# Kernel function, $q$-integral transformation and $q$-Heun equations

Kouichi Takemura  Department of Mathematics, Ochanomizu University, 2-1-1
Otsuka, Bunkyo-ku, Tokyo 112-8610, Japan<EMAIL_ADDRESS>
###### Abstract.

We find kernel functions of the $q$-Heun equation and its variants. We apply
them to obtain $q$-integral transformations of solutions to the $q$-Heun
equation and its variants. We discuss special solutions of the $q$-Heun
equation from the perspective of the $q$-integral transformation.

###### Key words and phrases:

Kernel function, Jackson integral, Heun equation, $q$-Heun equation,
Ruijsenaars system

###### 2020 Mathematics Subject Classification:

39A13,44A20

##  1\. Introduction

Heun’s differential equation is given by

(1.1)
$\frac{d^{2}y}{dz^{2}}+\left(\frac{\gamma}{z}+\frac{\delta}{z-1}+\frac{\epsilon}{z-t}\right)\frac{dy}{dz}+\frac{\alpha\beta
z-B}{z(z-1)(z-t)}y=0,$

with the condition $\gamma+\delta+\epsilon=\alpha+\beta+1$. It sometimes
appears in the study of mathematical physics. Relationship with the Kerr black
holes is remarkable among them. The parameter $B$ in equation Eq. (1.1) is
called the accessory parameter, and it plays special roles in the analysis of
the Heun equation.

It is known that Heun’s differential equation admits integral transformations.
Kazakov and Slavyanov established Euler integral transformations of Heun’s
differential equation in [8]. Set

(1.2)
$\displaystyle\gamma^{\prime}=\gamma+1-\alpha,\delta^{\prime}=\delta+1-\alpha,\;\epsilon^{\prime}=\epsilon+1-\alpha,\;\alpha^{\prime}=2-\alpha,\;\beta^{\prime}=-\alpha+\beta+1,$
$\displaystyle B^{\prime}=B+(1-\alpha)(\epsilon+\delta
t+(\gamma-\alpha)(t+1))$

and assume that the function $v(w)$ is a solution to the Heun equation with
the parameters
$\gamma^{\prime},\delta^{\prime},\epsilon^{\prime},\alpha^{\prime},\beta^{\prime},B^{\prime}$,
i.e.

(1.3)
$\frac{d^{2}v}{dw^{2}}+\left(\frac{\gamma^{\prime}}{w}+\frac{\delta^{\prime}}{w-1}+\frac{\epsilon^{\prime}}{w-t}\right)\frac{dv}{dw}+\frac{\alpha^{\prime}\beta^{\prime}w-B^{\prime}}{w(w-1)(w-t)}v=0,$

then it was established in [8] that the function

(1.4) $y(z)=\int_{C}v(w)(z-w)^{-\alpha}dw$

is a solution to Eq. (1.1) for a suitable cycle $C$. Examples of the cycle
were described explicitly in [8, 7]. Note that the integral transformation was
also obtained in [19, 20] by considering the middle convolution for some
special system of Fuchsian equations. As an application of the integral
transformation of the Heun equation, a correspondence of special solutions of
the Heun equation was obtained in [22]. Namely the polynomial-type solutions
correspond to the solutions written as a finite sum of the hypergeometric
functions by the integral transformation.

Some of integral transformations are related with the kernel function. Let
${\bf x}=(x_{1},\dots,x_{m})$ and ${\bf y}=(y_{1},\dots,y_{n})$ be the
variables and $({\mathcal{A}}_{\bf x},{\mathcal{B}}_{\bf y})$ be a pair of
operators which act on meromorphic functions in x and y respectively. In [10],
$\Phi({\bf x};{\bf y})$ is called a kernel function for the pair
$({\mathcal{A}}_{\bf x},{\mathcal{B}}_{\bf y})$, if it satisfies a functional
equation of the form

(1.5) ${\mathcal{A}}_{\bf x}\Phi({\bf x};{\bf y})={\mathcal{B}}_{\bf
y}\Phi({\bf x};{\bf y}).$

Langmann studied the kernel functions systematically in the analysis of
eigenfunctions for quantum integrable systems such as the Calogero-Moser-
Sutherland system and the Inozemtsev system [11, 12]. Note that we can regard
the eigenvalue problem of the Inozemtsev system as a multi-variable
generalization of Heun’s differential equation (see [21] for a review). By
using the kernel function identity for the Inozemtsev system, we can obtain
integral transformations of eigenfunctions of the Inozemtsev system and
related systems [12]. The Ruijsenaars-van Diejen system (see [13, 25]) is a
relativistic (or discrete) quantum integrable system of the Inozemtsev type,
and kernel functions of them were studied by Ruijsenaars [14, 15] and Komori,
Noumi and Shiraishi [10]. Atai and Noumi [2] applied the kernel functions of
the Ruijsenaars-van Diejen system to the integral transformations of the
eigenfunctions.

In this paper, we obtain kernel function identities for $q$-deformations of
Heun’s differential equations and apply them to integral transformations.
Here, the $q$-Heun equation was introduced by Hahn [6] as the form

(1.6)
$\displaystyle\\{a_{2}x^{2}+a_{1}x+a_{0}\\}g(x/q)-\\{b_{2}x^{2}+b_{1}x+b_{0}\\}g(x)+\\{c_{2}x^{2}+c_{1}x+c_{0}\\}g(xq)=0,$

with the condition $a_{2}a_{0}c_{2}c_{0}\neq 0$. It was rediscovered in [23]
by considering degeneration of the Ruijsenaars-van Diejen system four times.
The $q$-Heun equation was obtained in [23] as an eigenfunction of the fourth
degeneration of the Ruijsenaars-van Diejen operator of one variable

(1.7) $\displaystyle A^{\langle
4\rangle}(x;h_{1},h_{2},l_{1},l_{2},\alpha_{1},\alpha_{2},\beta)$
$\displaystyle=x^{-1}(x-q^{h_{1}+1/2}t_{1})(x-q^{h_{2}+1/2}t_{2})T_{x}^{-1}+q^{\alpha_{1}+\alpha_{2}}x^{-1}(x-q^{l_{1}-1/2}t_{1})(x-q^{l_{2}-1/2}t_{2})T_{x}$
$\displaystyle-\\{(q^{\alpha_{1}}+q^{\alpha_{2}})x+q^{(h_{1}+h_{2}+l_{1}+l_{2}+\alpha_{1}+\alpha_{2})/2}(q^{\beta/2}+q^{-\beta/2})t_{1}t_{2}x^{-1}\\},$

where $T_{x}^{\pm 1}g(x)=g(q^{\pm 1}x)$. Namely, Eq. (1.6) admits an
expression as

(1.8) $A^{\langle
4\rangle}(x;h_{1},h_{2},l_{1},l_{2},\alpha_{1},\alpha_{2},\beta)g(x)=Eg(x),$

where $E$ is an arbitrary complex number. The eigenvalue $E$ corresponds to
the parameter $b_{1}$ in Eq. (1.6), and it plays a role of the accessory
parameter. As $q\to 1$, we essentially obtain the differential equation

(1.9)
$\displaystyle\frac{d^{2}y}{dz^{2}}+\left(\frac{\gamma}{z}+\frac{\delta}{z-t_{1}}+\frac{\epsilon}{z-t_{2}}\right)\frac{dy}{dz}+\frac{\alpha\beta
z-B}{z(z-t_{1})(z-t_{2})}y=0,$

which is equivalent to Heun’s differential equation. Other $q$-deformations of
Heun’s differential equations were introduced in [24] by focusing on the third
degenerate Ruijsenaars-van Diejen operator $A^{\langle
3\rangle}(x;h_{1},h_{2},h_{3},l_{1},l_{2},l_{3},\alpha,\beta)$ (see Eq.
(2.14)) and the second degenerate one $A^{\langle
2\rangle}(x;h_{1},h_{2},h_{3},h_{4},l_{1},l_{2},l_{3},l_{4},\alpha)$ (see Eq.
(2.15)), and they were called the variants of the $q$-Heun equation.

