dots,N$, $i\neq j$, we have the determinantal
identity

$\displaystyle
T_{q,t_{p}}\left(\frac{\tau_{ij}}{\tau}\right)-\frac{\tau_{ij}}{\tau}=\frac{q^{-N\sum_{k=1}^{p}\theta_{k}+\theta_{p}}t_{p}D_{ij}}{T_{q,t_{p}}(\tau)\tau^{N-1}}\left|\begin{matrix}\tilde{\tau}_{11}&\tilde{\tau}_{12}&\cdots&\tilde{\tau}_{1N}\\\
\vdots&\vdots&&\vdots\\\
\tilde{\tau}_{j-11}&\tilde{\tau}_{j-12}&\cdots&\tilde{\tau}_{j-1N}\\\\[2.15277pt]
T_{q,t_{p}}\left(\tilde{\tau}_{i1}\right)&T_{q,t_{p}}\left(\tilde{\tau}_{i2}\right)&\cdots&T_{q,t_{p}}\left(\tilde{\tau}_{iN}\right)\\\\[2.15277pt]
\tilde{\tau}_{j+11}&\tilde{\tau}_{j+12}&\cdots&\tilde{\tau}_{j+1N}\\\
\vdots&\vdots&&\vdots\\\
\tilde{\tau}_{N1}&\tilde{\tau}_{N2}&\cdots&\tilde{\tau}_{NN}\end{matrix}\right|,$
(3.18)

where

$\displaystyle
D_{ij}=(-1)^{(m+1)(N-1)+1}q^{N(N-1)\sum_{k=1}^{m+1}\theta_{k}/2+\theta_{\infty}^{(i)}-\theta_{\infty}^{(j)}+\sum_{k=1}^{N}k(\theta_{\infty}^{(k)}-\theta_{0}^{(k)})}\big{[}1-\theta_{\infty}^{(i)}+\theta_{\infty}^{(j)}\big{]}$
$\displaystyle\hphantom{D_{ij}=}{}\times\prod_{k=1,\atop k\neq
j}^{N}\big{[}\theta_{\infty}^{(j)}-\theta_{\infty}^{(k)}\big{]}\prod_{k<k^{\prime}}\frac{1}{\big{[}\theta_{\infty}^{(k^{\prime})}-\theta_{\infty}^{(k)}\big{]}\big{[}\theta_{0}^{(k)}-\theta_{0}^{(k^{\prime})}\big{]}}\,s\prod_{k=1}^{m}s_{k,N}^{-N}.$

###### Proof.

By the relation (3.17), the $(i,j)$ entry of $B_{p,0}(t)$ for $i\neq j$
satisfies

$T_{q,t_{p}}\left(Y_{1}(t)_{ij}\right)-Y_{1}(t)_{ij}=-\frac{q^{-N\sum_{k=1}^{p}\theta_{k}}t_{p}}{\det
G(t)}\sum_{k=1}^{N}T_{q,t_{p}}\left(G(t)_{ik}\right)\widetilde{G(t)}_{jk},$
(3.19)

where $\widetilde{G(t)}$ is the cofactor matrix of $G(t)$.

By the definition of $Y_{1}(t)$ and $G(t)$ associated with $Y^{(0,1)}(x,t)$
and $Y^{(m+1,m+2)}(x,t)$, we have for $i\neq j$

$\displaystyle(Y_{1}(t))_{ij}=\frac{1}{\big{[}1-\theta_{\infty}^{(i)}+\theta_{\infty}^{(j)}\big{]}}\prod_{k=1,\,k\neq
i}^{N}\Gamma_{q}\big{(}\theta_{\infty}^{(i)}-\theta_{\infty}^{(k)}\big{)}\prod_{k=1}^{N}\frac{1}{\Gamma_{q}\big{(}1+\theta_{\infty}^{(j)}-\theta_{\infty}^{(k)}\big{)}}\,\frac{\tau_{ij}}{\tau},$
(3.20) $\displaystyle
G(t)_{ij}=q^{C_{ij}}\prod_{k=1}^{m+1}t_{k}^{\theta_{k}}\prod_{k=1,\,k\neq
i}^{N}\Gamma_{q}\big{(}\theta_{\infty}^{(i)}-\theta_{\infty}^{(k)}\big{)}\prod_{k=1}^{N}\frac{1}{\Gamma_{q}\big{(}1+\theta_{0}^{(k)}-\theta_{0}^{(j)}\big{)}}\,\frac{\tilde{\tau}_{ij}}{\tau},$
(3.21)

where

$\displaystyle
C_{ij}=\Delta_{\theta_{0}}-\Delta_{\theta_{\infty}}-\theta_{0}^{(j)}+\theta_{\infty}^{(i)}-\frac{N}{2}\bigg{(}\sum_{k=1}^{m+1}\theta_{k}\bigg{)}^{2}+\bigg{(}\frac{N}{2}-1-N\theta_{0}^{(j)}\bigg{)}\sum_{k=1}^{m+1}\theta_{k}.$

