$\psi$ is non-zero almost everywhere, then these two pairs are the same. The
equivalent statement holds for density matrices.

###### Proof.

First we note that if both Hamiltonians $H[v_{1},\mathbf{J}_{\rm ext,1}]$ and
$H[v_{2},\mathbf{J}_{\rm ext,2}]$ share a common eigenstate then due to Eq.
(28) we have $\mathbf{J}_{\rm ext,1}=\mathbf{J}_{\rm ext,2}$. Thus we are left
with the two equations

$\displaystyle\left(H_{0}+V[v_{1}]\right)\psi$
$\displaystyle=E[v_{1},\mathbf{J}_{\mathrm{ext}}]\psi,$
$\displaystyle\left(H_{0}+V[v_{2}]\right)\psi$
$\displaystyle=E[v_{2},\mathbf{J}_{\mathrm{ext}}]\psi,$

and we can follow the standard HK2 proof of Theorem 2 of Part I. We can
further use Corollary 3 of Part I to find the equivalent statement for the
density matrices. ∎

At that point we see again how powerful the abstract formulation of HK1 and
HK2 as presented at the end of Part I and then repeated at the beginning of
Part II of this review is. It allows to re-use many results of standard
electronic DFT for other settings as well. We finally note that for the
corresponding KS system in QEDFT one commonly uses non-interacting electrons
and photons, which leads to electronic Pauli–Kohn–Sham equations coupled to a
static inhomogeneous Maxwell equation of the form of Eq. (28) [73].

##  X Summary

flavor | density variables | potential variables | HK1 | HK2
---|---|---|---|---
DFT | $\rho$ | $v$ | yes | yes
LDFT | $\rho,\mathbf{p},\mathbf{L}$ | $v,\mathbf{A}=\mathbf{a}+\tfrac{1}{2}\mathbf{B}\times\mathbf{r}$ | yes | yes, if $\rho$ asym.
SDFT (col.) | $\rho,m_{z}=\rho_{\uparrow}-\rho_{\downarrow}$ | $v,B^{\prime}_{z}$ | yes | debated
SDFT (noncol.) | $\rho,\mathbf{m}$ | $v,\mathbf{B}^{\prime}$ | yes | no
CDFT (no spin) | $\rho,\mathbf{j}^{\mathrm{p}}$ | $v,\mathbf{A}$ | yes | no
CDFT (with spin curr.) | $\rho,\mathbf{j}^{\mathrm{m}}=\mathbf{j}^{\mathrm{p}}+\nabla\times\mathbf{m}$ | $v,\mathbf{A}$ | yes | no
CDFT (with spin dens.) | $\rho,\mathbf{m},\mathbf{j}^{\mathrm{p}}$ | $v,\mathbf{A}$ | yes | no
CDFT (with spin dens.) | $\rho,\mathbf{m},\mathbf{j}^{\mathrm{p}}$ | $v,\mathbf{B}^{\prime},\mathbf{A}$ | yes | no
MDFT | $\rho,\mathbf{B}_{\mathrm{tot}}$ | $v,\mathbf{B}$ | yes | yes
QEDFT | $\rho,\boldsymbol{A}$ | $v,\mathbf{J}_{\mathrm{ext}}$ | yes | yes
Table 1: The status of HK1 and HK2 within some flavors of DFT.

Many flavors of density-functional theory exist besides standard DFT. All
flavors considered here capture some aspect of spin and orbital magnetism.
They can be characterized in terms of constraints on the, at the outset, very
general Hamiltonian given in Eq. (2)

$\displaystyle H[v,\mathbf{B}^{\prime},\mathbf{A}]$
$\displaystyle=\frac{1}{2}\sum_{k=1}^{N}(-\mathrm{i}\nabla_{k}+\mathbf{A}(\mathbf{r}_{k}))^{2}+\sum_{k=1}^{N}\mathbf{B}^{\prime}(\mathbf{r}_{k})\cdot\mathbf{S}_{k}$
$\displaystyle\quad+\sum_{k}v(\mathbf{r}_{k})+\lambda\sum_{k<l}\frac{1}{r_{kl}}.$

As noted there, we allow for the case that the magnetic field appearing in the
spin-Zeeman term is unrelated to the vector potential that affects the orbital
degrees of freedom, i.e., $\mathbf{B}^{\prime}\neq\nabla\times\mathbf{A}$.

Noncollinear SDFT is obtained when orbital effects are neglected by setting
$\mathbf{A}=\mathbf{0}$. The external magnetic field $\mathbf{B}^{\prime}$ is
then paired with the spin density $\mathbf{m}$. Both of these are general,
noncollinear vector fields. Yet, most practical approximate functionals are
constructed for the collinear case when only collinear magnetic fields
$\mathbf{B}^{\prime}=(0,0,B^{\prime}_{z})$ along, say, the $z$-axis are
allowed. There is a global spin quantization axis and only the component
$m_{z}=\rho_{\uparrow}-\rho_{\downarrow}$ of the vector field $\mathbf{m}$ is
needed. The non-uniqueness of potentials (i.e., the lack of a HK2 result in
our terminology) in collinear SDFT has been discussed by several authors, with
different conclusions. The situation is summarized in Ayers and Yang [26].

In paramagnetic CDFT, the orbital effects are retained. Different flavors of
CDFT are possible depending on how the spin-Zeeman term is treated. The
simplest flavor, treated here in great detail, simply neglects it
($\mathbf{B}^{\prime}=\mathbf{0}$). Alternatively, in the physically natural
case where $\mathbf{B}^{\prime}=\nabla\times\mathbf{A}$, a partial integration
turns the spin-Zeeman term into an interaction between $\mathbf{A}$ and the
spin current density. The latter is absorbed into the paramagnetic current
density to form
$\mathbf{j}^{\mathrm{m}}=\mathbf{j}^{\mathrm{p}}+\nabla\times\mathbf{m}$.
Retaining $\mathbf{B}^{\prime}$ as an independent variable, unrelated to
$\nabla\times\mathbf{A}$, yields the most flexible setting with the triple
$(\rho,\mathbf{m},\mathbf{j}^{\mathrm{p}})$ as the basic density variables.
Loosely speaking, in a CDFT formulation analogous to Lieb’s formulation of
standard DFT, the triple of independent density variables must have a triple
of independent potential variables. Hence, $\mathbf{B}^{\prime}$ needs to be
retained as an independent variable if $\mathbf{m}$ is to be an independent
density. However, when Lieb-like formulation is not required, nothing prevents
the introduction of additional constraints in a constrained-search
formulation. In this sense, a CDFT formulation with a triple of density
variables $(\rho,\mathbf{m},\mathbf{j}^{\mathrm{p}})$ and a pair potential
variables $(v,\mathbf{A})$, with $\mathbf{B}^{\prime}=\nabla\times\mathbf{A}$,
also exists.

With regard to the Hohenberg–Kohn theorem in CDFT, the inclusion or exclusion
of spin effects makes no difference: HK1 holds and HK2 does not. As already
noted in Part I, the HK1 result does not only hold for standard DFT, but holds
for all variants of extended DFTs that offer the required structure.
Paramagnetic CDFT has this structure and is arguably the most natural CDFT
formulation as far as the mathematical results are concerned. At the same
time, this theory is not invariant under gauge transformations and a HK2
cannot hold. On the other hand, for the formulation of CDFT that uses the
total (physical) current it is unfortunate that in general

$\displaystyle\langle\psi|H[v,\mathbf{A}]|\psi\rangle$
$\displaystyle\neq\inf_{\psi\mapsto(\rho,\mathbf{j})}\\{\langle\psi|H_{0}|\psi\rangle\\}+\langle\mathbf{A},\mathbf{j}\rangle+\langle
v-\tfrac{1}{2}|\mathbf{A}|^{2},\rho\rangle.$

Equality only holds for $\psi$ such that
$\mathbf{a}[\psi;\mathbf{j}]=\mathbf{A}$ (where $\mathbf{a}[\psi;\mathbf{j}]$
was defined in Eq. (16)), then a HK1 result follows. As can be seen by Eq.
(20), it is not evident how to obtain an HK1 result since minimization of just
$H_{0}$ over wave functions then leaves out $\psi$-dependent terms. Note that
HK2 does not hold, since we know that a shared eigenstate of magnetic
Hamiltonians does not imply that the potentials are equal (up to a gauge).
Also note that, if one fixes $\mathbf{a}=\mathbf{A}$, then one effectively has
a paramagnetic formulation of CDFT again.

Furthermore, what could be stressed from the above discussion is that,
regardless of the status of a full HK result, we have no HK variational
principle in total CDFT [24]. Thus, even if the question of a HK result for
the total current could be answered in the positive, the formulation would be
restricted to $v$-representable densities and thereby excluding the usual
approach of utilizing constrained-search functionals on $N$-representable
densities. This has stopped the mathematical development of total CDFT as
compared to the paramagnetic variant.

We have seen that by going beyond the usual density-functional setting, when
new density and corresponding potential variables are included, problems arise
mostly with respect to HK2. This is compactly highlighted in Table 1. While
the mathematical reasons have been discussed in detail in the preceding
sections, there are often also simple physical reasons. This holds
specifically in the context when magnetic fields are included and associated
densities are considered. The non-uniqueness results, discussed in Section
IV.1, arise because the back-reaction of the current on the external field and
the change in Maxwell energy is not taken into account. Doing so by also
including the induced Maxwell field in a self-consistent manner, as discussed
in Section VIII, avoids some of these issues and a HK2 theorem becomes
available. Hence, the density-functional theory based on the
Maxwell–Schrödinger model (MDFT) is a type of total CDFT with a full HK
result. This intuitive result, however, raises the question why we do not need
to include the Maxwell field energy also in the usual (standard) DFT of only
scalar external potentials. The answer to this question is given in Section IX
with the help of QED. We saw that the Coulomb interaction of the usual
Schrödinger equation is actually taking the self-consistent longitudinal
photon energy into account upon interaction with matter. Therefore it seems
natural to also take the transverse photon energy into account. In the context
of low-energy QED, where both contributions are considered self-consistently,
we therefore again find a HK2 result.

This formal discussion has shown that promoting the Maxwell field to a
quantized system allows to recover a DFT formulation that is very close to the
original electronic DFT. And by approximating the Pauli–Fierz theory we
obtain, in the mean-field coupling limit, the Maxwell–Schrödinger equation,
and by discarding the transverse part of the Maxwell field altogether, we find
standard electronic DFT. Yet, apart from this nice consistency and the simple
form of a DFT, is there any other reason to consider QEDFT and the Pauli–Fierz
theory? The answer is ‘yes’ and lies in the emerging fields of polaritonic
chemistry and materials science as well as ab initio QED [75; 76]. In these
fields, photonic structures, such as optical cavities, change locally the
vacuum modes that couple to the matter subsystem and hence present a novel
control knob to influence chemical and material properties. There are by now
many seminal experimental results that show that upon reaching strong matter-
photon coupling in photonic structures, we can indeed modify and control such
properties. Consequently, methods that can approximately solve the Pauli–Fierz
field theory become increasingly important.

## Acknowledgement

We are indebted to our two referees for numerous comments and careful
corrections that helped to greatly improve the paper. EIT, MAC and AL thank
the Research Council of Norway (RCN) under CoE (Hylleraas Centre) Grant No.
262695, for AL and MAC also CCerror Grant No. 287906 and for EIT also
“Magnetic Chemistry” Grant No. 287950, and MR acknowledges the Cluster of
Excellence “CUI: Advanced Imaging of Matter” of the Deutsche
Forschungsgemeinschaft (DFG), EXC 2056, project ID 390715994. AL and MAC were
also supported by the ERC through StG REGAL under agreement No. 101041487. The
authors thank Centre for Advanced Studies (CAS) in Oslo, since this work
includes insights gathered at the YoungCAS workshop “Do Electron Current
Densities Determine All There Is to Know?”, held July 9-13, 2018, in Oslo,
Norway.

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$q$-Middle Convolution and $q$-Painlevé Equation

$\boldsymbol{q}$-Middle Convolution and $\boldsymbol{q}$-Painlevé Equation

Shoko SASAKI a, Shun TAKAGI a and Kouichi TAKEMURA b

S. Sasaki, S. Takagi and K. Takemura

a) Department of Mathematics, Faculty of Science and Engineering, Chuo
University,
a) 1-13-27 Kasuga, Bunkyo-ku, Tokyo 112-8551, Japan

b) Department of Mathematics, Ochanomizu University,
b) 2-1-1 Otsuka, Bunkyo-ku, Tokyo 112-8610, Japan<EMAIL_ADDRESS>
Received January 31, 2022, in final form July 08, 2022; Published online July
20, 2022

A $q$-deformation of the middle convolution was introduced by Sakai and
Yamaguchi. We apply it to a linear $q$-difference equation associated with the
$q$-Painlevé VI equation. Then we obtain integral transformations. We
investigate the $q$-middle convolution in terms of the affine Weyl group
symmetry of the $q$-Painlevé VI equation. We deduce an integral transformation
on the $q$-Heun equation.

$q$-Painlevé equation; $q$-Heun equation; middle convolution; integral
transformation

33E10; 34M55; 39A13

##  1 Introduction

The middle convolution was introduced by Katz [8] for local systems on a
punctured Riemann sphere, and Dettweiler and Reiter [3, 4] reformulated it for
the Fuchsian system of differential equations. Here the Fuchsian system of
differential equations is the system of linear differential equations written
as

$\displaystyle\frac{{\rm d}Y}{{\rm
d}x}=\bigg{(}\frac{A_{1}}{x-t_{1}}+\frac{A_{2}}{x-t_{2}}+\dots+\frac{A_{r}}{x-t_{r}}\bigg{)}Y,$
(1.1)

where $Y$ is a column vector with $n$ entries and $A_{1},A_{2},\dots,A_{r}$
are constant matrices of size $n\times n$. We review briefly the definition of
the middle convolution for equation (1.1) (or the tuple of the matrices
$(A_{1},\dots,A_{r})$). Let $\lambda\in\mathbb{C}$ and $F_{i}$, $i=1,\dots,r$,
be the matrix of size $nr\times nr$ of the form

$\displaystyle F_{i}=\begin{pmatrix}O&\cdots&O&\cdots&O\\\
\vdots&&\vdots&&\vdots\\\ A_{1}&\cdots&A_{i}+\lambda I_{n}&\cdots&A_{r}\\\
\vdots&&\vdots&&\vdots\\\ O&\cdots&O&\cdots&O\end{pmatrix}{\scriptstyle(i)},$
(1.2)

where $I_{n}$ is the identity matrix of size $n$. Then the correspondence of
the tuple of matrices $(A_{1},\dots,A_{r})\mapsto(F_{1},\dots,F_{r})$ (or the
correspondence of the associated Fuchsian system) is called the convolution.
The convolution does not preserve the irreducibility in general. It is shown
that the following subspaces $\mathcal{K},\mathcal{L}$ of $\mathbb{C}^{nr}$
are preserved by the action of $F_{i}$, $i=1,\dots,r$,

$\displaystyle\mathcal{K}=\begin{pmatrix}\ker A_{1}\\\ \vdots\\\ \ker
A_{r}\end{pmatrix}\\!,\qquad\mathcal{L}=\ker(F_{1}+F_{2}+\cdots+F_{r}).$

We denote the linear transformation induced from the action of $F_{i}$ on the
quotient space $\mathbb{C}^{nr}/(\mathcal{K}+\mathcal{L})$ by
$\overline{F}_{i}$. The correspondence of the tuple of matrices
$(A_{1},\dots,A_{r})\mapsto\big{(}\overline{F}_{1},\dots,\overline{F}_{r}\big{)}$
(or the correspondence of the associated Fuchsian system) is called the middle
convolution. It was shown in [3] that the convolution is related with Euler’s
integral transformation. Let $Y(x)$ be a solution of equation (1.1). Set

$\displaystyle W_{j}(x)=\frac{Y(x)}{x-t_{j}},\qquad
W(x)=\begin{pmatrix}W_{1}(x)\\\ \vdots\\\ W_{r}(x)\end{pmatrix}\\!.$

Then $W(x)$ is a column vector with $nr$ entries. We apply Euler’s integral
transformation for each entry of $W(x)$, i.e., we set

$\displaystyle G(x)=\int_{\Delta}W(s)(x-s)^{\lambda}\,{\rm d}s,$

where $\Delta$ is an appropriate cycle in $\mathbb{C}$ with the variable $s$.
Then the function $G(x)$ satisfies the following Fuchsian system of
differential equation

$\displaystyle\frac{{\rm d}Y}{{\rm
d}x}=\bigg{(}\frac{F_{1}}{x-t_{1}}+\frac{F_{2}}{x-t_{2}}+\dots+\frac{F_{r}}{x-t_{r}}\bigg{)}Y,$

where $F_{1},\dots,F_{r}$ were defined in equation (1.2).