In this paper, we obtain kernel function identities for the $q$-Heun equation
and its variants. Set

(1.10)
$P^{(1)}_{\mu,\mu_{0}}(x,s)=\frac{(q^{\mu}s/x;q)_{\infty}}{(q^{\mu_{0}}s/x;q)_{\infty}},\quad(a;q)_{\infty}=\prod_{j=0}^{\infty}(1-q^{j}a).$

We obtain a kernel function identity for the $q$-Heun equation as follows;

######  Theorem 1.1.

If the parameters satisfy

(1.11)
$\displaystyle\chi=(\tilde{h}_{1}+\tilde{h}_{2}-\tilde{l}_{1}-\tilde{l}_{2}+\tilde{\alpha}_{1}-\tilde{\alpha}_{2}-\tilde{\beta})/2,\;\nu=\mu_{0}+\alpha_{1}-\tilde{\alpha}_{2},\;\mu=\mu_{0}+\chi+1,$
$\displaystyle\beta=-\tilde{\beta}-\chi,\;\alpha_{2}=\alpha_{1}+\tilde{\alpha}_{1}-\tilde{\alpha}_{2}-\chi,\;l_{i}=\tilde{h}_{i}+\mu_{0},\;h_{i}=\tilde{l}_{i}+\mu_{0}+\chi,\;(i=1,2),$

then the function
$\Phi(x,s)=x^{-\alpha_{1}}s^{1+\chi-\tilde{\alpha}_{1}}P_{\mu,\mu_{0}}(x,s)$
satisfies

(1.12) $\displaystyle A^{\langle
4\rangle}(x;h_{1},h_{2},l_{1},l_{2},\alpha_{1},\alpha_{2},\beta)\Phi(x,s)=q^{\nu}A^{\langle
4\rangle}(s;\tilde{h}_{1},\tilde{h}_{2},\tilde{l}_{1},\tilde{l}_{2},\tilde{\alpha}_{1},\tilde{\alpha}_{2},\tilde{\beta})\Phi(x,s).$

We also obtain kernel function identities for the variants of the $q$-Heun
equation in Theorems 2.2 and 2.3. As an application, we have $q$-integral
transformations of solutions to the $q$-Heun equation and its variants. A
$q$-integral transformation for the $q$-Heun equation was already obtained in
[18] by using the $q$-middle convolution which was introduced by Sakai and
Yamaguchi [16]. However, convergence of the $q$-integral transformation
associated with the $q$-middle convolution was not discussed in [16]. In this
paper, we obtain $q$-integral transformation without using the $q$-middle
convolution and we discuss the convergence directly. It seems that
$q$-integral transformations for the variants of the $q$-Heun equation have
not been known before, and we obtain them in this paper.

This paper is organized as follows. In section 2, we obtain identities for
kernel functions related to the $q$-Heun equation and its variants. In section
3, we formulate a method of finding a $q$-integral transformation by using a
kernel function and its identity. In section 4, we obtain $q$-integral
transformations for the $q$-Heun equation and its variants. In section 5, we
discuss special solutions of the $q$-Heun equation from the perspective of the
$q$-integral transformation. In section 6, we give concluding remarks.

Throughout this paper, we assume that $q$ is a complex number such that
$0<|q|<1$.

##  2\. Kernel functions related to $q$-Heun equations

We introduce functions which appear in the kernel function identities. Recall
that the function $P^{(1)}_{\mu,\mu_{0}}(x,s)$ was defined as

(2.1)
$P^{(1)}_{\mu,\mu_{0}}(x,s)=\frac{(q^{\mu}s/x;q)_{\infty}}{(q^{\mu_{0}}s/x;q)_{\infty}}.$

It satisfies

(2.2) $\displaystyle
P_{\mu,\mu_{0}}(x,s/q)=P_{\mu,\mu_{0}}(qx,s)=\frac{x-q^{\mu-1}s}{x-q^{\mu_{0}-1}s}P_{\mu,\mu_{0}}(x,s),$
$\displaystyle
P_{\mu,\mu_{0}}(x,qs)=P_{\mu,\mu_{0}}(x/q,s)=\frac{x-q^{\mu_{0}}s}{x-q^{\mu}s}P_{\mu,\mu_{0}}(x,s).$

The function

(2.3)
$P^{(2)}_{\mu,\mu_{0}}(x,s)=(x/s)^{\mu-\mu_{0}}\frac{(q^{-\mu_{0}+1}x/s;q)_{\infty}}{(q^{-\mu+1}x/s;q)_{\infty}}$

also satisfies Eq. (2.2). Set
$\vartheta_{q}(t)=(t,q/t,q;q)_{\infty}(=(t;q)_{\infty}(q/t;q)_{\infty}(q;q)_{\infty})$.
Then we have

(2.4)
$\frac{P^{(2)}_{\mu,\mu_{0}}(x,q^{n}\xi)}{P^{(1)}_{\mu,\mu_{0}}(x,q^{n}\xi)}=\frac{P^{(2)}_{\mu,\mu_{0}}(x,\xi)}{P^{(1)}_{\mu,\mu_{0}}(x,\xi)}=(x/\xi)^{\mu-\mu_{0}}\frac{\vartheta_{q}(q^{-\mu_{0}+1}x/\xi)}{\vartheta_{q}(q^{-\mu+1}x/\xi)}$

for $n\in\mathbb{Z}$. On the limit with respect to the variable $s$, we have

(2.5) $\lim_{s\to
0}P^{(1)}_{\mu,\mu_{0}}(x,s)=1,\;\lim_{s\to\infty}s^{\mu-\mu_{0}}P^{(2)}_{\mu,\mu_{0}}(x,s)=x^{\mu-\mu_{0}},$

and it follows from Eq. (2.4) that

(2.6)
$\displaystyle\lim_{L\to+\infty}P^{(2)}_{\mu,\mu_{0}}(x,s)|_{s=q^{L}\xi}=(x/\xi)^{\mu-\mu_{0}}\frac{\vartheta_{q}(q^{-\mu_{0}+1}x/\xi)}{\vartheta_{q}(q^{-\mu+1}x/\xi)},$
$\displaystyle\lim_{K\to-\infty}s^{\mu-\mu_{0}}P^{(1)}_{\mu,\mu_{0}}(x,s)|_{s=q^{K-1}\xi}=\xi^{\mu-\mu_{0}}\frac{\vartheta_{q}(q^{-\mu+1}x/\xi)}{\vartheta_{q}(q^{-\mu_{0}+1}x/\xi)}.$

Recall that the fourth degeneration of the Ruijsenaars-van Diejen operator of
one variable ia written as

(2.7) $\displaystyle A^{\langle
4\rangle}(x;h_{1},h_{2},l_{1},l_{2},\alpha_{1},\alpha_{2},\beta)$
$\displaystyle=x^{-1}(x-q^{h_{1}+1/2}t_{1})(x-q^{h_{2}+1/2}t_{2})T_{x}^{-1}+q^{\alpha_{1}+\alpha_{2}}x^{-1}(x-q^{l_{1}-1/2}t_{1})(x-q^{l_{2}-1/2}t_{2})T_{x}$
$\displaystyle-\\{(q^{\alpha_{1}}+q^{\alpha_{2}})x+q^{(h_{1}+h_{2}+l_{1}+l_{2}+\alpha_{1}+\alpha_{2})/2}(q^{\beta/2}+q^{-\beta/2})t_{1}t_{2}x^{-1}\\},$

where $T_{x}^{\pm 1}g(x)=g(q^{\pm 1}x)$, and it is related with the $q$-Heun
equation as explained in the introduction.

We obtain a kernel function and its identity for the operator $A^{\langle
4\rangle}(x;h_{1},h_{2},l_{1},l_{2},\alpha_{1},\alpha_{2},\beta)$ as follows.

######  Theorem 2.1.

If the parameters satisfy

(2.8)
$\displaystyle\chi=(\tilde{h}_{1}+\tilde{h}_{2}-\tilde{l}_{1}-\tilde{l}_{2}+\tilde{\alpha}_{1}-\tilde{\alpha}_{2}-\tilde{\beta})/2,\;\nu=\mu_{0}+\alpha_{1}-\tilde{\alpha}_{2},\;\mu=\mu_{0}+\chi+1,$
$\displaystyle\beta=-\tilde{\beta}-\chi,\;\alpha_{2}=\alpha_{1}+\tilde{\alpha}_{1}-\tilde{\alpha}_{2}-\chi,\;l_{i}=\tilde{h}_{i}+\mu_{0},\;h_{i}=\tilde{l}_{i}+\mu_{0}+\chi,\;(i=1,2),$

then we have

(2.9) $\displaystyle A^{\langle
4\rangle}(x;h_{1},h_{2},l_{1},l_{2},\alpha_{1},\alpha_{2},\beta)x^{-\alpha_{1}}s^{1+\chi-\tilde{\alpha}_{1}}P_{\mu,\mu_{0}}(x,s)$
$\displaystyle=q^{\nu}A^{\langle
4\rangle}(s;\tilde{h}_{1},\tilde{h}_{2},\tilde{l}_{1},\tilde{l}_{2},\tilde{\alpha}_{1},\tilde{\alpha}_{2},\tilde{\beta})x^{-\alpha_{1}}s^{1+\chi-\tilde{\alpha}_{1}}P_{\mu,\mu_{0}}(x,s),$

where $P_{\mu,\mu_{0}}(x,s)$ is a function which satisfies Eq. (2.2).