Substituting (3.20) and (3.21) into (3.19) together with the expression

$\displaystyle\det
G(t)^{-1}=(-1)^{(m+1)(N-1)}q^{N\sum_{i=1}^{m+1}\theta_{i}(N\sum_{i=1}^{m+1}\theta_{i}+1)/2}$
$\displaystyle\hphantom{\det G(t)^{-1}=}{}\times\prod_{1\leq i<j\leq
N}\frac{\vartheta\big{(}\theta_{\infty}^{(i)}-\theta_{\infty}^{(j)}\big{)}}{\vartheta\big{(}\theta_{0}^{(i)}-\theta_{0}^{(j)}\big{)}}\prod_{k=0}^{m}s_{k}^{\sum_{i=1}^{N-1}(h_{i}-h_{N})}t_{k+1}^{-N\theta_{k+1}}$

of $\det(G(t))$, we obtain the formula (3.18). ∎

###  3.2 Schlesinger transformations

Integer shifts on the characteristic exponents of Fuchsian systems preserving
their monodromy are called the Schlesinger transformations [33]. In [19], the
authors studied the Schlesinger transformations of monodromy preserving
deformation including irregular singular case and showed that the ratio
$\tau^{\prime}/\tau$ is a determinant of a matrix whose elements are given in
terms of the coefficients of fundamental solutions, where $\tau$ is the
Jimbo–Miwa–Ueno tau function introduced in [18] and $\tau^{\prime}$ is a
Schlesinger transform of the original tau function. We also mention a recent
work [17] where the authors presented another determinant formula for the
ratio of $\tau$-functions by using relations between Hermite’s two
approximation problems and particular Schlesinger transformation.

In our case, the connection matrix has periodicities (2.5) and (2.6) with
shifts of the parameters. This fact implies that the following shifts of the
parameters

$\displaystyle\theta_{\infty}\to\theta_{\infty}-h_{i},\qquad\theta_{0}\to\theta_{0}-h_{j},\qquad
1\leq i,j\leq n,$ $\displaystyle\theta_{i}\to\theta_{i}+\frac{1}{N},\qquad
1\leq i\leq m+1,$

can be regarded as $q$-Schlesinger transformations and yield $q$-difference
linear equations with rational coefficients satisfied by $Y^{(k,k+1)}(x,t)$,
$k=0,1,\dots,m+1$.

Let transformations $r_{i}$, $i=1,\dots,m+1$, and $p$ on the parameters be
defined as follows.

$\displaystyle r_{i}(\theta_{\infty})=\theta_{\infty}-h_{1},\quad
r_{i}(\sigma_{j})=\sigma_{j}-h_{1},\quad j=1,\dots,i-1,\quad\ \
r_{i}(\sigma_{j})=\sigma_{j},\quad j=i,\dots,m,$ $\displaystyle
r_{i}(\theta_{0})=\theta_{0},\qquad
r_{i}(\theta_{j})=\theta_{j}+\frac{1}{N}\delta_{i,j},\qquad j=1,\dots,m+1,$
$\displaystyle r_{i}(t_{j})=t_{j},\qquad j=1,\dots,i,\qquad
r_{i}(t_{j})=qt_{j},\qquad j=i+1,\dots,m+1,$ $\displaystyle
p(\theta_{\infty})=\theta_{\infty}-h_{1},\qquad
p(\sigma_{j})=\sigma_{j}-h_{1},\qquad j=1,\dots,m,\qquad
p(\theta_{0})=\theta_{0}-h_{1}.$

We show below that from the $q$-Schlesinger transformations $r_{i}$ and $p$,
we obtain determinantal identities for the tau functions.