Sakai and Yamaguchi [10] constructed a theory of a $q$-deformation of the
middle convolution for systems of $q$-difference equations. Here the system is
described as

$\displaystyle
Y(qx)=\bigg{(}B_{\infty}+\frac{B_{1}}{1-x/t_{1}}+\dots+\frac{B_{r}}{1-x/t_{r}}\bigg{)}Y(x),$

where $Y(x)$ is a column vector with $n$ entries and
$B_{\infty},B_{1},\dots,B_{r}$ are constant matrices of size $n\times n$. The
construction of the $q$-middle convolution is similar to the case of the
Fuchsian system of differential equations. For details, see Section 2.

In this paper we apply the $q$-middle convolution to linear $q$-difference
equations which are related to the $q$-Painlevé VI equation

$\displaystyle\frac{y\overline{y}}{a_{3}a_{4}}=\frac{(\overline{z}-tb_{1})(\overline{z}-tb_{2})}{(\overline{z}-b_{3})(\overline{z}-b_{4})},\qquad\frac{z\overline{z}}{b_{3}b_{4}}=\frac{(y-ta_{1})(y-ta_{2})}{(y-a_{3})(y-a_{4})},$
(1.3)

with the constraint $b_{1}b_{2}a_{3}a_{4}=qa_{1}a_{2}b_{3}b_{4}$. Here
$\overline{y}$ and $\overline{z}$ denotes the time evolution $t\mapsto qt$ of
$y$ and $z$, and the parameters $a_{1},\dots,a_{4},b_{1},\dots,b_{4}$ are
time-independent. The $q$-Painlevé VI equation was introduced by Jimbo and
Sakai [6] as a $q$-deformation of the Painlevé VI equation. They obtained
equation (1.3) by introducing a $q$-analogue of the monodromy preserving
deformation, and it is related to the linear $q$-difference equations

$\displaystyle Y(qx)=A(x)Y(x),$ (1.4)

where $A(x)$ is a $2\times 2$ matrix with polynomial entries (see equation
(3.2) for details). The linear $q$-difference equation which we apply the
$q$-middle convolution is not equation (1.4) but the transformed equation

$\displaystyle Y(qx)=B(x)Y(x),$ $\displaystyle
B(x)=\frac{A(x)}{c_{0}(x-ta_{1})(x-ta_{2})}=B_{\infty}+\frac{B_{1}}{1-x/(ta_{1})}+\frac{B_{2}}{1-x/(ta_{2})}$
(1.5)

for some constant $c_{0}$. By applying the $q$-middle convolution with the
parameter $\lambda$, we obtain $2\times 2$ matrices, if we choose the
constants $c_{0}$ and $\lambda$ suitably. Then we obtain a correspondence of
the parameters, and we may regard it as a correspondence of the $q$-Painlevé
VI equations. On the other hand, it is known that the $q$-Painlevé VI equation
has a symmetry of the affine Weyl group of type $D^{(1)}_{5}$, and a
realization of the symmetry was described in the review [7] of Kajiwara, Noumi
and Yamada. In this paper, we express the symmetry by the $q$-middle
convolution in terms of the generators of the affine Weyl group.

In [11], a relationship between the $q$-Painlevé VI equation and the $q$-Heun
equation

$\displaystyle\big{(}x-h_{1}q^{1/2}\big{)}\big{(}x-h_{2}q^{1/2}\big{)}g(x/q)+l_{3}l_{4}\big{(}x-l_{1}q^{-1/2}\big{)}\big{(}x-l_{2}q^{-1/2}\big{)}g(qx)$
$\displaystyle\qquad{}-\big{\\{}(l_{3}+l_{4})x^{2}+Ex+(l_{1}l_{2}l_{3}l_{4}h_{1}h_{2})^{1/2}\big{(}h_{3}^{1/2}+h_{3}^{-1/2}\big{)}\big{\\}}g(x)=0$

was studied from a viewpoint of the initial value space. In particular, the
$q$-Heun equation was obtained from the linear $q$-difference equation
associated to the $q$-Painlevé VI equation by specializing the parameters. On
the other hand, the $q$-middle convolution induces an integral transformation
of the linear $q$-difference equation. By considering a particular
specialization, the linear $q$-difference equation turns out to be the
$q$-Heun equation and we obtain an integral transformation of $q$-Heun
equation.

This paper is organized as follows. In Section 2, we review a part of the
theory of the $q$-middle convolution established by Sakai and Yamaguchi [10].
In Section 3, we recall the linear $q$-difference equation associated to the
$q$-Painlevé VI equation and calculate the $q$-middle convolution for it. In
Section 4, we investigate the symmetry by the $q$-middle convolution in terms
of the Weyl group symmetry of the $q$-Painlevé VI equation. For this purpose,
we clarify a relationship between equation (1.5) and the Lax pair in [7]. In
Section 5, we obtain an integral transformation on the $q$-Heun equation. In
Section 6, we give concluding remarks.

##  2 $\boldsymbol{q}$-middle convolution

We recall the $q$-convolution and the $q$-middle convolution introduced by
Sakai and Yamaguchi [10].

Let $\mathbf{B}=(B_{\infty};B_{1},\dots,B_{N})$ be the tuple of the square
matrices of the same size and $\mathbf{b}=(b_{1},b_{2},\dots,b_{N})$ be the
tuple of the non-zero complex numbers which are different from one another. We
denote by $E_{\mathbf{B},\mathbf{b}}$ the linear $q$-difference equations

$\displaystyle Y(qx)=B(x)Y(x),\qquad
B(x)=B_{\infty}+\sum^{N}_{i=1}\frac{B_{i}}{1-x/b_{i}}.$

######  Definition 2.1 ($q$-convolution, [10]).

Let $\mathbf{B}=(B_{\infty};B_{1},\dots,B_{N})$ be the tuple of $m\times m$
matrices and $(b_{1},b_{2},\dots,b_{N})$ be the tuple of the non-zero complex
numbers which are different one another. Set
$B_{0}=I_{m}-B_{\infty}-B_{1}-\dots-B_{N}$, We define the $q$-convolution
$c_{\lambda}\colon(B_{\infty};B_{1},\dots,B_{N})\mapsto(F_{\infty};F_{1},\dots,F_{N})$
as follows:

$\displaystyle\mbox{\boldmath$F$}=(F_{\infty};F_{1},\dots,F_{N})\mbox{ \rm is
a tuple of $(N+1)m\times(N+1)m$ matrices,}$ $\displaystyle
F_{i}=\begin{pmatrix}{}&{}&O&{}&{}\\\\[5.0pt]
B_{0}&\cdots&B_{i}-\big{(}1-q^{\lambda}\big{)}I_{m}&\cdots&B_{N}\\\\[5.0pt]
{}&{}&O&{}&{}\end{pmatrix}{\scriptstyle(i+1)},\qquad 1\leq i\leq N,$
$\displaystyle
F_{\infty}=I_{(N+1)m}-\widehat{F},\qquad\widehat{F}=\begin{pmatrix}B_{0}&\cdots&B_{N}\\\\[5.0pt]
\vdots&\ddots&\vdots\\\\[5.0pt] B_{0}&\cdots&B_{N}\end{pmatrix}\\!.$

Let $\xi\in{\mathbb{C}}\setminus\\{0\\}$. The $q$-convolution induces the
$q$-analogue of the Euler’s integral transformation in terms of the Jackson
integral

$\displaystyle\int^{\xi\infty}_{0}f(x)\,{\rm
d}_{q}x=(1-q)\sum^{\infty}_{n=-\infty}q^{n}\xi f(q^{n}\xi)$

for the solutions of the $q$-difference equations. Note that the value of the
Jackson integral may depend on the value $\xi$.

######  Theorem 2.2 ([10, Theorem 2.1]).

Let $Y(x)$ be a solution of $E_{\mathbf{B},\mathbf{b}}$. Set $b_{0}=0$ and

$\displaystyle
P_{\lambda}(x,s)=\frac{\big{(}q^{\lambda+1}s/x;q\big{)}_{\infty}}{(qs/x;q)_{\infty}}=\prod^{\infty}_{i=0}\frac{x-q^{i+\lambda+1}s}{x-q^{i+1}s}.$

Define the function $\widehat{Y}(x)$ by

$\displaystyle\widehat{Y}_{i}(x)=\int^{\xi\infty}_{0}\frac{P_{\lambda}(x,s)}{s-b_{i}}Y(s)\,{\rm
d}_{q}s,\quad
i=0,\dots,N,\qquad\widehat{Y}(x)=\begin{pmatrix}\widehat{Y}_{0}(x)\\\
\vdots\\\ \widehat{Y}_{N}(x)\end{pmatrix}\\!.$

Then the function $\widehat{Y}(x)$ satisfies the equation
$E_{\mathbf{F},\mathbf{b}}$, i.e.,

$\displaystyle\widehat{Y}(qx)=\bigg{(}F_{\infty}+\sum^{N}_{i=1}\frac{F_{i}}{1-x/b_{i}}\bigg{)}\widehat{Y}(x).$

Although the original theorem by Sakai and Yamaguchi was restricted to the
case $\xi=1$ in the Jackson integral, we may just extend it to the case
$\xi\in{\mathbb{C}}\setminus\\{0\\}$, which was motivated by the theory of the
Jackson integral due to Aomoto [1]. The convergence of the Jackson integrals
$\widehat{Y}_{i}(x)$, $i=0,\dots,N$, would not be considered in [10]. In this
paper, we discuss the Jackson integrals formally and we do not consider the
convergence in details. Namely, we discuss the Jackson integrals under the
assumption that the integrals converge absolutely and Theorem 2.2 holds true
with the convergence. Thus we use the phrasing “formally” in theorems which
are related to the Jackson integral. An aspect of convergence on Theorem 2.2
will be discussed in [2].

The $q$-middle convolution is defined by considering an appropriate quotient
space.

######  Definition 2.3 ($q$-middle convolution, [10]).

We define the $\boldsymbol{F}$-invariant subspaces $\mathcal{K}$ and
$\mathcal{L}$ of $(\mathbb{C}^{m})^{N+1}$ as follows

$\displaystyle\mathcal{K}=\mathcal{K}_{\mathcal{V}}=\begin{pmatrix}\ker
B_{0}\\\ \vdots\\\ \ker
B_{N}\end{pmatrix}\\!,\qquad\mathcal{L}=\mathcal{L}_{\mathcal{V}}(\lambda)=\ker\big{(}\widehat{F}-\big{(}1-q^{\lambda}\big{)}I_{(N+1)m}\big{)}.$

We denote the action of $F_{k}$ on the quotient space
$(\mathbb{C}^{m})^{N+1}/(\mathcal{K}+\mathcal{L})$ by $\overline{F}_{k}$,
$k=\infty,1,\dots,N$. Then the $q$-middle convolution $mc_{\lambda}$ is
defined by the correspondence $E_{\mathbf{B},\mathbf{b}}\mapsto
E_{\overline{\mathbf{F}},\mathbf{b}}$, where
$\overline{\mathbf{F}}=\big{(}\overline{F}_{\infty};\overline{F}_{1},\dots,\overline{F}_{N}\big{)}$.

The $q$-middle convolution $mc_{\lambda}$ would induce the integral
transformation of the solutions by applying the integral transformation on the
$q$-convolution, although it would be necessary to consider the subspace
$\mathcal{K}+\mathcal{L}\subset(\mathbb{C}^{m})^{N+1}$.

##  3 Linear $\boldsymbol{q}$-difference equation associated
to $\boldsymbol{q}$-Painlevé VI equation and $\boldsymbol{q}$-middle
convolution

We recall the linear $q$-difference equation

$\displaystyle Y(qx)=A(x)Y(x),$ (3.1)

which was discussed by Jimbo and Sakai [6] to obtain the $q$-Painlevé VI
equation by the connection preserving deformation.

We take the $2\times 2$ matrix $A(x)$ in equation (3.1) to be of the form

$\displaystyle A(x)=A_{0}(t)+A_{1}(t)x+A_{2}x^{2},\qquad
A_{2}=\begin{pmatrix}\chi_{1}&0\\\ 0&\chi_{2}\end{pmatrix}\\!,$ $\displaystyle
A_{0}(t)\mbox{ has eigenvalues }t\theta_{1},\ t\theta_{2},$ $\displaystyle\det
A(x)=\chi_{1}\chi_{2}(x-ta_{1})(x-ta_{2})(x-a_{3})(x-a_{4}).$ (3.2)

Then we have the following relation

$\displaystyle\chi_{1}\chi_{2}a_{1}a_{2}a_{3}a_{4}=\theta_{1}\theta_{2}.$
(3.3)

Note that the relations to the parameter of the $q$-Painlevé VI equation in
equation (1.3) are given by $b_{1}=a_{1}a_{2}/\theta_{1}$,
$b_{2}=a_{1}a_{2}/\theta_{2}$, $b_{3}=1/(q\chi_{1})$, $b_{4}=1/\chi_{2}$.