###### Proof.

Set

$\displaystyle a_{0}(x)=x^{-1}(x-q^{h_{1}+1/2}t_{1})(x-q^{h_{2}+1/2}t_{2}),$
$\displaystyle
b_{0}(x)=-(q^{\alpha_{1}}+q^{\alpha_{2}})x-q^{(h_{1}+h_{2}+l_{1}+l_{2}+\alpha_{1}+\alpha_{2})/2}(q^{\beta/2}+q^{-\beta/2})t_{1}t_{2}x^{-1},$
$\displaystyle
c_{0}(x)=q^{\alpha_{1}+\alpha_{2}}x^{-1}(x-q^{l_{1}-1/2}t_{1})(x-q^{l_{2}-1/2}t_{2}),$
$\displaystyle\tilde{a}_{0}(s)=s^{-1}(s-q^{\tilde{h}_{1}+1/2}t_{1})(s-q^{\tilde{h}_{2}+1/2}t_{2}),$
$\displaystyle\tilde{b}_{0}(s)=-(q^{\tilde{\alpha}_{1}}+q^{\tilde{\alpha}_{2}})s-q^{(\tilde{h}_{1}+\tilde{h}_{2}+\tilde{l}_{1}+\tilde{l}_{2}+\tilde{\alpha}_{1}+\tilde{\alpha}_{2})/2}(q^{\tilde{\beta}/2}+q^{-\tilde{\beta}/2})t_{1}t_{2}s^{-1},$
$\displaystyle\tilde{c}_{0}(s)=q^{\tilde{\alpha}_{1}+\tilde{\alpha}_{2}}s^{-1}(s-q^{\tilde{l}_{1}-1/2}t_{1})(s-q^{\tilde{l}_{2}-1/2}t_{2}).$

It follows from Eq. (2.2) that Eq. (2.9) is equivalent to the equation

(2.10)
$\displaystyle(q^{\alpha_{1}}a_{0}(x)-q^{\nu+1+\chi-\tilde{\alpha}_{1}}\tilde{c}_{0}(s))\frac{x-q^{\mu_{0}}s}{x-q^{\mu}s}$
$\displaystyle+(q^{-\alpha_{1}}c_{0}(x)-q^{\nu-1-\chi+\tilde{\alpha}_{1}}\tilde{a}_{0}(s))\frac{x-q^{\mu-1}s}{x-q^{\mu_{0}-1}s}+b_{0}(x)-q^{\nu}\tilde{b}_{0}(s)=0.$

By using the relations in Eq. (2.8), we have

(2.11) $\displaystyle
q^{-\alpha_{1}}c_{0}(x)-q^{\nu-1-\chi+\tilde{\alpha}_{1}}\tilde{a}_{0}(s)$
$\displaystyle=q^{\alpha_{2}}\\{x^{-1}(x-q^{l_{1}-1/2}t_{1})(x-q^{l_{2}-1/2}t_{2})-q^{\mu_{0}-1}s^{-1}(s-q^{l_{1}-\mu_{0}+1/2}t_{1})(s-q^{l_{2}-\mu_{0}+1/2}t_{2})\\}$
$\displaystyle=q^{\alpha_{2}}(x-q^{\mu_{0}-1}s)(1-q^{l_{1}+l_{2}-\mu_{0}}t_{1}t_{2}x^{-1}s^{-1}),$
$\displaystyle
q^{\alpha_{1}}a_{0}(x)-q^{\nu+1+\chi-\tilde{\alpha}_{1}}\tilde{c}_{0}(s)$
$\displaystyle=q^{\alpha_{1}}\\{x^{-1}(x-q^{h_{1}+1/2}t_{1})(x-q^{h_{2}+1/2}t_{2})-q^{\mu}s^{-1}(s-q^{h_{1}+1/2-\mu}t_{1})(s-q^{h_{2}+1/2-\mu}t_{2})\\}$
$\displaystyle=q^{\alpha_{1}}(x-q^{\mu}s)(1-q^{h_{1}+h_{2}+1-\mu}t_{1}t_{2}x^{-1}s^{-1}).$

Hence

(2.12)
$\displaystyle(q^{\alpha_{1}}a_{0}(x)-q^{\nu+1+\chi-\tilde{\alpha}_{1}}\tilde{c}_{0}(s))\frac{x-q^{\mu_{0}}s}{x-q^{\mu}s}+(q^{-\alpha_{1}}c_{0}(x)-q^{\nu-1-\chi+\tilde{\alpha}_{1}}\tilde{a}_{0}(s))\frac{x-q^{\mu-1}s}{x-q^{\mu_{0}-1}s}$
$\displaystyle=q^{\alpha_{1}}(x-q^{\mu_{0}}s)(1-q^{h_{1}+h_{2}+1-\mu}t_{1}t_{2}x^{-1}s^{-1})+q^{\alpha_{2}}(x-q^{\mu-1}s)(1-q^{l_{1}+l_{2}-\mu_{0}}t_{1}t_{2}x^{-1}s^{-1})$
$\displaystyle=(q^{\alpha_{1}}+q^{\alpha_{2}})x+(q^{\alpha_{1}+h_{1}+h_{2}+1+\mu_{0}-\mu}+q^{\alpha_{2}+l_{1}+l_{2}-\mu_{0}+\mu-1})t_{1}t_{2}x^{-1}$
$\displaystyle\quad-(q^{\alpha_{1}+\mu_{0}}+q^{\alpha_{2}+\mu-1})s-(q^{\alpha_{1}+h_{1}+h_{2}+1-\mu}+q^{\alpha_{2}+l_{1}+l_{2}-\mu_{0}})t_{1}t_{2}s^{-1},$

and we obtain Eq. (2.10) by using Eq. (2.8). ∎

Note that Eq. (2.8) is equivalent to

(2.13)
$\displaystyle\chi=(h_{1}+h_{2}-l_{1}-l_{2}+\alpha_{1}-\alpha_{2}-\beta)/2,\;\nu=\mu_{0}+\alpha_{1}-\tilde{\alpha}_{2},\;\mu=\mu_{0}+\chi+1,$
$\displaystyle\tilde{\beta}=-\beta-\chi,\;\tilde{\alpha}_{1}=\tilde{\alpha}_{2}+\alpha_{2}-\alpha_{1}+\chi,\;\tilde{h}_{i}=l_{i}-\mu_{0},\;\tilde{l}_{i}=h_{i}-\mu_{0}-\chi,\;(i=1,2).$

Theorem 1.1 in the introduction follows immediately from Theorem 2.1, because
the function $P^{(1)}_{\mu,\mu_{0}}(x,s)$ satisfies Eq. (2.2).

In order to define the variants of the $q$-Heun equation, we introduce the
following operators

(2.14) $\displaystyle A^{\langle
3\rangle}(x;h_{1},h_{2},h_{3},l_{1},l_{2},l_{3},\alpha,\beta)$
$\displaystyle=x^{-1}\prod_{n=1}^{3}(x-q^{h_{n}+1/2}t_{n})T_{x}^{-1}+q^{2\alpha+1}x^{-1}\prod_{n=1}^{3}(x-q^{l_{n}-1/2}t_{n})T_{x}$
$\displaystyle\quad+q^{\alpha+1/2}[-(q^{1/2}+q^{-1/2})x^{2}+\sum_{n=1}^{3}(q^{h_{n}}+q^{l_{n}})t_{n}x$
$\displaystyle\quad+q^{(l_{1}+l_{2}+l_{3}+h_{1}+h_{2}+h_{3})/2}(q^{\beta/2}+q^{-\beta/2})t_{1}t_{2}t_{3}x^{-1}],$
(2.15) $\displaystyle A^{\langle
2\rangle}(x;h_{1},h_{2},h_{3},h_{4},l_{1},l_{2},l_{3},l_{4},\alpha)$
$\displaystyle=x^{-2}\prod_{n=1}^{4}(x-q^{h_{n}+1/2}t_{n})T_{x}^{-1}+q^{2\alpha+1}x^{-2}\prod_{n=1}^{4}(x-q^{l_{n}-1/2}t_{n})T_{x}$
$\displaystyle\quad+q^{\alpha+1/2}\Big{[}-(q^{1/2}+q^{-1/2})x^{2}+\sum_{n=1}^{4}(q^{h_{n}}+q^{l_{n}})t_{n}x$
$\displaystyle\qquad+\prod_{n=1}^{4}q^{(h_{n}+l_{n})/2}t_{n}\Big{\\{}-(q^{1/2}+q^{-1/2})x^{-2}+\sum_{n=1}^{4}\Big{(}\frac{1}{q^{h_{n}}t_{n}}+\frac{1}{q^{l_{n}}t_{n}}\Big{)}x^{-1}\Big{\\}}\Big{]}.$

The third and the second degenerate Ruijsenaars-van Diejen operators of one
variable in [24] are realized as the case $\alpha=-1/2$ in Eqs. (2.14) and
(2.15), and conversely Eqs. (2.14) and (2.15) are obtained from the third and
the second degenerate Ruijsenaars-van Diejen operators of one variable in [24]
by appropriate gauge transformations. The variant of the $q$-Heun equation of
degree three is written as

(2.16) $\displaystyle A^{\langle
3\rangle}(x;h_{1},h_{2},h_{3},l_{1},l_{2},l_{3},\alpha,\beta)g(x)=Eg(x),\quad(E\in\mathbb{C}),$

and the variant of the $q$-Heun equation of degree three is written as

(2.17) $\displaystyle A^{\langle
2\rangle}(x;h_{1},h_{2},h_{3},h_{4},l_{1},l_{2},l_{3},l_{4},\alpha)g(x)=Eg(x),\quad(E\in\mathbb{C}).$

Note that the variant of the $q$-Heun equation of degree three is a
$q$-deformation of the second order Fuchsian differential equation with four
singularities $\\{t_{1},t_{2},t_{3},\infty\\}$ written as

(2.18)
$\displaystyle\frac{d^{2}y}{dz^{2}}+\left(\frac{\gamma}{z-t_{1}}+\frac{\delta}{z-t_{2}}+\frac{\epsilon}{z-t_{3}}\right)\frac{dy}{dz}+\frac{\alpha\beta
z-B}{(z-t_{1})(z-t_{2})(z-t_{3})}y=0,$

and the variant of the $q$-Heun equation of degree four is a $q$-deformation
of the second order Fuchsian differential equation with four singularities
$\\{t_{1},t_{2},t_{3},t_{4}\\}$.

We obtain a kernel function and its identity for the operator $A^{\langle
3\rangle}(x;h_{1},h_{2},h_{3},l_{1},l_{2},l_{3},\alpha,\beta)$ as follows.