######  Proposition 3.3.

The functions

$\displaystyle(-1)^{N\sum_{j=1}^{k}\theta_{j}}Y^{(k,k+1)}(x,t)x^{\theta_{\infty}^{(N)}},\qquad
k=0,1,\dots,m+1,$

satisfy the $q$-difference linear equation

$\displaystyle r_{i}(Y(x,t))=R_{i}(x,t)Y(x,t)$

with

$\displaystyle\det R_{i}(x,t)=x-q^{-N\sum_{j=1}^{i}\theta_{j}}t_{i},\quad
R_{i}(x,t)=R_{0}x+R_{i,0}(t),\quad R_{0}=\mathop{\rm diag}(1,0,\dots,0),$
$\displaystyle
R_{i,0}(x,t)=r_{i}(Y_{1}(t))R_{0}+I-R_{0}-R_{0}Y_{1}(t)=-r_{i}(G(t))G(t)^{-1}.$
(3.22)

###### Proof.

Put

$\displaystyle
R_{i}(x,t)=r_{i}\big{(}Y^{(0,1)}(x,t)\big{)}Y^{(0,1)}(x,t)^{-1}x^{1/N}.$

Then we have

$\displaystyle
R_{i}(x,t)=\begin{cases}r_{i}\big{(}Y^{(k,k+1)}(x,t)\big{)}Y^{(k,k+1)}(x,t)^{-1}x^{1/N},&k=1,\dots,i-1,\\\
-r_{i}\big{(}Y^{(k,k+1)}(x,t)\big{)}Y^{(k,k+1)}(x,t)^{-1}x^{1/N},&k=i,\dots,m+1,\end{cases}$

due to the connection relations (3.1) and the periodicities (2.5), (2.6) of
the connection matrix.

By the definition of $q$-conformal blocks, $R_{i}(x,t)$ is a meromorphic
function on $\mathbb{P}^{1}$, and hence it is a rational function of $x$. The
determinant of $R_{i}(x,t)$ is computed from $\det Y^{(0,1)}(x,t)$ (3.15).
Then, the form of $R_{i}(x,t)$ is obtained from the asymptotic expansions of
$Y^{(0,1)}(x,t)$ and $Y^{(m+1,m+2)}(x,t)$. ∎

Since we can express $R_{i,0}(x,t)$ in two different way (3.22), we obtain the
following theorem.

######  Theorem 3.4.

Let $1\leq i\leq m+1$.

  1. $1.$

For $2\leq a,b\leq N$, we have

$\displaystyle
t_{i}^{1/N}q^{\theta_{\infty}^{(1)}-\sum_{j=1}^{i}\theta_{j}}\frac{D_{ab}}{r_{i}(\tau)\tau^{N-1}}\left|\begin{matrix}\tilde{\tau}_{11}&\tilde{\tau}_{12}&\cdots&\tilde{\tau}_{1N}\\\
\vdots&\vdots&&\vdots\\\
\tilde{\tau}_{b-11}&\tilde{\tau}_{b-12}&\cdots&\tilde{\tau}_{b-1N}\\\\[2.15277pt]
q^{-\theta_{0}^{(1)}}r_{i}\left(\tilde{\tau}_{a1}\right)&q^{-\theta_{0}^{(2)}}r_{i}\left(\tilde{\tau}_{a2}\right)&\cdots&q^{-\theta_{0}^{(N)}}r_{i}\left(\tilde{\tau}_{aN}\right)\\\
\tilde{\tau}_{b+11}&\tilde{\tau}_{b+12}&\cdots&\tilde{\tau}_{b+1N}\\\
\vdots&\vdots&&\vdots\\\
\tilde{\tau}_{N1}&\tilde{\tau}_{N2}&\cdots&\tilde{\tau}_{NN}\end{matrix}\right|$
$\displaystyle\qquad{}=-\delta_{ab}\frac{\prod_{j=1}^{N}\Gamma_{q}\big{(}1+\theta_{\infty}^{(b)}-\theta_{\infty}^{(j)}\big{)}}{\prod_{j=1,\,j\neq
a}^{N}\Gamma_{q}\big{(}\theta_{\infty}^{(a)}-\theta_{\infty}^{(j)}\big{)}}\frac{\big{[}1-\theta_{\infty}^{(a)}+\theta_{\infty}^{(b)}\big{]}}{\big{[}\theta_{\infty}^{(a)}-\theta_{\infty}^{(1)}\big{]}}.$