We need accessory parameters to determine uniquely the elements of the matrix
$A(x)$. Write

$\displaystyle A(x)=\begin{pmatrix}a_{11}(x)&a_{12}(x)\\\
a_{21}(x)&a_{22}(x)\end{pmatrix}\\!.$

Then $a_{12}(x)$ is a linear polynomial. We introduce the parameters $w$, $y$,
$z$ and impose the condition

$\displaystyle a_{12}(x)=\chi_{2}w(x-y),\qquad
a_{11}(x)|_{x=y}=(y-ta_{1})(y-ta_{2})/(qz).$ (3.4)

Then the elements of $A(x)$ are determined as

$\displaystyle
A(x)=\begin{pmatrix}\chi_{1}((x-y)(x-\alpha)+z_{1})&\chi_{2}w(x-y)\\\
\chi_{1}w^{-1}(\gamma
x+\delta)&\chi_{2}((x-y)(x-\beta)+z_{2})\end{pmatrix}\\!,$ (3.5)

where

$\displaystyle\alpha=\frac{1}{\chi_{1}-\chi_{2}}\big{[}y^{-1}((\theta_{1}+\theta_{2})t-\chi_{1}z_{1}-\chi_{2}z_{2})-\chi_{2}((a_{1}+a_{2})t+a_{3}+a_{4}-2y)\big{]},$
$\displaystyle\beta=\frac{1}{\chi_{1}-\chi_{2}}\big{[}{-}y^{-1}((\theta_{1}+\theta_{2})t-\chi_{1}z_{1}-\chi_{2}z_{2})+\chi_{1}((a_{1}+a_{2})t+a_{3}+a_{4}-2y)\big{]},$
$\displaystyle\gamma=z_{1}+z_{2}+(y+\alpha)(y+\beta)+(\alpha+\beta)y-a_{1}a_{2}t^{2}-(a_{1}+a_{2})(a_{3}+a_{4})t-a_{3}a_{4},$
$\displaystyle\delta=y^{-1}\big{(}a_{1}a_{2}a_{3}a_{4}t^{2}-(\alpha
y+z_{1})(\beta y+z_{2})\big{)}$

and

$\displaystyle z_{1}=\frac{(y-ta_{1})(y-ta_{2})}{q\chi_{1}z},\qquad
z_{2}=q\chi_{1}(y-a_{3})(y-a_{4})z.$

We consider the $q$-middle convolution for the $q$-difference equation

$\displaystyle Y(qx)=B(x)Y(x),\qquad
B(x)=\frac{A(x)}{c_{0}(x-ta_{1})(x-ta_{2})},$ (3.6)

where $c_{0}$ is a constant which will be fixed later. Note that, if
$\tilde{Y}(x)$ is a solution of equation (3.1) and the parameter $\mu$
satisfies $q^{\mu}=1/\big{(}c_{0}a_{1}a_{2}t^{2}\big{)}$, then the function

$\displaystyle
Y(x)=x^{\mu}(x/(ta_{1});q)_{\infty}(x/(ta_{2});q)_{\infty}\tilde{Y}(x)$

satisfies equation (3.6). Write

$\displaystyle
B(x)=B_{\infty}+\frac{B_{1}}{1-x/(ta_{1})}+\frac{B_{2}}{1-x/(ta_{2})}=\begin{pmatrix}b_{11}(x)&b_{12}(x)\\\
b_{21}(x)&b_{22}(x)\end{pmatrix}\\!.$ (3.7)

Then

$\displaystyle B_{\infty}=\frac{1}{c_{0}}\begin{pmatrix}\chi_{1}&0\\\
0&\chi_{2}\end{pmatrix}\\!,\qquad B_{2}=B_{1}|_{a_{1}\leftrightarrow a_{2}},$
(3.8) $\displaystyle
B_{1}=\frac{a_{2}}{t\theta_{1}(a_{1}-a_{2})}\begin{pmatrix}(y-ta_{1})b_{1}^{[1]}/(qyz(\chi_{1}-\chi_{2}))&w\chi_{2}(y-ta_{1})\\\\[4.30554pt]
-b_{1}^{[1]}b_{2}^{[1]}/\big{(}q^{2}wy^{2}z^{2}(\chi_{1}-\chi_{2})^{2}\big{)}&-\chi_{2}b_{2}^{[1]}/(qyz(\chi_{1}-\chi_{2}))\end{pmatrix}\\!,$

where

$\displaystyle b_{1}^{[1]}$
$\displaystyle=q^{2}\chi_{1}^{2}\chi_{2}(y-a_{3})(y-a_{4})z^{2}+(y-ta_{2})(\chi_{2}y-\chi_{1}ta_{1})$
$\displaystyle\phantom{=}-q\chi_{1}\big{\\{}2\chi_{2}y^{2}-(\chi_{1}ta_{1}+\chi_{2}ta_{2}+\chi_{2}a_{3}+\chi_{2}a_{4})y+t(\theta_{1}+\theta_{2})\big{\\}}z,$
$\displaystyle b_{2}^{[1]}$
$\displaystyle=q^{2}\chi_{1}(y-a_{3})(y-a_{4})(\chi_{1}y-\chi_{2}ta_{1})z^{2}+(y-ta_{1})^{2}(y-ta_{2})$
$\displaystyle\phantom{=}-q(y-ta_{1})\big{\\{}2\chi_{1}y^{2}-(\chi_{2}ta_{1}+\chi_{1}ta_{2}+\chi_{1}a_{3}+\chi_{1}a_{4})y+t(\theta_{1}+\theta_{2})\big{\\}}z.$

It is shown directly that $\det B_{1}=0$ and $\det B_{2}=0$. Set
$B_{0}=I_{2}-B_{\infty}-B_{1}-B_{2}(=I_{2}-B(0))$. Then the condition $\det
B_{0}=0$ is equivalent to $c_{0}=\theta_{1}/(ta_{1}a_{2})$ or
$c_{0}=\theta_{2}/(ta_{1}a_{2})$. We now impose the condition $\det B_{0}=0$.
For this purpose, we restrict to the case $c_{0}=\theta_{1}/(ta_{1}a_{2})$.
Note that the case $c_{0}=\theta_{2}/(ta_{1}a_{2})$ can be discussed by
replacing the parameters as $\theta_{1}\leftrightarrow\theta_{2}$. We
eliminate the parameter $\theta_{2}$ by equation (3.3). It follows from $\det
B_{0}=0$, $\det B_{1}=0$ and $\det B_{2}=0$ that there exists non-zero vectors
$\left(\begin{smallmatrix}v_{01}\\\ v_{02}\end{smallmatrix}\right)$,
$\left(\begin{smallmatrix}v_{11}\\\ v_{12}\end{smallmatrix}\right)$,
$\left(\begin{smallmatrix}v_{21}\\\ v_{22}\end{smallmatrix}\right)$ such that

$\displaystyle B_{0}\begin{pmatrix}v_{01}\\\
v_{02}\end{pmatrix}=\begin{pmatrix}0\\\ 0\end{pmatrix}\\!,\qquad
B_{1}\begin{pmatrix}v_{11}\\\ v_{12}\end{pmatrix}=\begin{pmatrix}0\\\
0\end{pmatrix}\\!,\qquad B_{2}\begin{pmatrix}v_{21}\\\
v_{22}\end{pmatrix}=\begin{pmatrix}0\\\ 0\end{pmatrix}\\!.$ (3.9)

We normalize the vectors by setting

$\displaystyle v_{01}=qwyz\theta_{1}(\chi_{1}-\chi_{2}),\quad\
v_{11}=qwyz\theta_{1}\chi_{2}(\chi_{1}-\chi_{2}),\quad\
v_{21}=qwyz\theta_{1}\chi_{2}(\chi_{1}-\chi_{2}).$ (3.10)

We now apply Definition 2.1. Namely we set

$\displaystyle F_{1}=\begin{pmatrix}{O}&O&{O}\\\
B_{0}&B_{1}-\big{(}1-q^{\lambda}\big{)}I_{2}&B_{2}\\\
{O}&O&{O}\end{pmatrix}\\!,\qquad F_{2}=\begin{pmatrix}{O}&O&{O}\\\
{O}&O&{O}\\\
B_{0}&B_{1}&B_{2}-\big{(}1-q^{\lambda}\big{)}I_{2}\end{pmatrix}\\!,$
$\displaystyle\widehat{F}=\begin{pmatrix}B_{0}&B_{1}&B_{2}\\\
B_{0}&B_{1}&B_{2}\\\ B_{0}&B_{1}&B_{2}\end{pmatrix}\\!,\qquad
F_{\infty}=I_{6}-\widehat{F}.$ (3.11)

The invariant subspaces $\mathcal{K}$ and $\mathcal{L}$ are described as

$\displaystyle\mathcal{K}=\mathcal{K}_{\mathcal{V}}=\begin{pmatrix}\ker
B_{0}\\\ \ker B_{1}\\\ \ker
B_{2}\end{pmatrix}\\!,\qquad\mathcal{L}=\mathcal{L}_{\mathcal{V}}(\lambda)=\ker\big{(}\widehat{F}-\big{(}1-q^{\lambda}\big{)}I_{6}\big{)}.$
(3.12)

Hence a basis of the space $\mathcal{K}$ is

$\displaystyle\left\\{\\!\begin{pmatrix}v_{01}\\\ v_{02}\\\ 0\\\ 0\\\ 0\\\
0\end{pmatrix}\\!,\begin{pmatrix}0\\\ 0\\\ v_{11}\\\ v_{12}\\\ 0\\\
0\end{pmatrix}\\!,\begin{pmatrix}0\\\ 0\\\ 0\\\ 0\\\ v_{21}\\\ v_{22}\\\
\end{pmatrix}\\!\right\\}\\!.$

If $q^{\lambda}=\chi_{2}ta_{1}a_{2}/\theta_{1}$ (resp.
$q^{\lambda}=\chi_{1}ta_{1}a_{2}/\theta_{1}$) then $\dim(\mathcal{L})=1$ and
the vector ${}^{t}(0,1,0,1,0,1)$ (resp. ${}^{t}(1,0,1,0,1,0)$) is a basis of
the space $\mathcal{L}$. Here we continue the discussion by setting
$q^{\lambda}=\chi_{2}ta_{1}a_{2}/\theta_{1}$.

We introduce the matrix $P$ by

$\displaystyle P=\begin{pmatrix}0&0&0&v_{01}&0&0\\\
g_{1}&g_{3}&1&v_{02}&0&0\\\ 0&0&0&0&v_{11}&0\\\ g_{2}&g_{4}&1&0&v_{12}&0\\\
0&0&0&0&0&v_{21}\\\ 0&0&1&0&0&v_{22}\\\
\end{pmatrix}\\!,\qquad\begin{array}[]{l}\displaystyle
g_{1}=qz\theta_{1}+ya_{2}-qyz\chi_{1}a_{2}-ta_{1}a_{2},\\\\[2.15277pt]
\displaystyle g_{2}=y(qz\chi_{1}-1)(a_{1}-a_{2}),\\\\[2.15277pt] \displaystyle
g_{3}=-a_{2}(y-ta_{1}),\\\\[2.15277pt] \displaystyle
g_{4}=y(a_{1}-a_{2}).\end{array}$ (3.17)

Then $\det
P=-q^{4}w^{3}y^{4}z^{4}\theta_{1}^{3}\chi_{2}^{2}(\chi_{1}-\chi_{2})^{3}(a_{1}-a_{2})(\chi_{1}ta_{1}a_{2}-\theta_{1})$,
and the matrix $P$ is invertible if $\det P\neq 0$. Set

$\displaystyle\widetilde{F}_{1}=P^{-1}F_{1}P,\qquad\widetilde{F}_{2}=P^{-1}F_{2}P,\qquad\widetilde{F}_{\infty}=P^{-1}{F}_{\infty}P.$

Then it follows from the invariance of the space $\mathcal{K}+\mathcal{L}$
that the $i.j$ elements of these matrices for $i\in\\{1,2\\}$ and
$j\in\\{3,4,5,6\\}$ are equal to zero. Thus, they admit the following
expression:

$\displaystyle\widetilde{F}_{1}=\begin{pmatrix}\overline{F}_{1}&O\\\
&*\end{pmatrix}\\!,\qquad\widetilde{F}_{2}=\begin{pmatrix}\overline{F}_{2}&O\\\
*&*\end{pmatrix},\qquad\widetilde{F}_{\infty}=\begin{pmatrix}\overline{F}_{\infty}&O\\\
&*\end{pmatrix}\\!,$

where $\overline{F}_{1}$, $\overline{F}_{2}$, $\overline{F}_{\infty}$ are
$2\times 2$ matrices. The matrix $\overline{F}_{\infty}$ is diagonal, which
follows from the choice of the parameters $g_{1},\dots,g_{4}$. We can restrict
the $q$-difference equation

$\displaystyle\check{Y}(qx)=\bigg{(}\widetilde{F}_{\infty}+\frac{\widetilde{F}_{1}}{1-x/(ta_{1})}+\frac{\widetilde{F}_{2}}{1-x/(ta_{2})}\bigg{)}\check{Y}(x)$
(3.18)

of size $6$ to that of size $2$ by choosing the first two components and we
write

$\displaystyle\overline{Y}(qx)=\overline{F}(x)\overline{Y}(x),\qquad\overline{F}(x)=\overline{F}_{\infty}+\frac{\overline{F}_{1}}{1-x/(ta_{1})}+\frac{\overline{F}_{2}}{1-x/(ta_{2})}.$
(3.19)

Then

$\displaystyle\overline{F}_{\infty}=\begin{pmatrix}\chi_{1}ta_{1}a_{2}/\theta_{1}&0\\\
0&1\end{pmatrix}\\!,\qquad\overline{F}_{2}=\overline{F}_{1}|_{a_{1}\leftrightarrow
a_{2}},$
$\displaystyle\overline{F}_{1}=\frac{a_{1}}{qyz\theta_{1}(a_{1}-a_{2})(\theta_{1}-\chi_{1}ta_{1}a_{2})}\begin{pmatrix}-(y-ta_{1})f_{1}^{[1]}a_{2}^{2}&-f_{2}^{[1]}(y-ta_{1})a_{2}\\\\[2.15277pt]
f_{1}^{[1]}f_{3}^{[1]}a_{2}&f_{2}^{[1]}f_{3}^{[1]}\end{pmatrix}\\!,$ (3.20)

where

$\displaystyle
f_{1}^{[1]}=q^{2}\chi_{1}^{2}\chi_{2}(y-a_{3})(y-a_{4})z^{2}+(y-ta_{2})(y\chi_{2}-\chi_{1}ta_{1})$
$\displaystyle\phantom{f_{1}^{[1]}=}{}-q\chi_{1}\\{2\chi_{2}y^{2}-(\chi_{1}ta_{1}+\chi_{2}ta_{2}+\chi_{2}a_{3}+\chi_{2}a_{4})y+t(\theta_{1}+\theta_{2})\\}z,$
$\displaystyle
f_{2}^{[1]}=q\chi_{1}\chi_{2}a_{2}(y-a_{3})(y-a_{4})z-(y-ta_{2})(a_{2}\chi_{2}y-\theta_{1}),$
$\displaystyle f_{3}^{[1]}=q(ya_{2}\chi_{1}-\theta_{1})z-a_{2}(y-ta_{1}).$

It is expected that the $q$-middle convolution induces an integral
transformation. Let $Y(x)$ be a solution to $Y(qx)=B(x)Y(x)$ in equation (3.6)
and write

$\displaystyle Y(x)=\begin{pmatrix}y_{1}(x)\\\ y_{2}(x)\end{pmatrix}\\!.$

It follows from Theorem 2.2 that the $q$-difference equation

$\displaystyle\widehat{Y}(qx)=F(x)\widehat{Y}(x),\qquad
F(x)=F_{\infty}+\frac{F_{1}}{1-x/(ta_{1})}+\frac{F_{2}}{1-x/(ta_{2})}$

has a solution written as

$\displaystyle\widehat{Y}(x)=\begin{pmatrix}\widehat{Y}_{0}(x)\\\
\widehat{Y}_{1}(x)\\\
\widehat{Y}_{2}(x)\end{pmatrix}=\begin{pmatrix}\int^{\xi\infty}_{0}s^{-1}P_{\lambda}(x,s)y_{1}(s)\,{\rm
d}_{q}s\\\\[5.0pt] \int^{\xi\infty}_{0}s^{-1}P_{\lambda}(x,s)y_{2}(s)\,{\rm
d}_{q}s\\\\[5.0pt]
\int^{\xi\infty}_{0}(s-ta_{1})^{-1}P_{\lambda}(x,s)y_{1}(s)\,{\rm
d}_{q}s\\\\[5.0pt]
\int^{\xi\infty}_{0}(s-ta_{1})^{-1}P_{\lambda}(x,s)y_{2}(s)\,{\rm
d}_{q}s\\\\[5.0pt]
\int^{\xi\infty}_{0}(s-ta_{2})^{-1}P_{\lambda}(x,s)y_{1}(s)\,{\rm
d}_{q}s\\\\[5.0pt]
\int^{\xi\infty}_{0}(s-ta_{2})^{-1}P_{\lambda}(x,s)y_{2}(s)\,{\rm
d}_{q}s\end{pmatrix}\\!.$ (3.21)