######  Theorem 2.2.

If the parameters satisfy

(2.19)
$\displaystyle\chi=(\tilde{h}_{1}+\tilde{h}_{2}+\tilde{h}_{3}-\tilde{l}_{1}-\tilde{l}_{2}-\tilde{l}_{3}-\tilde{\beta})/2,\;\nu=2\mu_{0}+\alpha-\tilde{\alpha}+\chi,\;\mu=\mu_{0}+\chi+1,$
$\displaystyle\beta=-\tilde{\beta}-\chi,\;l_{i}=\tilde{h}_{i}+\mu_{0},\;h_{i}=\tilde{l}_{i}+\mu_{0}+\chi,\;(i=1,2,3),$

then we have

(2.20) $\displaystyle A^{\langle
3\rangle}(x;h_{1},h_{2},h_{3},l_{1},l_{2},l_{3},\alpha,\beta)x^{-\alpha}s^{\chi+1-\tilde{\alpha}}P_{\mu,\mu_{0}}(x,s)$
$\displaystyle=q^{\nu}A^{\langle
3\rangle}(s;\tilde{h}_{1},\tilde{h}_{2},\tilde{h}_{3},\tilde{l}_{1},\tilde{l}_{2},\tilde{l}_{3},\tilde{\alpha},\tilde{\beta})x^{-\alpha}s^{\chi+1-\tilde{\alpha}}P_{\mu,\mu_{0}}(x,s).$

Theorem 2.2 is shown similarly to Theorem 2.1. Note that Eq. (2.19) is
equivalent to

(2.21)
$\displaystyle\chi=(h_{1}+h_{2}+h_{3}-l_{1}-l_{2}-l_{3}-\beta)/2,\;\nu=2\mu_{0}+\alpha-\tilde{\alpha}+\chi,\;\mu=\mu_{0}+\chi+1,$
$\displaystyle\tilde{\beta}=-\beta-\chi,\;\tilde{h}_{i}=l_{i}-\mu_{0},\;\tilde{l}_{i}=h_{i}-\mu_{0}-\chi,\;(i=1,2,3).$

We also obtain a kernel function and its identity for the operator $A^{\langle
2\rangle}(x;h_{1},h_{2},h_{3},h_{4},l_{1},l_{2},l_{3},l_{4},\alpha)$ as
follows.

######  Theorem 2.3.

If the parameters satisfy

(2.22)
$\displaystyle\chi=(\tilde{h}_{1}+\tilde{h}_{2}+\tilde{h}_{3}+\tilde{h}_{4}-\tilde{l}_{1}-\tilde{l}_{2}-\tilde{l}_{3}-\tilde{l}_{4})/2,\;\nu=2\mu_{0}+\alpha-\tilde{\alpha}+\chi,\;\mu=\mu_{0}+\chi+1,$
$\displaystyle
l_{i}=\tilde{h}_{i}+\mu_{0},\;h_{i}=\tilde{l}_{i}+\mu_{0}+\chi,\;(i=1,2,3,4),$

then we have

(2.23) $\displaystyle A^{\langle
2\rangle}(x;h_{1},h_{2},h_{3},h_{4},l_{1},l_{2},l_{3},l_{4},\alpha)x^{-\alpha}s^{\chi+1-\tilde{\alpha}}P_{\mu,\mu_{0}}(x,s)$
$\displaystyle=q^{\nu}A^{\langle
2\rangle}(s;\tilde{h}_{1},\tilde{h}_{2},\tilde{h}_{3},\tilde{h}_{4},\tilde{l}_{1},\tilde{l}_{2},\tilde{l}_{3},\tilde{l}_{4},\tilde{\alpha})x^{-\alpha}s^{\chi+1-\tilde{\alpha}}P_{\mu,\mu_{0}}(x,s).$

Theorem 2.3 is also shown similarly to Theorem 2.1. Note that Eq. (2.22) is
equivalent to

(2.24)
$\displaystyle\chi=(h_{1}+h_{2}+h_{3}+h_{4}-l_{1}-l_{2}-l_{3}-l_{4})/2,\;\nu=2\mu_{0}+\alpha-\tilde{\alpha}+\chi,\;\mu=\mu_{0}+\chi+1,$
$\displaystyle\tilde{h}_{i}=l_{i}-\mu_{0},\;\tilde{l}_{i}=h_{i}-\mu_{0}-\chi,\;(i=1,2,3,4).$

##  3\. Kernel function and $q$-integral transformation

In this section, we obtain $q$-integral transformations by using the kernel
function.

Let $\xi\in\mathbb{C}\setminus\\{0\\}$. The infinite sum defined by integral

(3.1) $\int^{\xi\infty}_{0}f(s)\,d_{q}s=(1-q)\sum^{\infty}_{n=-\infty}q^{n}\xi
f(q^{n}\xi),$

is called the Jackson integral. It is known that the usual integral over
$(0,+\infty)$ is recovered as $q\to 1$ (see e.g. [4]).

######  Theorem 3.1.

Assume that the function $\Phi(x,s)$ satisfies

(3.2)
$\\{a(x)T_{x}^{-1}+b(x)+c(x)T_{x}-(\tilde{a}(s)T_{s}^{-1}+\tilde{b}(s)+\tilde{c}(s)T_{s})\\}\Phi(x,s)=0.$

If the function $h(s)$ satisfies

(3.3)
$\\{q\tilde{a}(qs)T_{s}+\tilde{b}(s)+q^{-1}\tilde{c}(s/q)T_{s}^{-1}\\}h(s)=0,$

the Jackson integral

(3.4) $g(x):=\int^{\xi\infty}_{0}h(s)\Phi(x,s)\,d_{q}s$

and $g(q^{\pm 1}x)$ converge, the limits

(3.5) $\displaystyle
g_{1}(x):=\lim_{L\to+\infty}qs\tilde{a}(qs)h(qs)\Phi(x,s)-s\tilde{c}(s)h(s)\Phi(x,qs)|_{s=q^{L}\xi}$
$\displaystyle
g_{2}(x):=\lim_{K\to-\infty}qs\tilde{a}(qs)h(qs)\Phi(x,s)-s\tilde{c}(s)h(s)\Phi(x,qs)|_{s=q^{K-1}\xi}$

converge, and the variable $\xi$ is independent of the variable $x$ or it is
proportional to $x$ (i.e. $\xi=Ax$ where $A$ is independent of $x$), then the
function $g(x)$ in Eq. (3.4) satisfies

(3.6)
$\displaystyle\\{a(x)T_{x}^{-1}+b(x)+c(x)T_{x}\\}g(x)=(1-q)(g_{2}(x)-g_{1}(x)).$

To obtain the theorem, we use the following proposition.

######  Proposition 3.2.

Assume that the function $\Phi(x,s)$ satisfies Eq. (3.2) and the function
$h(s)$ satisfies Eq. (3.3). Set

(3.7) $g^{[K,L]}(x)=(1-q)\sum^{L}_{n=K}sh(s)\Phi(x,s)|_{s=q^{n}\xi}.$

(i) If the variable $\xi$ is independent of the variable $x$, then we have

(3.8) $\displaystyle\\{a(x)T_{x}^{-1}+b(x)+c(x)T_{x}\\}g^{[K,L]}(x)$
$\displaystyle=-(1-q)[qs\tilde{a}(qs)h(qs)\Phi(x,s)-s\tilde{c}(s)h(s)\Phi(x,qs)]|_{s=q^{L}\xi}$
$\displaystyle\quad+(1-q)[qs\tilde{a}(qs)h(qs)\Phi(x,s)-s\tilde{c}(s)h(s)\Phi(x,qs)]|_{s=q^{K-1}\xi}.$

(ii) If $\xi=Ax$ and $A$ is independent of $x$, then we have

(3.9) $\displaystyle\\{a(x)T_{x}^{-1}+b(x)+c(x)T_{x}\\}g^{[K,L]}(x)$
$\displaystyle=-(1-q)[sa(x)h(s)\Phi(x/q,s)-qsc(x)h(qs)\Phi(qx,qs)]_{s=q^{L}Ax}$
$\displaystyle\quad+(1-q)[sa(x)h(s)\Phi(x/q,s)-qsc(x)h(qs)\Phi(qx,qs)]_{s=q^{K-1}Ax}$
$\displaystyle\quad-(1-q)[qs\tilde{a}(qs)h(qs)\Phi(x,s)-s\tilde{c}(s)h(s)\Phi(x,qs)]|_{s=q^{L}Ax}$
$\displaystyle\quad+(1-q)[qs\tilde{a}(qs)h(qs)\Phi(x,s)-s\tilde{c}(s)h(s)\Phi(x,qs)]|_{s=q^{K-1}Ax}$

###### Proof.

It follows from Eq. (3.2) that

(3.10)
$\displaystyle[a(x)\Phi(x/q,s)+b(x)\Phi(x,s)+c(x)\Phi(qx,s)]_{s=q^{n}\xi}$
$\displaystyle\quad=[\tilde{a}(s)\Phi(x,s/q)+\tilde{b}(s)\Phi(x,s)+\tilde{c}(s)\Phi(x,qs)]_{s=q^{n}\xi}.$