  2. $2.$

For $2\leq b\leq N$, we have

$\displaystyle
t_{i}^{1/N}q^{\theta_{\infty}^{(1)}-1-\sum_{j=1}^{i}\theta_{j}}\frac{D_{1b}}{r_{i}(\tau)\tau^{N-1}}\left|\begin{matrix}\tilde{\tau}_{11}&\tilde{\tau}_{12}&\cdots&\tilde{\tau}_{1N}\\\
\vdots&\vdots&&\vdots\\\
\tilde{\tau}_{b-11}&\tilde{\tau}_{b-12}&\cdots&\tilde{\tau}_{b-1N}\\\\[2.15277pt]
q^{-\theta_{0}^{(1)}}r_{i}\left(\tilde{\tau}_{11}\right)&q^{-\theta_{0}^{(2)}}r_{i}\left(\tilde{\tau}_{12}\right)&\cdots&q^{-\theta_{0}^{(N)}}r_{i}\left(\tilde{\tau}_{1N}\right)\\\
\tilde{\tau}_{b+11}&\tilde{\tau}_{b+12}&\cdots&\tilde{\tau}_{b+1N}\\\
\vdots&\vdots&&\vdots\\\
\tilde{\tau}_{N1}&\tilde{\tau}_{N2}&\cdots&\tilde{\tau}_{NN}\end{matrix}\right|$
$\displaystyle\qquad{}=\frac{\tau_{1b}}{\tau}\prod_{j=2}^{N}\big{[}\theta_{\infty}^{(1)}-\theta_{\infty}^{(j)}-1\big{]}.$

  3. $3.$

For $2\leq a\leq N$, we have

$\displaystyle
t_{i}^{1/N}q^{\theta_{\infty}^{(1)}-\sum_{j=1}^{i}\theta_{j}}\frac{D_{a1}}{r_{i}(\tau)\tau^{N-1}}\left|\begin{matrix}q^{-\theta_{0}^{(1)}}r_{i}\left(\tilde{\tau}_{a1}\right)&q^{-\theta_{0}^{(2)}}r_{i}\left(\tilde{\tau}_{a2}\right)&\cdots&q^{-\theta_{0}^{(N)}}r_{i}\left(\tilde{\tau}_{aN}\right)\\\
\tilde{\tau}_{21}&\tilde{\tau}_{22}&\cdots&\tilde{\tau}_{2N}\\\
\vdots&\vdots&&\vdots\\\
\tilde{\tau}_{N1}&\tilde{\tau}_{N2}&\cdots&\tilde{\tau}_{NN}\end{matrix}\right|$
$\displaystyle\qquad{}=-r_{i}\bigg{(}\frac{\tau_{a1}}{\tau}\bigg{)}\big{[}1-\theta_{\infty}^{(a)}+\theta_{\infty}^{(1)}\big{]}\prod_{j=2,\,j\neq
a}^{N}\big{[}\theta_{\infty}^{(1)}-\theta_{\infty}^{(j)}\big{]}.$

Similarly to the $q$-Schlesinger transformation $r_{i}$, we have the following
proposition and theorem on the $q$-Schlesinger transformation $p$.

######  Proposition 3.5.

The functions $Y^{(k,k+1)}(x,t)$, $k=0,1,\dots,m+1$, satisfy the
$q$-difference linear equation

$\displaystyle p\left(Y(x,t)\right)=P(x,t)Y(x,t)x^{-1/N}$

with

$\displaystyle\det P(x,t)=x,\qquad P(x,t)=P_{0}x+P_{i,0}(t),\qquad
P_{0}=\mathop{\rm diag}(1,0,\dots,0),$ $\displaystyle
P_{i,0}(x,t)=p(Y_{1}(t))P_{0}+I-P_{0}-P_{0}Y_{1}(t)=p(G(t))G(t)^{-1}.$

######  Theorem 3.6.

  1. $1.$

For $2\leq a,b\leq N$, we have

$\displaystyle
q^{\theta_{\infty}^{(1)}-\theta_{0}^{(1)}-\sum_{j=1}^{N}\theta_{j}}\frac{D_{ab}}{p(\tau)\tau^{N-1}}\left|\begin{matrix}\tilde{\tau}_{11}&\tilde{\tau}_{12}&\cdots&\tilde{\tau}_{1N}\\\
\vdots&\vdots&&\vdots\\\
\tilde{\tau}_{b-11}&\tilde{\tau}_{b-12}&\cdots&\tilde{\tau}_{b-1N}\\\\[2.15277pt]
0&\big{[}\theta_{0}^{(1)}-\theta_{0}^{(2)}\big{]}p\big{(}\tilde{\tau}_{a2}\big{)}&\cdots&\big{[}\theta_{0}^{(1)}-\theta_{0}^{(N)}\big{]}p\big{(}\tilde{\tau}_{aN}\big{)}\\\
\tilde{\tau}_{b+11}&\tilde{\tau}_{b+12}&\cdots&\tilde{\tau}_{b+1N}\\\
\vdots&\vdots&&\vdots\\\
\tilde{\tau}_{N1}&\tilde{\tau}_{N2}&\cdots&\tilde{\tau}_{NN}\end{matrix}\right|$
$\displaystyle\qquad{}=\delta_{ab}\frac{\prod_{j=1}^{N}\Gamma_{q}\big{(}1+\theta_{\infty}^{(b)}-\theta_{\infty}^{(j)}\big{)}}{\prod_{j=1,\,j\neq
a}^{N}\Gamma_{q}\big{(}\theta_{\infty}^{(a)}-\theta_{\infty}^{(j)}\big{)}}\frac{\big{[}1-\theta_{\infty}^{(a)}+\theta_{\infty}^{(b)}\big{]}}{\big{[}\theta_{\infty}^{(a)}-\theta_{\infty}^{(1)}\big{]}}.$