Set $\check{Y}(x)=P^{-1}\widehat{Y}(x)$. Then it satisfies equation (3.18).
Write

$\displaystyle\check{Y}(x)=\begin{pmatrix}\check{y}_{1}(x)\\\
\check{y}_{2}(x)\\\ \vdots\\\ \check{y}_{6}(x)\end{pmatrix},\qquad
P^{-1}=\begin{pmatrix}p_{11}&p_{12}&\cdots&p_{16}\\\
p_{21}&p_{22}&\cdots&p_{26}\\\ \vdots&\vdots&\ddots&\vdots\\\
p_{61}&p_{62}&\cdots&p_{66}\end{pmatrix}\\!.$

Then the function
$\overline{Y}(x)=\left(\\!\begin{smallmatrix}\check{y}_{1}(x)\\\
\check{y}_{2}(x)\end{smallmatrix}\\!\right)$ satisfies equation (3.19). On the
other hand, it follows from equation (3.21) that

$\displaystyle\check{y}_{1}(x)$
$\displaystyle=\int^{\xi\infty}_{0}\bigg{\\{}\bigg{(}\frac{p_{11}}{s}+\frac{p_{13}}{s-ta_{1}}+\frac{p_{15}}{s-ta_{2}}\bigg{)}y_{1}(s)+\bigg{(}\frac{p_{12}}{s}+\frac{p_{14}}{s-ta_{1}}+\frac{p_{16}}{s-ta_{2}}\bigg{)}y_{2}(s)\bigg{\\}}$
$\displaystyle\hphantom{=}\times P_{\lambda}(x,s)\,{\rm d}_{q}s.$

By a straightforward calculation, we have

$\displaystyle\frac{p_{12}}{s}+\frac{p_{14}}{s-ta_{1}}+\frac{p_{16}}{s-ta_{2}}=\frac{t\theta_{1}}{qwyz\chi_{2}(\chi_{1}ta_{1}a_{2}-\theta_{1})s}b_{12}(s),$
$\displaystyle\frac{p_{11}}{s}+\frac{p_{13}}{s-ta_{1}}+\frac{p_{15}}{s-ta_{2}}=\frac{t\theta_{1}}{qwyz\chi_{2}(\chi_{1}ta_{1}a_{2}-\theta_{1})s}(b_{11}(s)-1),$

where $b_{12}(s)$ and $b_{11}(s)$ are elements of the matrix $B(s)$ in
equation (3.7). Therefore

$\displaystyle\bigg{(}\frac{p_{11}}{s}+\frac{p_{13}}{s-ta_{1}}+\frac{p_{15}}{s-ta_{2}}\bigg{)}y_{1}(s)+\bigg{(}\frac{p_{12}}{s}+\frac{p_{14}}{s-ta_{1}}+\frac{p_{16}}{s-ta_{2}}\bigg{)}y_{2}(s)$
$\displaystyle\qquad{}=\frac{t\theta_{1}}{qwyz\chi_{2}(\chi_{1}ta_{1}a_{2}-\theta_{1})s}\\{-y_{1}(s)+b_{11}(s)y_{1}(s)+b_{12}(s)y_{2}(s)\\}.$

Hence it follows from $y_{1}(qs)=b_{11}(s)y_{1}(s)+b_{12}(s)y_{2}(s)$ that

$\displaystyle\check{y}_{1}(x)=\frac{t\theta_{1}}{qwyz\chi_{2}(\chi_{1}ta_{1}a_{2}-\theta_{1})}\int^{\xi\infty}_{0}\frac{y_{1}(qs)-y_{1}(s)}{s}P_{\lambda}(x,s)\,{\rm
d}_{q}s.$

We are going to obtain the integral representation without using $y_{1}(qs)$.
It follows from the definitions of the Jackson integral and the function
$P_{\lambda}(x,s)$ that

$\displaystyle\int^{\xi\infty}_{0}y_{1}(qs)P_{\lambda}(x,s)\,\frac{{\rm
d}_{q}s}{s}=\int^{\xi\infty}_{0}y_{1}(s)P_{\lambda}(x,s/q)\frac{{\rm
d}_{q}s}{s}=\int^{\xi\infty}_{0}y_{1}(s)\,\frac{x-q^{\lambda}s}{x-s}P_{\lambda}(x,s)\,\frac{{\rm
d}_{q}s}{s}.$

See [12] for details. Hence

$\displaystyle\int^{\xi\infty}_{0}\frac{y_{1}(qs)-y_{1}(s)}{s}P_{\lambda}(x,s)\,{\rm
d}_{q}s$
$\displaystyle=\int^{\xi\infty}_{0}\frac{1}{s}\bigg{(}\frac{x-q^{\lambda}s}{x-s}-1\bigg{)}y_{1}(s)P_{\lambda}(x,s)\,{\rm
d}_{q}s$
$\displaystyle=(1-q^{\lambda})\int^{\xi\infty}_{0}\frac{y_{1}(s)}{x-s}P_{\lambda}(x,s)\,{\rm
d}_{q}s.$

We recall that $q^{\lambda}=\chi_{2}ta_{1}a_{2}/\theta_{1}$. Thus, we obtain

$\displaystyle\check{y}_{1}(x)=\frac{t(\chi_{2}ta_{1}a_{2}-\theta_{1})}{qwyz\chi_{2}(\chi_{1}ta_{1}a_{2}-\theta_{1})}\int^{\xi\infty}_{0}\frac{y_{1}(s)}{s-x}P_{\lambda}(x,s)\,{\rm
d}_{q}s.$ (3.22)

We also calculate $\check{y}_{2}(x)$. It follows from equation (3.21) that

$\displaystyle\check{y}_{2}(x)$
$\displaystyle=\int^{\xi\infty}_{0}\bigg{\\{}\bigg{(}\frac{p_{21}}{s}+\frac{p_{23}}{s-ta_{1}}+\frac{p_{25}}{s-ta_{2}}\bigg{)}y_{1}(s)+\bigg{(}\frac{p_{22}}{s}+\frac{p_{24}}{s-ta_{1}}+\frac{p_{26}}{s-ta_{2}}\bigg{)}y_{2}(s)\bigg{\\}}$
$\displaystyle\hphantom{=}\times P_{\lambda}(x,s)\,{\rm d}_{q}s.$

By a straightforward calculation, we have

$\displaystyle\frac{p_{22}}{s}+\frac{p_{24}}{s-ta_{1}}+\frac{p_{26}}{s-ta_{2}}=\frac{\theta_{1}\\{(ta_{1}a_{2}-qz\theta_{1})s+yta_{1}a_{2}(qz\chi_{1}-1)\\}}{qwyz\chi_{2}a_{1}a_{2}(\chi_{1}ta_{1}a_{2}-\theta_{1})(s-y)s}b_{12}(s),$
$\displaystyle\frac{p_{21}}{s}+\frac{p_{23}}{s-ta_{1}}+\frac{p_{25}}{s-ta_{2}}=\frac{\theta_{1}\\{(ta_{1}a_{2}-qz\theta_{1})s+yta_{1}a_{2}(qz\chi_{1}-1)\\}}{qwyz\chi_{2}a_{1}a_{2}(\chi_{1}ta_{1}a_{2}-\theta_{1})(s-y)s}b_{11}(s)$
$\displaystyle\hphantom{\frac{p_{21}}{s}+\frac{p_{23}}{s-ta_{1}}+\frac{p_{25}}{s-ta_{2}}=}{}-\frac{t\\{(ta_{1}a_{2}-qz\theta_{1})\chi_{1}s+y\theta_{1}(qz\chi_{1}-1)\\}}{qwyz\chi_{2}(\chi_{1}ta_{1}a_{2}-\theta_{1})(s-y)s}.$

Therefore it follows from $y_{1}(qs)=b_{11}(s)y_{1}(s)+b_{12}(s)y_{2}(s)$ that

$\displaystyle\bigg{(}\frac{p_{21}}{s}+\frac{p_{23}}{s-ta_{1}}+\frac{p_{25}}{s-ta_{2}}\bigg{)}y_{1}(s)+\bigg{(}\frac{p_{22}}{s}+\frac{p_{24}}{s-ta_{1}}+\frac{p_{26}}{s-ta_{2}}\bigg{)}y_{2}(s)$
$\displaystyle\qquad{}=\frac{\theta_{1}\\{(ta_{1}a_{2}-qz\theta_{1})s+yta_{1}a_{2}(qz\chi_{1}-1)\\}}{qwyz\chi_{2}a_{1}a_{2}(\chi_{1}ta_{1}a_{2}-\theta_{1})(s-y)s}y_{1}(qs)$
$\displaystyle\qquad\phantom{=}{}-\frac{t\\{(ta_{1}a_{2}-qz\theta_{1})\chi_{1}s+y\theta_{1}(qz\chi_{1}-1)\\}}{qwyz\chi_{2}(\chi_{1}ta_{1}a_{2}-\theta_{1})(s-y)s}y_{1}(s).$

We obtain that

$\displaystyle\int^{\xi\infty}_{0}\frac{(ta_{1}a_{2}-qz\theta_{1})s+ta_{1}a_{2}y(qz\chi_{1}-1)}{s-y}y_{1}(qs)P_{\lambda}(x,s)\,\frac{{\rm
d}_{q}s}{s}$
$\displaystyle\qquad{}=\int^{\xi\infty}_{0}\frac{(ta_{1}a_{2}-qz\theta_{1})s+qta_{1}a_{2}y(qz\chi_{1}-1)}{s-qy}y_{1}(s)P_{\lambda}(x,s/q)\frac{{\rm
d}_{q}s}{s}$
$\displaystyle\qquad{}=\int^{\xi\infty}_{0}\frac{(ta_{1}a_{2}-qz\theta_{1})s+qta_{1}a_{2}y(qz\chi_{1}-1)}{s-qy}\bigg{(}1+\frac{(1-q^{\lambda})s}{x-s}\bigg{)}y_{1}(s)P_{\lambda}(x,s)\,\frac{{\rm
d}_{q}s}{s}.$

Hence

$\displaystyle\check{y}_{2}(x)$
$\displaystyle=\frac{1}{qwyz\chi_{2}a_{1}a_{2}}\int^{\xi\infty}_{0}\bigg{\\{}\frac{qz\theta_{1}}{s-qy}-\frac{ta_{1}a_{2}}{s-y}+\bigg{(}\frac{qz\theta_{1}-ta_{1}a_{2}}{\chi_{1}ta_{1}a_{2}-\theta_{1}}-\frac{q^{2}yz}{s-qy}\bigg{)}\frac{\chi_{2}ta_{1}a_{2}-\theta_{1}}{x-s}\bigg{\\}}$
$\displaystyle\hphantom{=}\times y_{1}(s)P_{\lambda}(x,s)\,{\rm d}_{q}s.$
(3.23)

In summary, we obtain the following theorem by the $q$-middle convolution.

######  Theorem 3.1.

Let $Y(x)$ be a solution to

$\displaystyle
Y(qx)=\bigg{(}B_{\infty}+\frac{B_{1}}{1-x/(ta_{1})}+\frac{B_{2}}{1-x/(ta_{2})}\bigg{)}Y(x),\qquad
Y(x)=\begin{pmatrix}y_{1}(x)\\\ y_{2}(x)\end{pmatrix}\\!,$ (3.24)

where $B_{\infty}$, $B_{1}$ and $B_{2}$ are defined in equation (3.8). The
function

$\displaystyle\overline{Y}(x)=\begin{pmatrix}\check{y}_{1}(x)\\\
\check{y}_{2}(x)\end{pmatrix}$

defined by equations (3.22) and (3.23) formally satisfies

$\displaystyle\overline{Y}(qx)=\bigg{(}\overline{F}_{\infty}+\frac{\overline{F}_{1}}{1-x/(ta_{1})}+\frac{\overline{F}_{2}}{1-x/(ta_{2})}\bigg{)}\overline{Y}(x),$
(3.25)

where $\overline{F}_{\infty}$, $\overline{F}_{1}$ and $\overline{F}_{2}$ are
defined in equation (3.20).

Thus, we obtain the correspondence of the systems of linear $q$-difference
equations associated with the $q$-Painlevé VI equation by the $q$-middle
convolution. To give the correspondence of the parameters by the $q$-middle
convolution in the form of the equation
$Y(qx)=\big{\\{}A_{0}(t)+A_{1}(t)x\allowbreak+A_{2}x^{2}\big{\\}}Y(x)$ in
equation (3.2), we need to transform equation (3.25).

Let $\widetilde{c}$ be a non-zero constant which will be fixed later. Set

$\displaystyle\widetilde{A}(x)=\widetilde{c}(x-ta_{1})(x-ta_{2})\bigg{(}\overline{F}_{\infty}+\frac{\overline{F}_{1}}{1-x/(ta_{1})}+\frac{\overline{F}_{2}}{1-x/(ta_{2})}\bigg{)}$
(3.26)

and write
$\widetilde{A}(x)=\widetilde{A}(x,t)=\widetilde{A}_{0}(t)+\widetilde{A}_{1}(t)x+\widetilde{A}_{2}x^{2}$.
Then we have

$\displaystyle\widetilde{A}_{2}=\begin{pmatrix}\widetilde{c}\chi_{1}ta_{1}a_{2}/\theta_{1}&0\\\
0&\widetilde{c}\end{pmatrix}\\!,$ $\displaystyle\widetilde{A}_{0}(t)\mbox{ has
the eigenvalues
}\frac{\widetilde{c}\chi_{2}t^{3}a_{1}^{2}a_{2}^{2}}{\theta_{1}}\mbox{ and
}\frac{\widetilde{c}t^{2}a_{1}a_{2}\theta_{2}}{\theta_{1}},$
$\displaystyle\det\widetilde{A}(x,t)=\frac{\widetilde{c}^{2}\chi_{1}ta_{1}a_{2}}{\theta_{1}}(x-ta_{1})(x-ta_{2})\bigg{(}x-\frac{\chi_{2}ta_{1}a_{2}a_{3}}{\theta_{1}}\bigg{)}\bigg{(}x-\frac{\chi_{2}ta_{1}a_{2}a_{4}}{\theta_{1}}\bigg{)}.$

Hence the action of the $q$-middle convolution to the parameters is described
as

$\displaystyle\chi_{1}\rightarrow\frac{\widetilde{c}\chi_{1}ta_{1}a_{2}}{\theta_{1}},\qquad\chi_{2}\rightarrow\widetilde{c},\qquad\\{t\theta_{1},t\theta_{2}\\}\rightarrow\bigg{\\{}\frac{\widetilde{c}t^{2}a_{1}a_{2}\theta_{2}}{\theta_{1}},\frac{\widetilde{c}\chi_{2}t^{3}a_{1}^{2}a_{2}^{2}}{\theta_{1}}\bigg{\\}},$
$\displaystyle\\{a_{1},a_{2}\\}\rightarrow\\{a_{1},a_{2}\\},\qquad\\{a_{3},a_{4}\\}\rightarrow\bigg{\\{}\frac{\chi_{2}ta_{1}a_{2}a_{3}}{\theta_{1}},\frac{\chi_{2}ta_{1}a_{2}a_{4}}{\theta_{1}}\bigg{\\}}.$
(3.27)