(i) If $\xi$ is independent $x$, then we have

(3.11)
$\displaystyle\\{a(x)T_{x}^{-1}+b(x)+c(x)T_{x}\\}\sum_{n=K}^{L}s\Phi(x,s)h(s)|_{s=q^{n}\xi}$
$\displaystyle=\sum_{n=K}^{L}[s\tilde{a}(s)\Phi(x,s/q)h(s)+s\tilde{b}(s)\Phi(x,s)h(s)+s\tilde{c}(s)\Phi(x,qs)h(s)]|_{s=q^{n}\xi}$
$\displaystyle=\sum_{n=K-1}^{L-1}qs\tilde{a}(qs)\Phi(x,s)h(qs)|_{s=q^{n}\xi}+\sum_{n=K}^{L}s\tilde{b}(s)\Phi(x,s)h(s)|_{s=q^{n}\xi}$
$\displaystyle\quad+\sum_{n=K+1}^{L+1}q^{-1}s\tilde{c}(s/q)\Phi(x,s)h(s/q)|_{s=q^{n}\xi}$
$\displaystyle=\sum_{n=K}^{L}[q\tilde{a}(qs)h(qs)+\tilde{b}(s)h(s)+q^{-1}\tilde{c}(s/q)h(s/q)]s\Phi(x,s)|_{s=q^{n}\xi}$
$\displaystyle\quad-
qs\tilde{a}(qs)\Phi(x,s)h(qs)|_{s=q^{L}\xi}+q^{-1}s\tilde{c}(s/q)\Phi(x,s)h(s/q)|_{s=q^{L+1}\xi}$
$\displaystyle\quad+qs\tilde{a}(qs)\Phi(x,s)h(qs)|_{s=q^{K-1}\xi}-q^{-1}s\tilde{c}(s/q)\Phi(x,s)h(s/q)|_{s=q^{K}\xi}.$

Therefore we obtain (i) by Eq. (3.3).

(ii) If $\xi=Ax$ and $A$ is independent of $x$, then it follows from Eq.
(3.10) that

(3.12)
$\displaystyle\\{a(x)T_{x}^{-1}+b(x)+c(x)T_{x}\\}\sum_{n=K}^{L}s\Phi(x,s)h(s)|_{s=q^{n}Ax}$
$\displaystyle=\sum_{n=K}^{L}\\{a(x)q^{n-1}Ax\Phi(x/q,q^{n-1}Ax)h(q^{n-1}Ax)+b(x)q^{n}Ax\Phi(x,q^{n}Ax)h(q^{n}Ax)$
$\displaystyle\qquad+c(x)q^{n+1}Ax\Phi(qx,q^{n+1}Ax)h(q^{n+1}Ax)\\}$
$\displaystyle=\sum_{n=K}^{L}sh(s)(a(x)\Phi(x/q,s)+b(x)\Phi(x,s)+c(x)\Phi(qx,s))|_{s=q^{n}Ax}$
$\displaystyle\quad+a(x)q^{K-1}Ax\Phi(x/q,q^{K-1}Ax)h(q^{K-1}Ax)-a(x)q^{L}Ax\Phi(x/q,q^{L}Ax)h(q^{L}Ax)$
$\displaystyle\quad-c(x)q^{K}Ax\Phi(qx,q^{K}Ax)h(q^{K}Ax)+c(x)q^{L+1}Ax\Phi(qx,q^{L+1}Ax)h(q^{L+1}Ax)$
$\displaystyle=\sum_{n=K}^{L}[s\tilde{a}(s)\Phi(x,s/q)h(s)+s\tilde{b}(s)\Phi(x,s)h(s)+s\tilde{c}(s)\Phi(x,qs)h(s)]|_{s=q^{n}Ax}$
$\displaystyle\quad-[sa(x)\Phi(x/q,s)h(s)-qsc(x)\Phi(qx,qs)h(qs)]_{s=q^{L}Ax}$
$\displaystyle\quad+[sa(x)\Phi(x/q,s)h(s)-qsc(x)\Phi(qx,qs)h(qs)]_{s=q^{K-1}Ax},$

where we applied Eq. (3.10) in the case $\xi=Ax$. Hence we obtain (ii) by
repeating the discussion in (i). ∎

We continue the proof of Theorem 3.1. If the variable $\xi$ is independent of
the variable $x$, then we obtain Theorem 3.1 from Eq. (3.8) as $L\to+\infty$
and $K\to-\infty$.

We consider the case $\xi=Ax$. The convergence of the Jackson integrals
$g(q^{\pm 1}x)$ is equivalent to the convergence of the summations

(3.13) $\displaystyle\sum_{n=0}^{L}s\Phi(q^{\pm
1}x,s)h(s)|_{s=q^{n}Ax},\quad\sum_{n=K}^{-1}s\Phi(q^{\pm
1}x,s)h(s)|_{s=q^{n}Ax}$

as $L\to+\infty$ and $K\to-\infty$, and it follows that

(3.14) $\displaystyle\lim_{L\to+\infty}s\Phi(q^{\pm
1}x,s)h(s)|_{s=q^{n}Ax}=0=\lim_{K\to-\infty}s\Phi(q^{\pm
1}x,s)h(s)|_{s=q^{n}Ax}.$

Therefore we have

(3.15)
$\displaystyle\lim_{n\to\pm\infty}[sa(x)\Phi(x/q,s)h(s)-qsc(x)\Phi(qx,qs)h(qs)]_{s=q^{n}Ax}=0,$

and we obtain Theorem 3.1 for the case $\xi=Ax$ from Eq. (3.9) as
$L\to+\infty$ and $K\to-\infty$.

##  4\. $q$-integral transformations for $q$-Heun equations and its variants

In this section, we obtain $q$-integral transformations for the $q$-Heun
equation and its variants by applying results in sections 2 and 3.

### 4.1.

Recall that the operator $A^{\langle
4\rangle}(x;h_{1},h_{2},l_{1},l_{2},\alpha_{1},\alpha_{2},\beta)$ was
introduced in Eq. (2.7), and the $q$-Heun equation was written as the equation
for the eigenfunction of it with the eigenvalue $E$. In Theorem 2.1, we
obtained the identity

(4.1) $\displaystyle A^{\langle
4\rangle}(x;h_{1},h_{2},l_{1},l_{2},\alpha_{1},\alpha_{2},\beta)x^{-\alpha_{1}}s^{1+\chi-\tilde{\alpha}_{1}}P_{\mu,\mu_{0}}(x,s)$
$\displaystyle=q^{\nu}A^{\langle
4\rangle}(s;\tilde{h}_{1},\tilde{h}_{2},\tilde{l}_{1},\tilde{l}_{2},\tilde{\alpha}_{1},\tilde{\alpha}_{2},\tilde{\beta})x^{-\alpha_{1}}s^{1+\chi-\tilde{\alpha}_{1}}P_{\mu,\mu_{0}}(x,s),$

where the parameters satisfy Eq. (2.8).

We now apply Theorem 3.1 for the $q$-Heun equation. Let $E$ and $\tilde{E}$ be
constants such that $E=q^{\nu}\tilde{E}$ and write

(4.2) $\displaystyle a(x)T_{x}^{-1}+b(x)+c(x)T_{x}=A^{\langle
4\rangle}(x;h_{1},h_{2},l_{1},l_{2},\alpha_{1},\alpha_{2},\beta)-E,$
$\displaystyle\tilde{a}(s)T_{s}^{-1}+\tilde{b}(s)+\tilde{c}(s)T_{s}=q^{\nu}\\{A^{\langle
4\rangle}(s;\tilde{h}_{1},\tilde{h}_{2},\tilde{l}_{1},\tilde{l}_{2},\tilde{\alpha}_{1},\tilde{\alpha}_{2},\tilde{\beta})-\tilde{E}\\}.$

Eq. (4.1) gives a realization of Eq. (3.2) by setting
$\Phi(x,s)=x^{-\alpha_{1}}s^{1+\chi-\tilde{\alpha}_{1}}P_{\mu,\mu_{0}}(x,s)$.
In this situation, the coefficients of the equation

(4.3)
$\\{q\tilde{a}(qs)T_{s}+\tilde{b}(s)+q^{-1}\tilde{c}(s/q)T_{s}^{-1}\\}h(s)=0$

is described as

(4.4) $\displaystyle
q\tilde{a}(qs)=q^{\nu+2}s^{-1}(s-q^{\tilde{h}_{1}-1/2}t_{1})(s-q^{\tilde{h}_{2}-1/2}t_{2}),$
$\displaystyle
q^{-1}\tilde{c}(s/q)=q^{\nu+\tilde{\alpha}_{1}+\tilde{\alpha}_{2}-2}s^{-1}(s-q^{\tilde{l}_{1}+1/2}t_{1})(s-q^{\tilde{l}_{2}+1/2}t_{2}),$
$\displaystyle\tilde{b}(s)=-q^{\nu}\\{(q^{\tilde{\alpha}_{1}}+q^{\tilde{\alpha}_{2}})s+q^{(\tilde{h}_{1}+\tilde{h}_{2}+\tilde{l}_{1}+\tilde{l}_{2}+\tilde{\alpha}_{1}+\tilde{\alpha}_{2})/2}(q^{\tilde{\beta}/2}+q^{-\tilde{\beta}/2})t_{1}t_{2}s^{-1}\\}-q^{\nu}\tilde{E}.$