  2. $2.$

For $2\leq b\leq N$, we have

$\displaystyle
q^{\theta_{\infty}^{(1)}-\theta_{0}^{(1)}-1-\sum_{j=1}^{N}\theta_{j}}\frac{D_{1b}}{p(\tau)\tau^{N-1}}\left|\begin{matrix}\tilde{\tau}_{11}&\tilde{\tau}_{12}&\cdots&\tilde{\tau}_{1N}\\\
\vdots&\vdots&&\vdots\\\
\tilde{\tau}_{b-11}&\tilde{\tau}_{b-12}&\cdots&\tilde{\tau}_{b-1N}\\\
0&\big{[}\theta_{0}^{(1)}-\theta_{0}^{(2)}\big{]}p\left(\tilde{\tau}_{12}\right)&\cdots&\big{[}\theta_{0}^{(1)}-\theta_{0}^{(N)}\big{]}p\left(\tilde{\tau}_{1N}\right)\\\
\tilde{\tau}_{b+11}&\tilde{\tau}_{b+12}&\cdots&\tilde{\tau}_{b+1N}\\\
\vdots&\vdots&&\vdots\\\
\tilde{\tau}_{N1}&\tilde{\tau}_{N2}&\cdots&\tilde{\tau}_{NN}\end{matrix}\right|$
$\displaystyle\qquad{}=-\frac{\tau_{1b}}{\tau}\prod_{j=2}^{N}\big{[}\theta_{\infty}^{(1)}-\theta_{\infty}^{(j)}-1\big{]}.$

  3. $3.$

For $2\leq a\leq N$, we have

$\displaystyle
q^{\theta_{\infty}^{(1)}-\theta_{0}^{(1)}-\sum_{j=1}^{N}\theta_{j}}\frac{D_{a1}}{p(\tau)\tau^{N-1}}\left|\begin{matrix}0&\big{[}\theta_{0}^{(1)}-\theta_{0}^{(2)}\big{]}p\left(\tilde{\tau}_{a2}\right)&\cdots&\big{[}\theta_{0}^{(1)}-\theta_{0}^{(N)}\big{]}p\left(\tilde{\tau}_{aN}\right)\\\
\tilde{\tau}_{21}&\tilde{\tau}_{22}&\cdots&\tilde{\tau}_{2N}\\\
\vdots&\vdots&&\vdots\\\
\tilde{\tau}_{N1}&\tilde{\tau}_{N2}&\cdots&\tilde{\tau}_{NN}\end{matrix}\right|$
$\displaystyle\qquad{}=p\bigg{(}\frac{\tau_{a1}}{\tau}\bigg{)}\big{[}1-\theta_{\infty}^{(a)}+\theta_{\infty}^{(1)}\big{]}\prod_{j=2,\,j\neq
a}^{N}\big{[}\theta_{\infty}^{(1)}-\theta_{\infty}^{(j)}\big{]}.$

######  Remark 3.7.

In the case of $N=2$, the determinantal identities above become bilinear
equations satisfied by the tau functions. On the other hand, in the previous
work [20], we gave conjectural bilinear equations of the tau functions for
$N=2$ and $m=1$. They are not equal to, and it is not clear whether the
bilinear equations obtained in this paper recover the conjectural bilinear
equations (equations (3.16)–(3.23) in [20]).

######  Remark 3.8.

When $N=2$ and $m>1$, the compatibility condition (3.16) of the Lax pair
yields the $q$-Garnier system. For its $q\to 1$ limit, namely, the Garnier
system, bilinear equations of the tau functions were presented in [34]. It is
easily seen from the transformation of the parameters by $r_{i}$ that the
bilinear equations in Theorem 3.4(2), (3) are $q$-analogs of (part of) the
Hirota–Miwa equations (equations (3.47) in [34]).

### Acknowledgements

This work is partially supported by JSPS KAKENHI Grant Number JP18K03326. The
author thanks the referees for valuable suggestions and comments.

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