We investigate the action to the parameters $y$ and $z$. Recall that $y$ and
$z$ are determined by equation (3.4). We denote the images of $y$ and $z$ by
$\widetilde{y}$ and $\widetilde{z}$. Let $\widetilde{a}_{11}(x)$ (resp.
$\widetilde{a}_{12}(x)$) be the upper left entry (resp. the upper right entry)
of the matrix $\widetilde{A}(x)$. Then the value $\widetilde{y}$ is the zero
of the linear function $\widetilde{a}_{12}(x)$, and we have

$\displaystyle\widetilde{y}$
$\displaystyle=\frac{\chi_{2}ta_{1}a_{2}\\{q\chi_{1}(y-a_{3})(y-a_{4})z-(y-ta_{1})(y-ta_{2})\\}}{q\chi_{1}\chi_{2}ta_{1}a_{2}(y-a_{3})(y-a_{4})z-\theta_{1}(y-ta_{1})(y-ta_{2})}\,y$
$\displaystyle=\frac{qz\dfrac{(y-a_{3})(y-a_{4})}{(y-ta_{1})(y-ta_{2})}-\dfrac{1}{\chi_{1}}}{qz\dfrac{(y-a_{3})(y-a_{4})}{(y-ta_{1})(y-ta_{2})}-\dfrac{\theta_{1}}{\chi_{1}\chi_{2}ta_{1}a_{2}}}\,y.$
(3.28)

The value $\widetilde{z}$ satisfies
$\widetilde{a}_{11}(x)|_{x=\widetilde{y}}=(\widetilde{y}-ta_{1})(\widetilde{y}-ta_{2})/(q\widetilde{z})$.
Since

$\displaystyle\widetilde{a}_{11}(x)|_{x=\widetilde{y}}=\frac{\widetilde{c}}{q\chi_{2}z}\frac{(y-ta_{1})(y-ta_{2})}{(y-a_{3})(y-a_{4})}\bigg{(}\widetilde{y}-\frac{\chi_{2}ta_{1}a_{2}a_{3}}{\theta_{1}}\bigg{)}\bigg{(}\widetilde{y}-\frac{\chi_{2}ta_{1}a_{2}a_{4}}{\theta_{1}}\bigg{)},$

we have

$\displaystyle\widetilde{z}=z\frac{\chi_{2}}{\widetilde{c}}\frac{(y-a_{3})(y-a_{4})}{(y-ta_{1})(y-ta_{2})}\frac{(\widetilde{y}-ta_{1})(\widetilde{y}-ta_{2})}{(\widetilde{y}-\chi_{2}ta_{1}a_{2}a_{3}/\theta_{1})(\widetilde{y}-\chi_{2}ta_{1}a_{2}a_{3}/\theta_{1})}.$
(3.29)

##  4 $\boldsymbol{q}$-middle convolution and Weyl group symmetry
of $\boldsymbol{q}$-Painlevé VI equation

We investigate the transformation of the parameters induced by the $q$-middle
convolution in terms of the Weyl group symmetry associated with the
$q$-Painlevé VI equation. Kajiwara, Noumi and Yamada gave a survey on discrete
Painlevé equations in [7]. They presented a list of the Weyl group symmetry
and a Lax pair for each discrete Painlevé equation. The $q$-Painlevé VI
equation by Jimbo and Sakai [6] corresponds to the equation
$q$-$P\big{(}D^{(1)}_{5}\big{)}$ in [7]. In this section, we make a
correspondence between the parameters of the $q$-Painlevé VI equation and
those of the equation $q$-$P\big{(}D^{(1)}_{5}\big{)}$.

The equation $q$-$P\big{(}D^{(1)}_{5}\big{)}$ was obtained by the
compatibility condition of the Lax pair $L_{1}$ and $L_{2}$ in [7]. The
operator $L_{1}$ is defined by

$\displaystyle L_{1}y(x)$
$\displaystyle=\bigg{\\{}\frac{x(g\nu_{1}-1)(g\nu_{2}-1)}{qg}-\frac{\nu_{1}\nu_{2}\nu_{3}\nu_{4}(g-\nu_{5}/\kappa_{2})(g-\nu_{6}/\kappa_{2})}{fg}\bigg{\\}}y(x)$
$\displaystyle\hphantom{=}+\frac{\nu_{1}\nu_{2}(x-q\nu_{3})(x-q\nu_{4})}{q(qf-x)}(gy(x)-y(x/q))$
$\displaystyle\hphantom{=}+\frac{(x-\kappa_{1}/\nu_{7})(x-\kappa_{1}/\nu_{8})}{q(f-x)}\bigg{(}\dfrac{1}{g}y(x)-y(qx)\bigg{)},$
(4.1)

which is independent from the time evolution. Here the parameters are
constrained by the relation

$\displaystyle\kappa_{1}^{2}\kappa_{2}^{2}=q\nu_{1}\nu_{2}\nu_{3}\nu_{4}\nu_{5}\nu_{6}\nu_{7}\nu_{8}.$
(4.2)

In this paper, we do not use the operator $L_{2}$, which contains the
operation of the time evolution.

The correspondence between the parameters of the $q$-Painlevé VI equation and
those of the equation $q$-$P\big{(}D^{(1)}_{5}\big{)}$ was made by considering
the linear $q$-difference equation $Y(qx)=A(x)Y(x)$ in equation (3.1) and
$L_{1}y(x)=0$ in equation (4.1).

For the system of $q$-difference equation

$\displaystyle Y(qx)=A(x)Y(x),\qquad
A(x)=\begin{pmatrix}a_{11}(x)&a_{12}(x)\\\
a_{21}(x)&a_{22}(x)\end{pmatrix}\\!,\qquad Y(x)=\begin{pmatrix}y_{1}(x)\\\
y_{2}(x)\end{pmatrix}\\!,$

we calculate the $q$-difference equation for $y_{1}(x)$ by eliminating
$y_{2}$. The system of the $q$-difference equation is written as

$\displaystyle y_{1}(qx)=a_{11}(x)y_{1}(x)+a_{12}(x)y_{2}(x),$ $\displaystyle
y_{2}(qx)=a_{21}(x)y_{1}(x)+a_{22}(x)y_{2}(x).$

We substitute $y_{2}(x)\\!=a_{21}(x/q)y_{1}(x/q)+a_{22}(x/q)y_{2}(x/q)$ into
$y_{1}(qx)\\!=a_{11}(x)y_{1}(x)+a_{12}(x)y_{2}(x)$. Then we have

$\displaystyle
y_{1}(qx)=a_{11}(x)y_{1}(x)+a_{12}(x)a_{21}(x/q)y_{1}(x/q)+a_{12}(x)a_{22}(x/q)y_{2}(x/q).$

We elimilate $y_{2}(x/q)$ by the relation
$y_{2}(x/q)=\\{y_{1}(x)-a_{11}(x/q)y_{1}(x/q)\\}/a_{12}(x/q)$. Thus,

$\displaystyle\frac{y_{1}(qx)}{a_{12}(x)}-\bigg{\\{}\frac{a_{11}(x)}{a_{12}(x)}+\frac{a_{22}(x/q)}{a_{12}(x/q)}\bigg{\\}}y_{1}(x)+\frac{a_{11}(x/q)a_{22}(x/q)-a_{12}(x/q)a_{21}(x/q)}{a_{12}(x/q)}y_{1}(x/q)=0.$

We restrict it to the case that the matrix elements are fixed to equation
(3.5). Then it follows from equations (3.2) and (3.5) that

$\displaystyle\frac{y_{1}(qx)}{x-y}-\bigg{\\{}\frac{a_{11}(x)}{x-y}+\frac{a_{22}(x/q)}{x/q-y}\bigg{\\}}y_{1}(x)$
$\displaystyle\qquad{}+\frac{\chi_{1}\chi_{2}(x/q-ta_{1})(x/q-ta_{2})(x/q-a_{3})(x/q-a_{4})}{x/q-y}y_{1}(x/q)=0.$
(4.3)

Let $\phi(x)$ be the function such that
$\phi(qx)=d_{0}(x-ta_{1})(x-ta_{2})\phi(x)$. Set $u(x)=y_{1}(x)/\phi(x)$. Then
it follows from equation (4.3) that the function $u(x)$ satisfies

$\displaystyle\frac{(x-ta_{1})(x-ta_{2})}{x-y}d_{0}u(qx)-\bigg{\\{}\frac{a_{11}(x)}{x-y}+\frac{a_{22}(x/q)}{x/q-y}\bigg{\\}}u(x)$
$\displaystyle\qquad{}+\frac{\chi_{1}\chi_{2}(x/q-a_{3})(x/q-a_{4})}{x/q-y}\frac{1}{d_{0}}u(x/q)=0.$

We rewrite it by using equation (3.5) as

$\displaystyle\frac{(x-ta_{1})(x-ta_{2})}{x-y}\bigg{\\{}d_{0}u(qx)-\frac{u(x)}{qz}\bigg{\\}}+\frac{\chi_{1}\chi_{2}(x/q-a_{3})(x/q-a_{4})}{x/q-y}\bigg{\\{}\frac{1}{d_{0}}u(x/q)-qzu(x)\bigg{\\}}$
$\displaystyle\qquad{}-\bigg{\\{}\chi_{1}(x-\alpha)+\frac{\chi_{1}z_{1}}{x-y}+\chi_{2}(x/q-\beta)+\frac{\chi_{2}z_{2}}{x/q-y}-\frac{(x-ta_{1})(x-ta_{2})}{qz(x-y)}$
$\displaystyle\qquad\hphantom{-\bigg{\\{}}{}-\frac{qz\chi_{1}\chi_{2}(x/q-a_{3})(x/q-a_{4})}{x/q-y}\bigg{\\}}u(x)=0.$
(4.4)

It is seen that the poles $x=y$ and $x=qy$ are cancelled on the coefficient of
$u(x)$. By a straightforward calculation (see [12] for details), equation
(4.4) is written as

$\displaystyle\bigg{\\{}\frac{x(qz\chi_{1}-1)(z\chi_{2}-1)}{q^{2}z}-\chi_{1}\chi_{2}a_{3}a_{4}\frac{(qz-
ta_{1}a_{2}/\theta_{1})(qz-ta_{1}a_{2}/\theta_{2})}{q^{2}yz}\bigg{\\}}u(x)$
$\displaystyle\qquad{}+\frac{\chi_{1}\chi_{2}(a_{3}-x/q)(a_{4}-x/q)}{qy-x}\bigg{\\{}qzu(x)-\frac{1}{d_{0}}u(x/q)\bigg{\\}}$
$\displaystyle\qquad{}+\frac{(x-ta_{1})(x-ta_{2})}{q(y-x)}\bigg{\\{}\frac{u(x)}{qz}-d_{0}u(qx)\bigg{\\}}=0.$
(4.5)

We compare equation (4.5) for the case $d_{0}=1$ with equation (4.1). Then we
obtain the following correspondence:

$\displaystyle qz=g,\qquad
y=f,\qquad\chi_{1}=\nu_{1},\qquad\chi_{2}=q\nu_{2},\qquad\frac{ta_{1}a_{2}}{\theta_{1}}=\frac{\nu_{5}}{\kappa_{2}},\qquad\frac{ta_{1}a_{2}}{\theta_{2}}=\frac{\nu_{6}}{\kappa_{2}},$
$\displaystyle a_{3}=\nu_{3},\qquad a_{4}=\nu_{4},\qquad
ta_{1}=\frac{\kappa_{1}}{\nu_{7}},\qquad ta_{2}=\frac{\kappa_{1}}{\nu_{8}}.$
(4.6)

On the relation of the parameters, the condition
$\chi_{1}\chi_{2}a_{1}a_{2}a_{3}a_{4}=\theta_{1}\theta_{2}$ is equivalent to
equation (4.2). Thus, we obtained a correspondence between the parameters of
the $q$-Painlevé VI equation in [6] and those of the equation
$q$-$P\big{(}D^{(1)}_{5}\big{)}$ in [7].

Sakai [9] established that each discrete Painlevé equation has the symmetry in
terms of the affine Weyl group, and the description of the symmetry was
reviewed explicitly by Kajiwara, Noumi and Yamada in [7]. The $q$-Painlevé VI
equation has the symmetry of the affine Weyl group of the type $D^{(1)}_{5}$.
We describe the action of the operators $s_{0},\dots,s_{5}$ for the parameters
$(\kappa_{1},\kappa_{2},\nu_{1},\dots,\nu_{8})\in(\mathbb{C}^{\times})^{10}$
and $(f,g)\in\mathbb{P}^{1}\times\mathbb{P}^{1}$ as follows

$\displaystyle s_{0}\colon\ \nu_{7}\leftrightarrow\nu_{8},\qquad s_{1}\colon\
\nu_{3}\leftrightarrow\nu_{4},\qquad s_{4}\colon\
\nu_{1}\leftrightarrow\nu_{2},\qquad s_{5}\colon\
\nu_{5}\leftrightarrow\nu_{6},$ $\displaystyle s_{2}\colon\
\nu_{3}\rightarrow\frac{k_{1}}{\nu_{7}},\qquad\nu_{7}\rightarrow\frac{k_{1}}{\nu_{3}},\qquad
k_{2}\rightarrow\frac{k_{1}k_{2}}{\nu_{3}\nu_{7}},\qquad g\rightarrow
g\,\frac{f-\nu_{3}}{f-k_{1}/\nu_{7}},$ $\displaystyle s_{3}\colon\
\nu_{1}\rightarrow\frac{k_{2}}{\nu_{5}},\qquad\nu_{5}\rightarrow\frac{k_{2}}{\nu_{1}},\qquad
k_{1}\rightarrow\frac{k_{1}k_{2}}{\nu_{1}\nu_{5}},\qquad f\rightarrow
f\,\frac{g-1/\nu_{1}}{g-\nu_{5}/k_{2}}.$ (4.7)

The omitted variables are invariant by the action, i.e., $s_{2}(f)=f$. Then we
can confirm that these operations satisfy the relations of the Weyl group
$W\big{(}D^{(1)}_{5}\big{)}$ whose Dynkin diagram is as follows

$0$$1$$4$$5$$2$$3$

On the other hand, the $q$-middle convolution induces the transformation of
the parameters of the $q$-Painlevé VI equation given in equation (3.27),
although there was an arbitrary parameter $\widetilde{c}$. We describe it in
terms of the Weyl group action by specializing the parameter $\widetilde{c}$
in equation (3.26).