Eq. (4.3) is equivalent to the equation $A^{\langle
4\rangle}(x;h^{\prime}_{1},h^{\prime}_{2},l^{\prime}_{1},l^{\prime}_{2},\alpha^{\prime}_{1},\alpha^{\prime}_{2},\beta^{\prime})h(s)=E^{\prime}h(s)$,
where

(4.5)
$\displaystyle\alpha^{\prime}_{i}=2-\tilde{\alpha}_{i},\;l^{\prime}_{i}=\tilde{h}_{i},\;h^{\prime}_{i}=\tilde{l}_{i},\;(i=1,2),\;\beta^{\prime}=\tilde{\beta},\;E^{\prime}=q^{2-\tilde{\alpha}_{1}-\tilde{\alpha}_{2}}\tilde{E}.$

Note that the exponents of Eq. (4.3) about $s=\infty$ are
$\\{\alpha^{\prime}_{1},\alpha^{\prime}_{2}\\}$ and the exponents about $s=0$
are $\\{\lambda^{\prime}_{+},\lambda^{\prime}_{-}\\}$, where
$\lambda^{\prime}_{\pm}=(h^{\prime}_{1}+h^{\prime}_{2}-l^{\prime}_{1}-l^{\prime}_{2}-\alpha^{\prime}_{1}-\alpha^{\prime}_{2}\pm\beta^{\prime}+2)/2$.
Hence, if $\alpha^{\prime}_{1}-\alpha^{\prime}_{2}\not\in\mathbb{Z}$ (resp.
$\beta^{\prime}\not\in\mathbb{Z}$), then there exist the solutions $h_{1}(s)$
and $h_{2}(s)$ (resp. $h_{3}(s)$ and $h_{4}(s)$) of Eq. (4.3) about $s=\infty$
(resp. $s=0$) such that $h_{1}(s)/s^{-\alpha^{\prime}_{1}}\to 1$ and
$h_{2}(s)/s^{-\alpha^{\prime}_{2}}\to 1$ as $s\to\infty$ (resp.
$h_{3}(s)/s^{\lambda^{\prime}_{+}}\to 1$ and
$h_{4}(s)/s^{\lambda^{\prime}_{-}}\to 1$ as $s\to 0$). See [24] for details of
the exponents. Then we obtain the following theorem from Theorem 3.1.

######  Theorem 4.1.

Assume that the function $h(s)$ satisfies

(4.6) $\displaystyle A^{\langle
4\rangle}(x;h^{\prime}_{1},h^{\prime}_{2},l^{\prime}_{1},l^{\prime}_{2},\alpha^{\prime}_{1},\alpha^{\prime}_{2},\beta^{\prime})h(s)=E^{\prime}h(s)$

and

(4.7)
$\displaystyle\lim_{L\to+\infty}\frac{h(s)}{s^{(h^{\prime}_{1}+h^{\prime}_{2}-l^{\prime}_{1}-l^{\prime}_{2}-\alpha^{\prime}_{1}-\alpha^{\prime}_{2}+\beta^{\prime}+2)/2}}\big{|}_{s=q^{L}\xi}=C_{1},\;\lim_{K\to-\infty}\frac{h(s)}{s^{-\alpha^{\prime}_{1}}}\big{|}_{s=q^{K}\xi}=C_{2}$

for some constants $E^{\prime},C_{1},C_{2}$. Then the Jackson integral

(4.8) $\displaystyle
g(x)=x^{-\alpha_{1}}\int^{\xi\infty}_{0}s^{-(h^{\prime}_{1}+h^{\prime}_{2}-l^{\prime}_{1}-l^{\prime}_{2}-\alpha^{\prime}_{1}-\alpha^{\prime}_{2}+\beta^{\prime}+2)/2}h(s)P^{(1)}_{\mu,\mu_{0}}(x,s)\,d_{q}s$

converges and it satisfies

(4.9) $\displaystyle A^{\langle
4\rangle}(x;h_{1},h_{2},l_{1},l_{2},\alpha_{1},\alpha_{2},\beta)g(x)=Eg(x)+(1-q)(g_{2}(x)-g_{1}(x)),$

where

(4.10) $\displaystyle
g_{1}(x)=C_{1}x^{-\alpha_{1}}q^{\mu_{0}+\alpha_{1}+h^{\prime}_{1}+h^{\prime}_{2}+\chi}(q^{\beta^{\prime}}-1)t_{1}t_{2},$
$\displaystyle
g_{2}(x)=C_{2}x^{-\alpha_{1}}\frac{\vartheta_{q}(q^{-\mu_{0}-\chi}x/\xi)}{\vartheta_{q}(q^{-\mu_{0}+1}x/\xi)}\xi^{\chi+1}q^{\mu_{0}+\alpha_{1}}(q^{\alpha^{\prime}_{2}-\alpha^{\prime}_{1}}-1),$
$\displaystyle\chi=(l^{\prime}_{1}+l^{\prime}_{2}-h^{\prime}_{1}-h^{\prime}_{2}-\alpha^{\prime}_{1}+\alpha^{\prime}_{2}-\beta^{\prime})/2,\;\mu=\mu_{0}+1+\chi,$
$\displaystyle
E=q^{\mu_{0}+\alpha_{1}-\alpha^{\prime}_{1}}E^{\prime},\;\beta=-\beta^{\prime}-\chi,\;\alpha_{2}=\alpha_{1}-\alpha^{\prime}_{1}+\alpha^{\prime}_{2}-\chi,$
$\displaystyle
l_{i}=l^{\prime}_{i}+\mu_{0},\;h_{i}=h^{\prime}_{i}+\mu_{0}+\chi,\;(i=1,2).$

In particular, if $C_{1}=C_{2}=0$, then we have

(4.11) $\displaystyle A^{\langle
4\rangle}(x;h_{1},h_{2},l_{1},l_{2},\alpha_{1},\alpha_{2},\beta)g(x)=Eg(x).$

###### Proof.

On the given parameters
$h^{\prime}_{1},h^{\prime}_{2},l^{\prime}_{1},l^{\prime}_{2},\alpha^{\prime}_{1},\alpha^{\prime}_{2},\beta^{\prime},E^{\prime}$,
we define the parameters
$\tilde{h}_{1},\tilde{h}_{2},\tilde{l}_{1},\tilde{l}_{2},\tilde{\alpha}_{1},\tilde{\alpha}_{2},\tilde{\beta},\tilde{E}$
by Eq. (4.5). Then it follows from Eq. (4.6) that the function $h(s)$
satisfies Eq. (4.3) with Eq. (4.4). By combining Eq. (4.10) with Eq. (4.5), we
obtain Eq. (2.8).

It follows from Eq. (4.7) that
$h(s)s^{1+\chi-\tilde{\alpha}_{1}}|_{s=q^{L}\xi}\to C_{1}$ as $L\to+\infty$.
On the first limit in Eq. (3.5), we have

(4.12) $\displaystyle
x^{-\alpha_{1}}[qs\tilde{a}(qs)h(qs)s^{1+\chi-\tilde{\alpha}_{1}}P^{(1)}_{\mu,\mu_{0}}(x,s)-s\tilde{c}(s)h(s)(qs)^{1+\chi-\tilde{\alpha}_{1}}P^{(1)}_{\mu,\mu_{0}}(x,qs)]|_{s=q^{L}\xi}$
$\displaystyle=q^{\nu}x^{-\alpha_{1}}[(qs-q^{\tilde{h}_{1}+1/2}t_{1})(qs-q^{\tilde{h}_{2}+1/2}t_{2})q^{-1-\chi+\tilde{\alpha}_{1}}P^{(1)}_{\mu,\mu_{0}}(x,s)h(qs)(qs)^{1+\chi-\tilde{\alpha}_{1}}$
$\displaystyle\quad-q^{\tilde{\alpha}_{1}+\tilde{\alpha}_{2}}(s-q^{\tilde{l}_{1}-1/2}t_{1})(s-q^{\tilde{l}_{2}-1/2}t_{2})q^{1+\chi-\tilde{\alpha}_{1}}P^{(1)}_{\mu,\mu_{0}}(x,qs)h(s)s^{1+\chi-\tilde{\alpha}_{1}}]|_{s=q^{L}\xi}$
$\displaystyle\to
C_{1}x^{-\alpha_{1}}q^{\nu}(q^{\tilde{h}_{1}+\tilde{h}_{2}-\chi+\tilde{\alpha}_{1}}-q^{\tilde{\alpha}_{1}+\tilde{\alpha}_{2}+\tilde{l}_{1}+\tilde{l}_{2}+\chi-\tilde{\alpha}_{1}})t_{1}t_{2}$
$\displaystyle=C_{1}x^{-\alpha_{1}}q^{\mu_{0}+\alpha_{1}+(\tilde{h}_{1}+\tilde{h}_{2}+\tilde{l}_{1}+\tilde{l}_{2}+\tilde{\alpha}_{1}-\tilde{\alpha}_{2})/2}(q^{\tilde{\beta}/2}-q^{-\tilde{\beta}/2})t_{1}t_{2}$

as $L\to+\infty$. Here we used $P^{(1)}_{\mu,\mu_{0}}(x,s)\to 1$ as $s\to 0$
(see Eq. (2.5)). Therefore the first limit in Eq. (3.5) converges and we have