######  Proposition 4.1.

We specify the parameter $\widetilde{c}$ in equation (3.26) by setting
$\widetilde{c}=\chi_{2}$. Then the transformation of the parameters of the
$q$-Painlevé VI equation which is induced by the $q$-middle convolution
coincides with the action

$\displaystyle s_{5}s_{2}s_{1}s_{0}s_{2}s_{3}s_{2}s_{0}s_{1}s_{2}$ (4.8)

by the generators of $W\big{(}D^{(1)}_{5}\big{)}$, and it is written as

$\displaystyle\nu_{1}\rightarrow
q\,\frac{\nu_{1}\nu_{2}\nu_{5}}{\kappa_{2}},\qquad\nu_{2}\rightarrow\nu_{2},\qquad\frac{\kappa_{1}}{\nu_{7}}\rightarrow\frac{\kappa_{1}}{\nu_{7}},\qquad\frac{\kappa_{1}}{\nu_{8}}\rightarrow\frac{\kappa_{1}}{\nu_{8}},$
$\displaystyle\nu_{3}\rightarrow
q\,\frac{\nu_{2}\nu_{3}\nu_{5}}{\kappa_{2}},\qquad\nu_{4}\rightarrow
q\,\frac{\nu_{2}\nu_{4}\nu_{5}}{\kappa_{2}},\qquad\frac{\kappa_{2}}{\nu_{5}}\rightarrow
q\,\frac{\nu_{2}\nu_{5}}{\nu_{6}},\qquad\frac{\kappa_{2}}{\nu_{6}}\rightarrow
q^{2}\,\frac{\nu_{2}^{2}\nu_{5}}{\kappa_{2}},$ $\displaystyle
f\rightarrow\widetilde{f}=\frac{\dfrac{(f-\nu_{3})(f-\nu_{4})}{(f-\kappa_{1}/\nu_{7})(f-\kappa_{1}/\nu_{8})}g-\dfrac{1}{\nu_{1}}}{\dfrac{(f-\nu_{3})(f-\nu_{4})}{(f-\kappa_{1}/\nu_{7})(f-\kappa_{1}/\nu_{8})}g-\dfrac{\kappa_{2}}{q\nu_{1}\nu_{2}\nu_{5}}}\,f,$
$\displaystyle
g\rightarrow\widetilde{g}=\frac{(f-\nu_{3})(f-\nu_{4})}{(f-\kappa_{1}/\nu_{7})(f-\kappa_{1}/\nu_{8})}\frac{(\widetilde{f}-\kappa_{1}/\nu_{7})(\widetilde{f}-\kappa_{1}/\nu_{8})}{(\widetilde{f}-q\nu_{2}\nu_{3}\nu_{5}/\kappa_{2})(\widetilde{f}-q\nu_{2}\nu_{4}\nu_{5}/\kappa_{2})}\,g.$
(4.9)

###### Proof.

Set $\widetilde{c}=\chi_{2}$. Then it follows from equations (3.27), (3.28)
and (3.29) that we may write the transformation of the parameters induced by
the $q$-middle convolution as

$\displaystyle\chi_{1}\rightarrow\frac{\chi_{1}\chi_{2}ta_{1}a_{2}}{\theta_{1}},\qquad\chi_{2}\rightarrow\chi_{2},\qquad
a_{1}\rightarrow a_{1},\qquad a_{2}\rightarrow a_{2},$ $\displaystyle
a_{3}\rightarrow\frac{\chi_{2}ta_{1}a_{2}a_{3}}{\theta_{1}},\qquad
a_{4}\rightarrow\frac{\chi_{2}ta_{1}a_{2}a_{4}}{\theta_{1}},\qquad
t\theta_{1}\rightarrow\frac{\chi_{2}t^{2}a_{1}a_{2}\theta_{2}}{\theta_{1}},\qquad
t\theta_{2}\rightarrow\frac{\chi_{2}^{2}t^{3}a_{1}^{2}a_{2}^{2}}{\theta_{1}},$
$\displaystyle
y\rightarrow\widetilde{y}=\frac{qz\dfrac{(y-a_{3})(y-a_{4})}{(y-ta_{1})(y-ta_{2})}-\dfrac{1}{\chi_{1}}}{qz\dfrac{(y-a_{3})(y-a_{4})}{(y-ta_{1})(y-ta_{2})}-\dfrac{\theta_{1}}{\chi_{1}\chi_{2}ta_{1}a_{2}}}\,y,$
$\displaystyle
z\rightarrow\widetilde{z}=\frac{(y-a_{3})(y-a_{4})}{(y-ta_{1})(y-ta_{2})}\frac{(\widetilde{y}-ta_{1})(\widetilde{y}-ta_{2})}{(\widetilde{y}-\chi_{2}ta_{1}a_{2}a_{3}/\theta_{1})(\widetilde{y}-\chi_{2}ta_{1}a_{2}a_{3}/\theta_{1})}z.$
(4.10)

By the correspondence in equation (4.6), it is rewritten as equation (4.9).

We show that the transformation of the parameters given in equation (4.9)
coincides with the consequence of the action given in equation (4.8). We apply
the operation $s_{2}s_{0}s_{1}s_{2}$ to $f$ and $g$. Then

$\displaystyle s_{2}s_{0}s_{1}s_{2}(f)=f,\qquad
s_{2}s_{0}s_{1}s_{2}(g)=\frac{(f-\nu_{3})(f-\nu_{4})}{(f-\kappa_{1}/\nu_{7})(f-\kappa_{1}/\nu_{8})}\,g.$

Note that we define the composition of the transformations as automorphisms of
the algebra (symbolical composition in [7, Remark 2.1]). Since
$s_{3}(f)=f(g-1/\nu_{1})/(g-\nu_{5}/k_{2})$, we have

$\displaystyle
s_{2}s_{0}s_{1}s_{2}s_{3}(f)=\frac{\dfrac{(f-\nu_{3})(f-\nu_{4})}{(f-\kappa_{1}/\nu_{7})(f-\kappa_{1}/\nu_{8})}g-\dfrac{1}{\nu_{1}}}{\dfrac{(f-\nu_{3})(f-\nu_{4})}{(f-\kappa_{1}/\nu_{7})(f-\kappa_{1}/\nu_{8})}g-\dfrac{\kappa_{2}}{q\nu_{1}\nu_{2}\nu_{6}}}\,f.$

Here we used equation (4.2). By comparing it with equation (4.9), we obtain

$\displaystyle s_{5}s_{2}s_{0}s_{1}s_{2}s_{3}(f)=\widetilde{f}.$

We consider the operation to $g$. Since $g$ is invariant under the actions of
$s_{3}$ and $s_{5}$, we have

$\displaystyle
s_{5}s_{2}s_{0}s_{1}s_{2}s_{3}(g)=s_{2}s_{0}s_{1}s_{2}(g)=\frac{(f-\nu_{3})(f-\nu_{4})}{(f-\kappa_{1}/\nu_{7})(f-\kappa_{1}/\nu_{8})}\,g.$

We set $s=s_{5}s_{2}s_{0}s_{1}s_{2}s_{3}$. Then

$\displaystyle s_{5}s_{2}s_{1}s_{0}s_{2}s_{3}s_{2}s_{0}s_{1}s_{2}(g)$
$\displaystyle=\frac{s(f)-s(\nu_{3})}{s(f)-s(\kappa_{1}/\nu_{7})}\frac{s(f)-s(\nu_{4})}{s(f)-s(\kappa_{1}/\nu_{8})}\,s(g)$
$\displaystyle=\frac{\widetilde{f}-s(\nu_{3})}{\widetilde{f}-s\kappa_{1}/\nu_{7})}\frac{\widetilde{f}-s(\nu_{4})}{\widetilde{f}-s(\kappa_{1}/\nu_{8})}\frac{f-\nu_{3}}{f-\kappa_{1}/\nu_{7}}\frac{f-\nu_{4}}{f-\kappa_{1}/\nu_{8}}\,g.$

It follows from equation (4.2) that

$\displaystyle
s\bigg{(}\frac{\kappa_{1}}{\nu_{7}}\bigg{)}=\frac{q\nu_{2}\nu_{4}\nu_{5}}{\kappa_{2}},\qquad
s\bigg{(}\frac{\kappa_{1}}{\nu_{8}}\bigg{)}=\frac{q\nu_{2}\nu_{3}\nu_{5}}{\kappa_{2}},\qquad
s(\nu_{3})=\frac{\kappa_{1}}{\nu_{8}},\qquad
s(\nu_{4})=\frac{\kappa_{1}}{\nu_{7}}.$

Hence

$\displaystyle
s_{5}s_{2}s_{1}s_{0}s_{2}s_{3}s_{2}s_{0}s_{1}s_{2}(g)=\widetilde{g},$
$\displaystyle
s_{5}s_{2}s_{1}s_{0}s_{2}s_{3}s_{2}s_{0}s_{1}s_{2}(f)=s_{5}s_{2}s_{0}s_{1}s_{2}s_{3}(f)=\widetilde{f}.$

The action of the operation
$s_{5}s_{2}s_{1}s_{0}s_{2}s_{3}s_{2}s_{0}s_{1}s_{2}$ to the other parameters
is given by

$\displaystyle\kappa_{1}\rightarrow\frac{q\kappa_{1}\nu_{2}\nu_{5}}{\kappa_{2}},\qquad\kappa_{2}\rightarrow\frac{q^{2}\nu_{2}^{2}\nu_{5}^{2}}{\kappa_{2}},\qquad$
$\displaystyle\nu_{1}\rightarrow\frac{q\nu_{1}\nu_{2}\nu_{5}}{\kappa_{2}},\qquad\nu_{2}\rightarrow\nu_{2},\qquad\nu_{3}\rightarrow\frac{q\nu_{2}\nu_{3}\nu_{5}}{\kappa_{2}},\qquad\nu_{4}\rightarrow\frac{q\nu_{2}\nu_{4}\nu_{5}}{\kappa_{2}},\qquad$
$\displaystyle\nu_{5}\rightarrow\frac{q\nu_{2}\nu_{5}\nu_{6}}{\kappa_{2}},\qquad\nu_{6}\rightarrow\nu_{5},\qquad\nu_{7}\rightarrow\frac{q\nu_{2}\nu_{5}\nu_{7}}{\kappa_{2}},\qquad\nu_{8}\rightarrow\frac{q\nu_{2}\nu_{5}\nu_{8}}{\kappa_{2}},$

and it recovers equation (4.9). ∎

##  5 Integral transformation on $\boldsymbol{q}$-Heun equation

We interpret the integral transformation in Theorem 3.1 as the one for
solutions of the single second-order linear $q$-difference equations.

######  Proposition 5.1.

  1. $(i)$

The function $y_{1}(x)$ in equation (3.24) satisfies

$\displaystyle\bigg{\\{}\frac{x(q\chi_{1}z-1)(\chi_{2}z-1)}{z}-\chi_{1}\chi_{2}a_{3}a_{4}\frac{(qz-
ta_{1}a_{2}/\theta_{1})(qz-ta_{1}a_{2}/\theta_{2})}{yz}\bigg{\\}}y_{1}(x)$
$\displaystyle\qquad{}+\frac{\chi_{1}\chi_{2}(x-qa_{3})(x-qa_{4})}{qy-x}\bigg{\\{}qzy_{1}(x)-\frac{ta_{1}a_{2}}{\theta_{1}}y_{1}(x/q)\bigg{\\}}$
$\displaystyle\qquad{}+\frac{q(x-ta_{1})(x-ta_{2})}{y-x}\bigg{\\{}\frac{y_{1}(x)}{qz}-\frac{\theta_{1}}{ta_{1}a_{2}}y_{1}(qx)\bigg{\\}}=0.$
(5.1)

  2. $(ii)$

The function $\check{y}_{1}(x)$ in equation (3.25) satisfies

$\displaystyle\bigg{\\{}\frac{x(qt\theta_{2}\widetilde{z}-a_{3}a_{4})(\chi_{2}\widetilde{z}-1)}{a_{3}a_{4}\widetilde{z}}-\frac{t^{2}a_{1}a_{2}(q\chi_{2}\theta_{2}\widetilde{z}-\theta_{1})\big{(}q\chi_{2}^{2}ta_{1}a_{2}\widetilde{z}-\theta_{1}\big{)}}{\theta_{1}^{2}\widetilde{y}\widetilde{z}}\bigg{\\}}\check{y}_{1}(x)$
$\displaystyle\qquad{}+\frac{\chi_{2}t\theta_{2}(x-qt\theta_{2}/(\chi_{1}a_{4}))(x-qt\theta_{2}/(\chi_{1}a_{3}))}{a_{3}a_{4}(q\widetilde{y}-x)}\bigg{\\{}q\widetilde{z}\check{y}_{1}(x)-\frac{1}{\chi_{2}}\check{y}_{1}(x/q)\bigg{\\}}$
$\displaystyle\qquad{}+\frac{q(x-ta_{1})(x-ta_{2})}{\widetilde{y}-x}\bigg{\\{}\frac{\check{y}_{1}(x)}{q\widetilde{z}}-\chi_{2}\check{y}_{1}(qx)\bigg{\\}}=0,$
(5.2)

where $\widetilde{y}$ and $\widetilde{z}$ are determined by equation (4.10).

###### Proof.

It follows from equation (3.6) that the function $y_{1}(x)$ in equation (3.24)
satisfies equation (4.5) with the condition
$d_{0}=c_{0}=\theta_{1}/(ta_{1}a_{2})$. Then we obtain $(i)$.

Equation (3.25) is obtained as the form of equation (3.24) by replacing the
parameters as $(y,z)\to(\widetilde{y},\widetilde{z})$ and equation (3.27) up
to the ambiguity of the parameter $w$, and the $q$-difference equation for
$\check{y}_{1}(x)$ does not depend on the parameter $w$. Hence $(ii)$ follows
from equation (3.26) with the condition $\widetilde{c}=\chi_{2}$. ∎

######  Theorem 5.2.

Assume that $y_{1}(x)$ is a solution to equation (5.1) and $\lambda$ satisfies
$q^{\lambda}=\chi_{2}a_{1}a_{2}t/\theta_{1}$. Then the function

$\displaystyle\check{y}_{1}(x)=\int^{\xi\infty}_{0}\frac{y_{1}(s)}{s-x}P_{\lambda}(x,s)\,{\rm
d}_{q}s$ (5.3)

formally satisfies equation (5.2).

###### Proof.

Let $b_{ij}(x)$ $(i,j\in\\{1,2\\})$ be the elements of the matrix $B(x)$ in
equation (3.7). The function $y_{1}(x)$ is given as a solution to equation
(5.1). Define the function $y_{2}(x)$ by
$y_{1}(qx)=b_{11}(x)y_{1}(x)+b_{12}(x)y_{2}(x)$. Since equation (5.1) is
written as

$\displaystyle\frac{y_{1}(qx)}{b_{12}(x)}-\bigg{\\{}\frac{b_{11}(x)}{b_{12}(x)}+\frac{b_{22}(x/q)}{b_{12}(x/q)}\bigg{\\}}y_{1}(x)+\frac{b_{11}(x/q)b_{22}(x/q)-b_{12}(x/q)b_{21}(x/q)}{b_{12}(x/q)}y_{1}(x/q)=0,$

we obtain the equality $y_{2}(x)=b_{21}(x/q)y_{1}(x/q)+b_{22}(x/q)y_{2}(x/q)$.
Then the function $Y(x)={}^{t}(y_{1}(x),y_{2}(x))$ satisfies equation (3.24).
Hence it follows from Theorem 2.2 that the function
$\check{Y}(x)={}^{t}(\check{y}_{1}(x),\check{y}_{2}(x))$ satisfies equation
(3.25). Therefore the function $\check{y}_{1}(x)$ in equation (5.3) satisfies
equation (5.2) by Proposition 5.1$(ii)$. ∎

######  Corollary 5.3.

Let $\mu$, $\mu^{\prime}$ and $\lambda$ be the constants such that
$q^{\mu}=\nu_{5}/\kappa_{2}$, $q^{\mu^{\prime}}=q\nu_{2}$ and
$q^{\lambda}=q\nu_{2}\nu_{5}/\kappa_{2}$. Let $y(x)$ be a solution to the
equation $L_{1}y(x)=0$ which was described in equation (4.1). Then the
function

$\displaystyle\check{y}(x)=x^{\mu^{\prime}}\int^{\xi\infty}_{0}\frac{y(s)}{s-x}s^{\mu}P_{\lambda}(x,s)\,{\rm
d}_{q}s$ (5.4)

formally satisfies the equation $\widetilde{L_{1}}\check{y}(x)=0$, where the
operator $\widetilde{L_{1}}$ is obtained from $L_{1}$ by replacing the
parameters in accordance with the action of
$s_{5}s_{2}s_{1}s_{0}s_{2}s_{3}s_{2}s_{0}s_{1}s_{2}$ $($see equation
$(\ref{eq:qmcKNY}))$.

###### Proof.

Let $y(x)$ be a solution to the equation $L_{1}y(x)=0$ (see equation (4.1)).
Set $y_{1}(x)=x^{\mu}y(x)$. Then function $y_{1}(x)$ satisfies

$\displaystyle\bigg{\\{}\frac{x(g\nu_{1}-1)(g\nu_{2}-1)}{qg}-\frac{\nu_{1}\nu_{2}\nu_{3}\nu_{4}(g-\nu_{5}/\kappa_{2})(g-\nu_{6}/\kappa_{2})}{fg}\bigg{\\}}y_{1}(x)$
$\displaystyle\qquad{}+\frac{\nu_{1}\nu_{2}(x-q\nu_{3})(x-q\nu_{4})}{q(qf-x)}\bigg{(}gy_{1}(x)-\frac{\nu_{5}}{\kappa_{2}}y_{1}(x/q)\bigg{)}$
$\displaystyle\qquad{}+\frac{(x-\kappa_{1}/\nu_{7})(x-\kappa_{1}/\nu_{8})}{q(f-x)}\bigg{(}\dfrac{1}{g}y_{1}(x)-\frac{\kappa_{2}}{\nu_{5}}y_{1}(qx)\bigg{)}=0,$

and it is written as equation (5.1) by the correspondence in equation (4.6).
It follows from Theorem 5.2 that the function

$\displaystyle\check{y}_{1}(x)=\int^{\xi\infty}_{0}\frac{y(s)}{s-x}s^{\mu}P_{\lambda}(x,s)\,{\rm
d}_{q}s$

satisfies equation (5.2). We apply the correspondence in equation (4.6) for
the parameters and set $\check{y}(x)=x^{\mu^{\prime}}\check{y}_{1}(x)$. Then
it is seen that the function $\check{y}(x)$ is written as equation (5.4) and
satifies the equation $\widetilde{L_{1}}\check{y}(x)=0$. ∎

We specialize the parameters $y$ and $z$ in Theorem 5.2 as

$\displaystyle y=a_{3},\qquad
z=\frac{(ta_{1}-a_{3})(ta_{2}-a_{3})}{qt(\theta_{1}+\theta_{2})+a_{3}^{2}(q\chi_{1}+\chi_{2})+Eq\theta_{1}a_{3}/(ta_{1}a_{2})}.$
(5.5)

Then equation (5.1) is written as

$\displaystyle(x-ta_{1})(x-ta_{2})y_{1}(qx)+\frac{\chi_{1}\chi_{2}t^{2}a_{1}^{2}a_{2}^{2}(x-a_{3})(x-qa_{4})}{q\theta_{1}^{2}}y_{1}(x/q)$
$\displaystyle\qquad{}-\bigg{\\{}\frac{ta_{1}a_{2}(\chi_{2}+q\chi_{1})}{q\theta_{1}}x^{2}+Ex+t^{2}a_{1}a_{2}\bigg{(}1+\frac{\theta_{2}}{\theta_{1}}\bigg{)}\bigg{\\}}y_{1}(x)=0,$
(5.6)

and equation (5.2) is written as

$\displaystyle(x-ta_{1})(x-ta_{2})\check{y}_{1}(qx)+\frac{\chi_{1}ta_{1}a_{2}}{q\theta_{1}}\bigg{(}x-\frac{t\theta_{2}}{\chi_{1}a_{4}}\bigg{)}\bigg{(}x-\frac{qt\theta_{2}}{\chi_{1}a_{3}}\bigg{)}\check{y}_{1}(x/q)$
$\displaystyle\qquad{}-\bigg{\\{}\frac{q\chi_{1}ta_{1}a_{2}+\theta_{1}}{q\theta_{1}}x^{2}+Ex+t^{2}a_{1}a_{2}\frac{\chi_{2}ta_{1}a_{2}+\theta_{2}}{\theta_{1}}\bigg{\\}}\check{y}_{1}(x)=0.$
(5.7)

Note that these equations are the $q$-Heun equation, and the specialization of
the parameters is related to the initial value space of
$q$-$P\big{(}D^{(1)}_{5}\big{)}$ (see [11]). By replacing the parameters, we
obtain the following theorem on the $q$-Heun equation.

######  Theorem 5.4.

Assume that $l_{1}l_{2}l_{3}l_{4}=h_{1}h_{2}h_{3}q^{2}$ and $g(x)$ satisfies
the $q$-Heun equation written as

$\displaystyle\big{(}x-h_{1}q^{1/2}\big{)}\big{(}x-h_{2}q^{1/2}\big{)}g(x/q)+l_{3}l_{4}\big{(}x-l_{1}q^{-1/2}\big{)}\big{(}x-l_{2}q^{-1/2}\big{)}g(qx)$
$\displaystyle\qquad{}-\big{\\{}(l_{3}+l_{4})x^{2}+l_{3}l_{4}Ex+(l_{1}l_{2}l_{3}l_{4}h_{1}h_{2})^{1/2}\big{(}h_{3}^{1/2}+h_{3}^{-1/2}\big{)}\big{\\}}g(x)=0.$
(5.8)

Let $\lambda$ be the value satisfying $q^{\lambda}=q/l_{4}$. Then the function

$\displaystyle\check{g}(x)=\int^{\xi\infty}_{0}\frac{g(s)}{s-x}P_{\lambda}(x,s)\,{\rm
d}_{q}s$

formally satisfies

$\displaystyle\big{(}x-h^{\prime}_{1}q^{1/2}\big{)}\big{(}x-h^{\prime}_{2}q^{1/2}\big{)}\check{g}(x/q)+l^{\prime}_{3}l^{\prime}_{4}\big{(}x-l^{\prime}_{1}q^{-1/2}\big{)}\big{(}x-l^{\prime}_{2}q^{-1/2}\big{)}\check{g}(qx)$
$\displaystyle\qquad{}-\big{\\{}(l^{\prime}_{3}+l^{\prime}_{4})x^{2}+l^{\prime}_{3}l^{\prime}_{4}Ex+(l^{\prime}_{1}l^{\prime}_{2}l^{\prime}_{3}l^{\prime}_{4}h^{\prime}_{1}h^{\prime}_{2})^{1/2}\big{(}(h^{\prime}_{3})^{1/2}+(h^{\prime}_{3})^{-1/2}\big{)}\big{\\}}\check{g}(x)=0,$
(5.9)

where

$\displaystyle l_{1}^{\prime}=l_{1},\quad\ l_{2}^{\prime}=l_{2},\quad\
l_{3}^{\prime}=l_{3},\quad\ l_{4}^{\prime}=q,\quad\
h_{1}^{\prime}=qh_{1}/l_{4},\quad\ h_{2}^{\prime}=qh_{2}/l_{4},\quad\
h_{3}^{\prime}=l_{4}/(qh_{3}).$

###### Proof.

Let $l_{1}$, $l_{2}$, $l_{3}$, $l_{4}$, $h_{1}$, $h_{2}$ and $h_{3}$ be the
values such that $l_{1}l_{2}l_{3}l_{4}=h_{1}h_{2}h_{3}q^{2}$. We evaluate the
values $a_{1},a_{2},a_{3}$ and $a_{4}$ as $a_{3}=h_{1}q^{1/2}$,
$a_{4}=h_{2}q^{-1/2}$, $ta_{1}=l_{1}q^{-1/2}$ and $ta_{2}=l_{2}q^{-1/2}$, and
fix the ratios $\theta_{1}/\chi_{1}$ and $\theta_{1}/\chi_{2}$ by
$l_{3}=\theta_{1}/(\chi_{1}ta_{1}a_{2})$ and
$l_{4}=q\theta_{1}/(\chi_{2}ta_{1}a_{2})$. It follows from the relation
$\chi_{1}\chi_{2}a_{1}a_{2}a_{3}a_{4}=\theta_{1}\theta_{2}$ in equation (3.3)
that
$h_{3}=l_{1}l_{2}l_{3}l_{4}/\big{(}h_{1}h_{2}q^{2}\big{)}=\theta_{1}/\theta_{2}$.
Then equation (5.8) is written as equation (5.6) by setting $g(x)=y_{1}(x)$.
We apply Theorem 5.2, where the parameters $y$ and $z$ are specialized as
equation (5.5). Then we obtain equation (5.7) by the $q$-integral
transformation. By rewriting the parameters $a_{1}$, $a_{2}$, $a_{3}$,
$a_{4}$, $\theta_{1}/\chi_{1}$, $\theta_{1}/\chi_{2}$ and
$\theta_{1}/\theta_{2}$, we obtain equation (5.9). ∎

##  6 Concluding remarks

In this paper, we applied the $q$-middle convolution to a linear
$q$-difference equation associated with the $q$-Painlevé VI equation, and we
obtain integral transformations as a consequence. We investigated the symmetry
by the $q$-middle convolution in terms of the affine Weyl group symmetry of
the $q$-Painlevé VI equation. As an application, we obtained an integral
transformation on the $q$-Heun equation. Note that our result is a
$q$-analogue of the results in [5, 13, 14] on the Painlevé VI equation, the
middle convolution and Heun’s differential equation.

In [10], the $q$-convolution was connected to the $q$-integral transformation
by the Jackson integral. On the other hand, the convolution for the system of
Fuchsian differential equations was connected to the Euler’s integral
transformation, and there are several choice of the cycles on the integration
by the Pochhammer contour. Thus, there is a problem to find more cycles on the
$q$-integral transformation associated with the $q$-convolution. Other
problems related with the general $q$-middle convolution may be found from our
explicit application of the specified $q$-middle convolution.

The $q$-Painlevé VI equation was denoted by $q$-$P\big{(}D^{(1)}_{5}\big{)}$
in [7], and the Weyl group symmetry and the Lax pair of discrete Painlevé
equations including $q$-$P\big{(}D^{(1)}_{5}\big{)}$ were reviewed in [7]. The
equations $q$-$P\big{(}E^{(1)}_{6}\big{)}$ and
$q$-$P\big{(}E^{(1)}_{7}\big{)}$ are also $q$-analogue of the Painlevé VI
equation. We hope to extend the symmetry of integral transformations to the
cases $q$-$P\big{(}E^{(1)}_{6}\big{)}$ and $q$-$P\big{(}E^{(1)}_{7}\big{)}$ by
using the $q$-middle convolution, which might be related with the variants of
$q$-Heun equation [15, 16]. There is also a problem to connect the degenerated
$q$-Painlevé equations (e.g., $q$-$P\big{(}A^{(1)}_{4}\big{)}$) with the
$q$-middle convolution. In this direction, the theory of the $q$-middle
convolution for the non-Fuchsian $q$-difference equation is anticipated.

##  Appendix A Middle convolution for other parameters

In Section 3, we discussed the middle convolution which is related to the
$q$-Painlevé VI equation. The space $\mathcal{L}$ was defined in equation
(3.12) and we imposed the condition
$q^{\lambda}=\chi_{2}a_{1}a_{2}t/\theta_{1}$ in Section 3, which induces
$\dim(\mathcal{L})=1$. In the appendix, we discuss the case
$q^{\lambda}=\chi_{1}a_{1}a_{2}t/\theta_{1}$, which also induces
$\dim(\mathcal{L})=1$.

We continue the argument in the case
$q^{\lambda}=\chi_{1}a_{1}a_{2}t/\theta_{1}$ by replacing the matrix $P$ in
equation (3.17) with

$\displaystyle P=\begin{pmatrix}0&0&1&v_{01}&0&0\\\
g_{1}&g_{3}&0&v_{02}&0&0\\\ 0&0&1&0&v_{11}&0\\\ g_{2}&g_{4}&0&0&v_{12}&0\\\
0&0&1&0&0&v_{21}\\\ 0&0&0&0&0&v_{22}\\\
\end{pmatrix}\\!,\qquad\begin{array}[]{l}g_{3}=-q(\chi_{1}a_{2}y-\theta_{1})z+a_{2}(y-ta_{1}),\\\\[2.15277pt]
g_{4}=y(a_{1}-a_{2})(q\chi_{1}z-1),\end{array}$ $\displaystyle
g_{1}=-\chi_{2}a_{2}\big{\\{}q^{2}\chi_{1}\theta_{1}(y-a_{3})(y-a_{4})(\chi_{1}y-\chi_{2}ta_{1})z^{2}+\theta_{1}(y-ta_{1})^{2}(y-ta_{2})$
$\displaystyle\hphantom{g_{1}=}{}-q(y\\!-ta_{1})\big{(}2\theta_{1}\chi_{1}y^{2}\\!-\theta_{1}(\chi_{2}ta_{1}\\!+\chi_{1}ta_{2}\\!+\chi_{1}a_{3}\\!+\chi_{1}a_{4})y\\!+t\big{(}\theta_{1}^{2}\\!+\chi_{1}\chi_{2}a_{1}a_{2}a_{3}a_{4}\big{)}\big{)}z\big{\\}},$
$\displaystyle
g_{2}=(a_{1}-a_{2})y\big{\\{}q^{2}\chi_{1}^{2}\chi_{2}\theta_{1}(y-a_{3})(y-a_{4})z^{2}+\theta_{1}\chi_{2}(y-ta_{1})(y-ta_{2})$
$\displaystyle\hphantom{g_{2}=}{}-q\chi_{1}\big{(}2\chi_{2}\theta_{1}y^{2}-\chi_{2}\theta_{1}(ta_{1}+ta_{2}+a_{3}+a_{4})y+t\big{(}\theta_{1}^{2}+\chi_{2}^{2}a_{1}a_{2}a_{3}a_{4}\big{)}\big{)}z\big{\\}}.$

Recall that $v_{01},\dots,v_{22}$ were defined by equations (3.9) and (3.10).
Then $\det
P=q^{3}w^{2}y^{3}z^{3}\times\chi_{2}(\chi_{1}-\chi_{2})^{2}(a_{1}-a_{2})\theta_{1}^{2}(\chi_{2}ta_{1}a_{2}-\theta_{1})v_{22}^{2}$,
and the matrix $P$ is invertible if $\det P\neq 0$. Set

$\displaystyle\widetilde{F}_{1}=P^{-1}F_{1}P,\qquad\widetilde{F}_{2}=P^{-1}F_{2}P,\qquad\widetilde{F}_{\infty}=P^{-1}{F}_{\infty}P,$

(see equation (3.11)). Then they admit the following expression:

$\displaystyle\widetilde{F}_{1}=\begin{pmatrix}\overline{F}_{1}&O\\\
*&*\end{pmatrix}\\!,\qquad\widetilde{F}_{2}=\begin{pmatrix}\overline{F}_{2}&O\\\
*&*\end{pmatrix}\\!,\qquad\widetilde{F}_{\infty}=\begin{pmatrix}\overline{F}_{\infty}&O\\\
*&*\end{pmatrix}\\!,$

where $\overline{F}_{1}$, $\overline{F}_{2}$, $\overline{F}_{\infty}$ are
$2\times 2$ matrices given by