(4.13)
$g_{1}(x)=C_{1}x^{-\alpha_{1}}q^{\mu_{0}+\alpha_{1}+(h^{\prime}_{1}+h^{\prime}_{2}+l^{\prime}_{1}+l^{\prime}_{2}-\alpha^{\prime}_{1}+\alpha^{\prime}_{2})/2}(q^{\beta^{\prime}/2}-q^{-\beta^{\prime}/2})t_{1}t_{2}.$

It also follows from Eq. (4.7) that
$h(s)s^{2-\tilde{\alpha}_{1}}|_{s=q^{K-1}\xi}\to C_{2}$ as $K\to-\infty$. On
the second limit in Eq. (3.5), we have

(4.14) $\displaystyle
x^{-\alpha_{1}}[qs\tilde{a}(qs)h(qs)s^{1+\chi-\tilde{\alpha}_{1}}P^{(1)}_{\mu,\mu_{0}}(x,s)-s\tilde{c}(s)h(s)(qs)^{1+\chi-\tilde{\alpha}_{1}}P^{(1)}_{\mu,\mu_{0}}(x,qs)]|_{s=q^{K-1}\xi}$
$\displaystyle=x^{-\alpha_{1}}(x/\xi)^{\mu_{0}-\mu}\frac{\vartheta_{q}(q^{-\mu+1}x/\xi)}{\vartheta_{q}(q^{-\mu_{0}+1}x/\xi)}[q^{\tilde{\alpha}_{1}-1}s^{-1}\tilde{a}(qs)s^{1+\chi}P^{(2)}_{\mu,\mu_{0}}(x,s)h(qs)(qs)^{2-\tilde{\alpha}_{1}}$
$\displaystyle\quad-s^{-1}\tilde{c}(s)q^{-\tilde{\alpha}_{1}}(qs)^{1+\chi}P^{(2)}_{\mu,\mu_{0}}(x,qs)h(s)s^{2-\tilde{\alpha}_{1}}]|_{s=q^{K-1}\xi}$
$\displaystyle\to
x^{-\alpha_{1}}(x/\xi)^{-1-\chi}\frac{\vartheta_{q}(q^{-\mu+1}x/\xi)}{\vartheta_{q}(q^{-\mu_{0}+1}x/\xi)}q^{\nu}(q^{\tilde{\alpha}_{1}}x^{1+\chi}C_{2}-q^{\tilde{\alpha}_{2}}x^{1+\chi}C_{2})$
$\displaystyle=C_{2}x^{-\alpha_{1}}\xi^{1+\chi}\frac{\vartheta_{q}(q^{-\mu_{0}-\chi}x/\xi)}{\vartheta_{q}(q^{-\mu_{0}+1}x/\xi)}q^{\mu_{0}+\alpha_{1}-\tilde{\alpha}_{2}}(q^{\tilde{\alpha}_{1}}-q^{\tilde{\alpha}_{2}})$

as $K\to-\infty$. Here we used Eq. (2.5). Therefore the second limit in Eq.
(3.5) converges and we have

(4.15)
$g_{2}(x)=C_{2}x^{-\alpha_{1}}\xi^{\chi+1}\frac{\vartheta_{q}(q^{-\mu_{0}-\chi}x/\xi)}{\vartheta_{q}(q^{-\mu_{0}+1}x/\xi)}q^{\mu_{0}+\alpha_{1}}(q^{\alpha^{\prime}_{2}-\alpha^{\prime}_{1}}-1).$

We show convergence of the Jackson integral in Eq. (4.8), which is equivalent
to convergence of the summation of the sequence $a_{n}$ over $n\in\mathbb{Z}$,
where

(4.16)
$a_{n}=sh(s)s^{1+\chi-\tilde{\alpha}_{1}}P^{(1)}_{\mu,\mu_{0}}(x,s)|_{s=q^{n}\xi}.$

Since $h(s)s^{1+\chi-\tilde{\alpha}_{1}}|_{s=q^{L}\xi}\to C_{1}$ as
$L\to+\infty$ and $P^{(1)}_{\mu,\mu_{0}}(x,s)\to 1$ as $s\to 0$, there exists
an integer $N_{1}$ and a positive number $C^{\prime}_{1}$ such that
$|a_{n}|\leq C^{\prime}_{1}q^{n}$ for any integer $n$ such that $n\geq N_{1}$.
Hence, we obtain convergence of the summation of $a_{n}$ over
$n\in\mathbb{Z}_{\geq 0}$. Eq. (4.16) is also expressed as

(4.17)
$a_{n}=(x/\xi)^{\mu_{0}-\mu}\frac{\vartheta_{q}(q^{-\mu+1}x/\xi)}{\vartheta_{q}(q^{-\mu_{0}+1}x/\xi)}s^{-1}h(s)s^{2-\tilde{\alpha}_{1}}s^{1+\chi}P^{(2)}_{\mu,\mu_{0}}(x,s)|_{s=q^{n}\xi}.$

Since $h(s)s^{2-\tilde{\alpha}_{1}}|_{s=q^{K-1}\xi}\to C_{2}$ as $K\to-\infty$
and $s^{\mu-\mu_{0}}P^{(2)}_{\mu,\mu_{0}}(x,s)\to x^{\mu-\mu_{0}}$ as
$s\to\infty$, there exists an integer $N_{2}$ and a positive number
$C^{\prime}_{2}$ such that $|a_{n}|\leq C^{\prime}_{2}q^{-n}$ for any integer
$n$ such that $n\leq N_{2}$. Hence, we obtain convergence of the summation of
$a_{n}$ over $n\in\mathbb{Z}_{\leq-1}$. Therefore, convergence of the Jackson
integral in Eq. (4.8) is shown.

Recall that Eq. (4.1) gives a realization of Eq. (3.2) and we have the
relation
$E=q^{\mu_{0}+\alpha_{1}-\alpha^{\prime}_{1}}E^{\prime}=q^{\nu}\tilde{E}$ by
Eqs. (4.5) and (4.10). Hence, we have confirmed the assumption of Theorem 3.1,
and we obtain that the function $g(x)$ satisfies Eq. (4.9). ∎

In Theorem 4.1, we can replace the Jackson integral in Eq. (4.8) with

(4.18) $\displaystyle
g(x)=x^{-\alpha_{1}}\int^{\xi\infty}_{0}s^{-(h^{\prime}_{1}+h^{\prime}_{2}-l^{\prime}_{1}-l^{\prime}_{2}-\alpha^{\prime}_{1}-\alpha^{\prime}_{2}+\beta^{\prime}+2)/2}h(s)P^{(2)}_{\mu,\mu_{0}}(x,s)\,d_{q}s.$

Namely, under the assumption of Theorem 4.1, the function $g(x)$ in Eq. (4.18)
converges and it satisfies

(4.19) $\displaystyle A^{\langle
4\rangle}(x;h_{1},h_{2},l_{1},l_{2},\alpha_{1},\alpha_{2},\beta)g(x)=Eg(x)+(1-q)(g_{2}(x)-g_{1}(x)),$

where

(4.20) $\displaystyle
g_{1}(x)=C_{1}x^{-\alpha_{1}+\chi+1}\xi^{-\chi-1}\frac{\vartheta_{q}(q^{-\mu_{0}+1}x/\xi)}{\vartheta_{q}(q^{-\mu_{0}-\chi}x/\xi)}q^{\mu_{0}+\alpha_{1}+h^{\prime}_{1}+h^{\prime}_{2}+\chi}(q^{\beta^{\prime}}-1)t_{1}t_{2},$
$\displaystyle
g_{2}(x)=C_{2}x^{-\alpha_{1}+\chi+1}q^{\mu_{0}+\alpha_{1}}(q^{\alpha^{\prime}_{2}-\alpha^{\prime}_{1}}-1),$
$\displaystyle\chi=(l^{\prime}_{1}+l^{\prime}_{2}-h^{\prime}_{1}-h^{\prime}_{2}-\alpha^{\prime}_{1}+\alpha^{\prime}_{2}-\beta^{\prime})/2,\;\mu=\mu_{0}+1+\chi,$
$\displaystyle
E=q^{\mu_{0}+\alpha_{1}-\alpha^{\prime}_{1}}E^{\prime},\;\beta=-\beta^{\prime}-\chi,\;\alpha_{2}=\alpha_{1}-\alpha^{\prime}_{1}+\alpha^{\prime}_{2}-\chi,$
$\displaystyle
l_{i}=l^{\prime}_{i}+\mu_{0},\;h_{i}=h^{\prime}_{i}+\mu_{0}+\chi,\;(i=1,2).$

This is proved by rewriting the proof of Theorem 4.1 while replacing
$P^{(2)}_{\mu,\mu_{0}}(x,s)$ with $P^{(1)}_{\mu,\mu_{0}}(x,s)$ by using Eq.
(2.4).

We specialize the parameters in Theorem 4.1 to reproduce Theorem 5.4 in [18].