$\displaystyle\overline{F}_{\infty}=\begin{pmatrix}1&0\\\
0&\chi_{2}ta_{1}a_{2}/\theta_{1}\end{pmatrix}\\!,\qquad\overline{F}_{2}=\overline{F}_{1}|_{a_{1}\leftrightarrow
a_{2}},$
$\displaystyle\overline{F}_{1}=\frac{a_{1}}{qyz\theta_{1}^{2}(a_{1}-a_{2})(\chi_{2}ta_{1}a_{2}-\theta_{1})}\begin{pmatrix}-\theta_{1}f_{1}^{[1]}f_{2}^{[1]}&-a_{2}f_{2}^{[1]}\\\\[2.15277pt]
\chi_{2}a_{2}\theta_{1}f_{1}^{[1]}f_{3}^{[1]}&\chi_{2}a_{2}^{2}f_{3}^{[1]}\end{pmatrix}\\!,$

where

$\displaystyle
f_{1}^{[1]}=q\chi_{1}\chi_{2}a_{2}(y-a_{3})(y-a_{4})z-(\chi_{2}a_{2}y-\theta_{1})(y-ta_{2}),$
$\displaystyle f_{2}^{[1]}=q(\chi_{1}a_{2}y-\theta_{1})z-a_{2}(y-ta_{1}),$
$\displaystyle
f_{3}^{[1]}=q^{2}\chi_{1}\theta_{1}(y-a_{3})(y-a_{4})(\chi_{1}y-\chi_{2}ta_{1})z^{2}+\theta_{1}(y-ta_{1})^{2}(y-ta_{2})$
$\displaystyle\hphantom{f_{3}^{[1]}=}{}-q(y\\!-ta_{1})\big{(}2\chi_{1}\theta_{1}y^{2}\\!-\theta_{1}(\chi_{2}ta_{1}\\!+\chi_{1}ta_{2}\\!+\chi_{1}a_{3}\\!+\chi_{1}a_{4})y\\!+t\big{(}\theta_{1}^{2}\\!+\chi_{1}\chi_{2}a_{1}a_{2}a_{3}a_{4}\big{)}\big{)}z.$

Write

$\displaystyle\overline{Y}(qx)=\overline{F}(x)\overline{Y}(x),\qquad\overline{F}(x)=\overline{F}_{\infty}+\frac{\overline{F}_{1}}{1-x/(ta_{1})}+\frac{\overline{F}_{2}}{1-x/(ta_{2})}.$
(A.1)

Note that the upper right entry of $\overline{F}(x)$ is written as

$\displaystyle\frac{ta_{1}a_{2}((ta_{1}a_{2}-q\theta_{1}z)x+ta_{1}a_{2}y(q\chi_{1}z-1))}{qyz\theta_{1}^{2}(x-ta_{1})(x-ta_{2})(\chi_{2}ta_{1}a_{2}-\theta_{1})}.$

As discussed in Section 3, equation (A.1) is related to an integral
transformation. Let $Y(x)$ be a solution to $Y(qx)=B(x)Y(x)$ in equation (3.6)
and write $Y(x)=\left(\\!\begin{smallmatrix}y_{1}(x)\\\
y_{2}(x)\end{smallmatrix}\\!\right)$. By applying Theorem 2.2, it is shown
that the function
$\overline{Y}(x)=\left(\\!\begin{smallmatrix}\check{y}_{1}(x)\\\
\check{y}_{2}(x)\end{smallmatrix}\\!\right)$ defined by

$\displaystyle\check{y}_{j}(x)$
$\displaystyle=\int^{\xi\infty}_{0}\bigg{\\{}\bigg{(}\frac{p_{j1}}{s}+\frac{p_{j3}}{s-ta_{1}}+\frac{p_{j5}}{s-ta_{2}}\bigg{)}y_{1}(s)+\bigg{(}\frac{p_{j2}}{s}+\frac{p_{j4}}{s-ta_{1}}+\frac{p_{j6}}{s-ta_{2}}\bigg{)}y_{2}(s)\bigg{\\}}$
$\displaystyle\hphantom{=}\times P_{\lambda}(x,s)\,{\rm d}_{q}s,\qquad j=1,2,$

formally satisfies equation (A.1), where $p_{jk}$ is the $(j,k)$-entry of the
matrix $P^{-1}$.

By a straightforward calculation, it is shown that

$\displaystyle\frac{p_{21}}{s}+\frac{p_{23}}{s-ta_{1}}+\frac{p_{25}}{s-ta_{2}}=\frac{-t\theta_{1}^{2}(\chi_{1}-\chi_{2})}{c(\chi_{2}ta_{1}a_{2}-\theta_{1})s}b_{21}(s),$
$\displaystyle\frac{p_{22}}{s}+\frac{p_{24}}{s-ta_{1}}+\frac{p_{26}}{s-ta_{2}}=\frac{-t\theta_{1}^{2}(\chi_{1}-\chi_{2})}{c(\chi_{2}ta_{1}a_{2}-\theta_{1})s}(b_{22}(s)-1),$
$\displaystyle\frac{p_{12}}{s}+\frac{p_{14}}{s-ta_{1}}+\frac{p_{16}}{s-ta_{2}}=\frac{\theta_{1}((ta_{1}a_{2}-qz\theta_{1})s+yta_{1}a_{2}(qz\chi_{1}-1))}{qcwyz\chi_{2}a_{1}a_{2}(\chi_{2}ta_{1}a_{2}-\theta_{1})(s-y)s}b_{12}(s),$
$\displaystyle\frac{p_{11}}{s}+\frac{p_{13}}{s-ta_{1}}+\frac{p_{15}}{s-ta_{2}}=\frac{\theta_{1}((ta_{1}a_{2}-qz\theta_{1})s+yta_{1}a_{2}(qz\chi_{1}-1))}{qcwyz\chi_{2}a_{1}a_{2}(\chi_{2}ta_{1}a_{2}-\theta_{1})(s-y)s}b_{11}(s)$
$\displaystyle\hphantom{\frac{p_{11}}{s}+\frac{p_{13}}{s-ta_{1}}+\frac{p_{15}}{s-ta_{2}}=}{}-\frac{t(\chi_{1}(ta_{1}a_{2}-qz\theta_{1})s+y\theta_{1}(qz\chi_{1}-1))}{qcwyz\chi_{2}(\chi_{2}ta_{1}a_{2}-\theta_{1})(s-y)s},$
$\displaystyle
c=q^{2}\chi_{1}^{2}\chi_{2}\theta_{1}(y-a_{3})(y-a_{4})z^{2}+\theta_{1}(y\chi_{2}-\chi_{1}ta_{2})(y-ta_{1})$
$\displaystyle\hphantom{c=}{}-q\chi_{1}\big{(}2\chi_{2}\theta_{1}y^{2}-\theta_{1}(\chi_{2}ta_{1}+\chi_{1}ta_{2}+\chi_{2}a_{3}+\chi_{2}a_{4})y+t\big{(}\chi_{1}\chi_{2}a_{1}a_{2}a_{3}a_{4}+\theta_{1}^{2}\big{)}\big{)}z,$

where $b_{jk}(s)$ are elements of the matrix $B(s)$ in equation (3.7). It
follows from $y_{1}(qs)=b_{11}(s)y_{1}(s)+b_{12}(s)y_{2}(s)$ that

$\displaystyle\check{y}_{1}(x)$
$\displaystyle=\frac{(ta_{1}a_{2}-qz\theta_{1})}{qcwyz\chi_{2}a_{1}a_{2}(\chi_{2}ta_{1}a_{2}-\theta_{1})}$
$\displaystyle\times\\!\\!\\!\\!\\!\phantom{=}\int^{\xi\infty}_{0}\\!\biggl{\\{}\\!-(\chi_{1}ta_{1}a_{2}\\!-\\!\theta_{1})y_{1}(s)\\!+\\!\theta_{1}\bigg{(}\\!1\\!+\\!\frac{yta_{1}a_{2}(qz\chi_{1}\\!-\\!1)}{(ta_{1}a_{2}\\!-\\!qz\theta_{1})s}\bigg{)}(y_{1}(qs)\\!-\\!y(s))\\!\biggr{\\}}\frac{P_{\lambda}(x,s)}{s-y}\,{\rm
d}_{q}s.$

Hence, the integral representation of $\check{y}_{1}(x)$ in the case
$q^{\lambda}=\chi_{1}ta_{1}a_{2}/\theta_{1}$ is more complicated than that in
the case $q^{\lambda}=\chi_{2}ta_{1}a_{2}/\theta_{1}$. On the other hand, it
follows from $y_{2}(qs)=b_{21}(s)y_{1}(s)+b_{22}(s)y_{2}(s)$ that

$\displaystyle\check{y}_{2}(x)=\frac{-t\theta_{1}^{2}(\chi_{1}-\chi_{2})}{c(\chi_{2}ta_{1}a_{2}-\theta_{1})}\int^{\xi\infty}_{0}\frac{y_{2}(qs)-y_{2}(s)}{s}P_{\lambda}(x,s)\,{\rm
d}_{q}s.$

To give the correspondence of the parameters by the $q$-middle convolution in
the form of the equation
$Y(qx)=\big{\\{}A_{0}(t)+A_{1}(t)x+A_{2}x^{2}\big{\\}}Y(x)$ in equation (3.2),
we need to transform equation (3.25).

Let $\widetilde{c}$ and $\widetilde{d}$ be a non-zero constant which will be
fixed later. Set $x=\widetilde{d}\widetilde{x}$,

$\displaystyle\widetilde{A}(\widetilde{x})=\widetilde{c}(x-ta_{1})(x-ta_{2})\bigg{(}\overline{F}_{\infty}+\frac{\overline{F}_{1}}{1-x/(ta_{1})}+\frac{\overline{F}_{2}}{1-x/(ta_{2})}\bigg{)}$

and write
$\widetilde{A}(\widetilde{x})=\widetilde{A}(\widetilde{x},t)=\widetilde{A}_{0}(t)+\widetilde{A}_{1}(t)\widetilde{x}+\widetilde{A}_{2}\widetilde{x}^{2}$.
Then we have

$\displaystyle\widetilde{A}_{2}=\begin{pmatrix}\widetilde{c}\widetilde{d}^{2}\theta_{1}/(ta_{1}a_{2})&0\\\
0&\widetilde{c}\widetilde{d}^{2}\chi_{2}\end{pmatrix}\\!,$
$\displaystyle\widetilde{A}_{0}(t)\mbox{ has the eigenvalues
}\widetilde{c}\chi_{1}t^{2}a_{1}a_{2}\mbox{ and }\widetilde{c}t\theta_{2},$
$\displaystyle\det\widetilde{A}(\widetilde{x},t)=\frac{\widetilde{c}^{2}\widetilde{d}^{4}\chi_{2}\theta_{1}}{ta_{1}a_{2}}\bigg{(}\widetilde{x}-\frac{ta_{1}}{\widetilde{d}}\bigg{)}\bigg{(}\widetilde{x}-\frac{ta_{2}}{\widetilde{d}}\bigg{)}\bigg{(}\widetilde{x}-\frac{\chi_{1}ta_{1}a_{2}a_{3}}{\widetilde{d}\theta_{1}}\bigg{)}\bigg{(}\widetilde{x}-\frac{\chi_{1}ta_{1}a_{2}a_{4}}{\widetilde{d}\theta_{1}}\bigg{)}.$

Hence the action of the $q$-middle convolution to the parameters in the case
$q^{\lambda}=\chi_{1}ta_{1}a_{2}/\theta_{1}$ is described as

$\displaystyle\chi_{1}\rightarrow\frac{\widetilde{c}\widetilde{d}^{2}\theta_{1}}{ta_{1}a_{2}},\qquad\chi_{2}\rightarrow\widetilde{c}\widetilde{d}^{2}\chi_{2},\qquad\\{a_{1},a_{2}\\}\rightarrow\bigg{\\{}\frac{a_{1}}{\widetilde{d}},\frac{a_{2}}{\widetilde{d}}\bigg{\\}},$
$\displaystyle\\{a_{3},a_{4}\\}\rightarrow\bigg{\\{}\frac{\chi_{1}ta_{1}a_{2}a_{3}}{\widetilde{d}\theta_{1}},\frac{\chi_{1}ta_{1}a_{2}a_{4}}{\widetilde{d}\theta_{1}}\bigg{\\}},\qquad\\{t\theta_{1},t\theta_{2}\\}\rightarrow\big{\\{}\widetilde{c}\chi_{1}t^{2}a_{1}a_{2},\widetilde{c}t\theta_{2}\big{\\}}.$

We investigate the action to the parameters $y$ and $z$. We denote the images
of $y$ and $z$ by $\widetilde{y}$ and $\widetilde{z}$. Let
$\widetilde{a}_{11}(\widetilde{x})$ (resp.
$\widetilde{a}_{12}(\widetilde{x})$) be the upper left entry (resp. the upper
right entry) of the matrix $\widetilde{A}(\widetilde{x})$. Then the value
$\widetilde{y}$ is the zero of the linear function
$\widetilde{a}_{12}(\widetilde{x})$, and we have

$\displaystyle\widetilde{y}=\frac{ta_{1}a_{2}(q\chi_{1}z-1)}{\widetilde{d}(q\theta_{1}z-ta_{1}a_{2})}\,y=\frac{\chi_{1}ta_{1}a_{2}(qz-1/\chi_{1})}{\widetilde{d}\theta_{1}(qz-
ta_{1}a_{2}/\theta_{1})}\,y.$

The value $\widetilde{z}$ satisfies
$\widetilde{a}_{11}(\widetilde{x})|_{\widetilde{x}=\widetilde{y}}=\big{(}\widetilde{y}-ta_{1}/\widetilde{d}\big{)}\big{(}\widetilde{y}-ta_{2}/\widetilde{d}\big{)}/(q\widetilde{z})$,
and we obtain

$\displaystyle\widetilde{z}=\frac{z}{\widetilde{c}\widetilde{d}^{2}}.$

Set $\widetilde{d}=\chi_{1}ta_{1}a_{2}/\theta_{1}$ and
$\widetilde{c}\widetilde{d}^{2}=1$. By applying the correspondence in equation
(4.6), the transformation of the parameters is written as

$\displaystyle\nu_{1}\rightarrow\frac{\kappa_{2}}{\nu_{5}},\qquad\nu_{2}\rightarrow\nu_{2},\qquad\bigg{\\{}\frac{\kappa_{1}}{\nu_{7}},\frac{\kappa_{1}}{\nu_{8}}\bigg{\\}}\rightarrow\bigg{\\{}\frac{\kappa_{1}\kappa_{2}}{\nu_{7}\nu_{1}\nu_{5}},\frac{\kappa_{1}\kappa_{2}}{\nu_{8}\nu_{1}\nu_{5}}\bigg{\\}},\qquad\\{\nu_{3},\nu_{4}\\}\rightarrow\\{\nu_{3},\nu_{4}\\},$
$\displaystyle\bigg{\\{}\frac{\kappa_{2}}{\nu_{5}},\frac{\kappa_{2}}{\nu_{6}}\bigg{\\}}\rightarrow\bigg{\\{}\nu_{1},\frac{\kappa_{2}}{\nu_{6}}\bigg{\\}},\qquad
f\rightarrow\frac{g-1/\nu_{1}}{g-\nu_{5}/\kappa_{2}}\,f,\qquad g\rightarrow
g.$

Hence we obtain the following proposition.

######  Proposition A.1.

We specify the parameters $\widetilde{c}$ and $\widetilde{d}$ by setting
$\widetilde{d}=\chi_{1}ta_{1}a_{2}/\theta_{1}$ and
$\widetilde{c}\widetilde{d}^{2}=1$. Then the transformation of the parameters
induced by the $q$-middle convolution in the case
$q^{\lambda}=\chi_{1}ta_{1}a_{2}/\theta_{1}$ is realized by the action of
$s_{3}$ given in equation (4.7).

### Acknowledgements

The authors are grateful to the referees for careful reading of the manuscript
and valuable comments. The third author was supported by JSPS KAKENHI Grant
Number JP18K03378.

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