######  Corollary 4.2.

Assume that
$h^{\prime}_{1}+h^{\prime}_{2}-l^{\prime}_{1}-l^{\prime}_{2}-\alpha^{\prime}_{1}-\alpha^{\prime}_{2}+\beta^{\prime}+2=0$
and the function $h(s)$ satisfies

(4.21) $\displaystyle A^{\langle
4\rangle}(x;h^{\prime}_{1},h^{\prime}_{2},l^{\prime}_{1},l^{\prime}_{2},\alpha^{\prime}_{1},\alpha^{\prime}_{2},\beta^{\prime})h(s)=E^{\prime}h(s)$

and

(4.22)
$\displaystyle\lim_{L\to+\infty}h(s)\big{|}_{s=q^{L}\xi}=0,\;\lim_{K\to-\infty}\frac{h(s)}{s^{-\alpha^{\prime}_{1}}}\big{|}_{s=q^{K}\xi}=0$

for some constants $E^{\prime}$. Then the Jackson integral

(4.23) $\displaystyle
g(x)=\int^{\xi\infty}_{0}h(s)P^{(1)}_{-\alpha^{\prime}_{1},0}(x,s)\,d_{q}s=\int^{\xi\infty}_{0}h(s)\frac{(q^{-\alpha^{\prime}_{1}}s/x;q)_{\infty}}{(s/x;q)_{\infty}}\,d_{q}s$

converges and it satisfies

(4.24) $\displaystyle A^{\langle
4\rangle}(x;h^{\prime}_{1}+1-\alpha^{\prime}_{1},h^{\prime}_{2}+1-\alpha^{\prime}_{1},l^{\prime}_{1},l^{\prime}_{2},0,\alpha^{\prime}_{2}-1,\beta^{\prime}+1-\alpha^{\prime}_{1})g(x)=q^{-\alpha^{\prime}_{1}}E^{\prime}g(x).$

###### Proof.

Set $\mu_{0}=0$ and $\alpha_{1}=0$ in Theorem 4.1. It follows from
$h^{\prime}_{1}+h^{\prime}_{2}-l^{\prime}_{1}-l^{\prime}_{2}-\alpha^{\prime}_{1}-\alpha^{\prime}_{2}+\beta^{\prime}+2=0$
that $\chi=1-\alpha^{\prime}_{1}$, and we obtain the corollary from Theorem
4.1. ∎

### 4.2.

We investigate a $q$-integral transformation for the variant of the $q$-Heun
equation of degree three. The operator $A^{\langle
3\rangle}(x;h_{1},h_{2},h_{3},l_{1},l_{2},l_{3},\alpha,\beta)$ in Eq. (2.14)
was used to write down the variant of the $q$-Heun equation of degree three.
In Theorem 2.2, we obtained the identity

(4.25) $\displaystyle A^{\langle
3\rangle}(x;h_{1},h_{2},h_{3},l_{1},l_{2},l_{3},\alpha,\beta)x^{-\alpha}s^{\chi+1-\tilde{\alpha}}P_{\mu,\mu_{0}}(x,s)$
$\displaystyle=q^{\nu}A^{\langle
3\rangle}(s;\tilde{h}_{1},\tilde{h}_{2},\tilde{h}_{3},\tilde{l}_{1},\tilde{l}_{2},\tilde{l}_{3},\tilde{\alpha},\tilde{\beta})x^{-\alpha}s^{\chi+1-\tilde{\alpha}}P_{\mu,\mu_{0}}(x,s),$

where the parameters satisfy Eq. (2.19). We now apply Theorem 3.1 for the
variant of the $q$-Heun equation of degree three. Let $E$ and $\tilde{E}$ be
constants such that $E=q^{\nu}\tilde{E}$ and write

(4.26) $\displaystyle a(x)T_{x}^{-1}+b(x)+c(x)T_{x}=A^{\langle
3\rangle}(x;h_{1},h_{2},h_{3},l_{1},l_{2},l_{3},\alpha,\beta)-E,$
$\displaystyle\tilde{a}(s)T_{s}^{-1}+\tilde{b}(s)+\tilde{c}(s)T_{s}=q^{\nu}\\{A^{\langle
3\rangle}(s;\tilde{h}_{1},\tilde{h}_{2},\tilde{h}_{3},\tilde{l}_{1},\tilde{l}_{2},\tilde{l}_{3},\tilde{\alpha},\tilde{\beta})-\tilde{E}\\}.$

Eq. (4.25) gives a realization of Eq. (3.2) by setting
$\Phi(x,s)=x^{-\alpha}s^{\chi+1-\tilde{\alpha}}P_{\mu,\mu_{0}}(x,s)$. In this
situation, the coefficients of the equation

(4.27)
$\\{q\tilde{a}(qs)T_{s}+\tilde{b}(s)+q^{-1}\tilde{c}(s/q)T_{s}^{-1}\\}h(s)=0$

are described as

(4.28) $\displaystyle
q\tilde{a}(qs)=q^{\nu+3}s^{-1}(s-q^{\tilde{h}_{1}-1/2}t_{1})(s-q^{\tilde{h}_{2}-1/2}t_{2})(s-q^{\tilde{h}_{3}-1/2}t_{3}),$
$\displaystyle
q^{-1}\tilde{c}(s/q)=q^{\nu+2\tilde{\alpha}-2}s^{-1}(s-q^{\tilde{l}_{1}+1/2}t_{1})(s-q^{\tilde{l}_{2}+1/2}t_{2})(s-q^{\tilde{l}_{3}+1/2}t_{3}),$
$\displaystyle\tilde{b}(s)=q^{\nu+\tilde{\alpha}+1/2}[-(q^{1/2}+q^{-1/2})s^{2}+\sum_{n=1}^{3}(q^{\tilde{h}_{n}}+q^{\tilde{l}_{n}})t_{n}s$
$\displaystyle\qquad+q^{(\tilde{l}_{1}+\tilde{l}_{2}+\tilde{l}_{3}+\tilde{h}_{1}+\tilde{h}_{2}+\tilde{h}_{3})/2}(q^{\tilde{\beta}/2}+q^{-\tilde{\beta}/2})t_{1}t_{2}t_{3}s^{-1}]-q^{\nu}\tilde{E}.$

Eq. (4.27) is equivalent to the equation $A^{\langle
3\rangle}(x;h^{\prime}_{1},h^{\prime}_{2},h^{\prime}_{3},l^{\prime}_{1},l^{\prime}_{2},l^{\prime}_{3},\alpha^{\prime},\beta^{\prime})h(s)=E^{\prime}h(s)$,
where

(4.29) $\displaystyle
l^{\prime}_{i}=\tilde{h}_{i},\;h^{\prime}_{i}=\tilde{l}_{i}\;(i=1,2,3),\;\alpha^{\prime}=2-\tilde{\alpha},\;\beta^{\prime}=\tilde{\beta},\;E^{\prime}=q^{2-2\tilde{\alpha}}\tilde{E}.$

Note that the exponents of Eq. (4.27) about $s=\infty$ are
$\\{\alpha^{\prime},\alpha^{\prime}+1\\}$ and the exponents about $s=0$ are
$\\{\lambda^{\prime}_{+},\lambda^{\prime}_{-}\\}$, where
$\lambda^{\prime}_{\pm}=(h^{\prime}_{1}+h^{\prime}_{2}+h^{\prime}_{3}-l^{\prime}_{1}-l^{\prime}_{2}-l^{\prime}_{3}-2\alpha^{\prime}\pm\beta^{\prime}+2)/2$.
The singular point $s=\infty$ is non-logarithmic. Then we obtain the following
theorem from Theorem 3.1.

######  Theorem 4.3.

Assume that the function $h(s)$ satisfies

(4.30) $\displaystyle A^{\langle
3\rangle}(x;h^{\prime}_{1},h^{\prime}_{2},h^{\prime}_{3},l^{\prime}_{1},l^{\prime}_{2},l^{\prime}_{3},\alpha^{\prime},\beta^{\prime})h(s)=E^{\prime}h(s)$

and

(4.31)
$\displaystyle\lim_{L\to+\infty}\frac{h(s)}{s^{(h^{\prime}_{1}+h^{\prime}_{2}+h^{\prime}_{3}-l^{\prime}_{1}-l^{\prime}_{2}-l^{\prime}_{3}-2\alpha^{\prime}+\beta^{\prime}+2)/2}}\big{|}_{s=q^{L}\xi}=0,\;\lim_{K\to-\infty}\frac{h(s)}{s^{-\alpha^{\prime}-1}}\big{|}_{s=q^{K}\xi}=0$

for some constant $E^{\prime}$. Then the Jackson integral

(4.32) $\displaystyle
g(x)=x^{-\alpha}\int^{\xi\infty}_{0}s^{-(h^{\prime}_{1}+h^{\prime}_{2}+h^{\prime}_{3}-l^{\prime}_{1}-l^{\prime}_{2}-l^{\prime}_{3}-2\alpha^{\prime}+\beta^{\prime}+2)/2}h(s)P^{(1)}_{\mu,\mu_{0}}(x,s)\,d_{q}s$

converges and it satisfies

(4.33) $\displaystyle A^{\langle
3\rangle}(x;h_{1},h_{2},h_{3},l_{1},l_{2},l_{3},\alpha,\beta)g(x)=Eg(x),$

where

(4.34)
$\displaystyle\chi=(l^{\prime}_{1}+l^{\prime}_{2}+l^{\prime}_{3}-h^{\prime}_{1}-h^{\prime}_{2}-h^{\prime}_{3}-\beta^{\prime})/2,\;E=q^{2\mu_{0}+\alpha-\alpha^{\prime}+\chi}E^{\prime},\;\mu=\mu_{0}+1+\chi,$
$\displaystyle\beta=-\beta^{\prime}-\chi,\;l_{i}=l^{\prime}_{i}+\mu_{0},\;h_{i}=h^{\prime}_{i}+\mu_{0}+\chi,\;(i=1,2,3).$

### 4.3.

We investigate a $q$-integral transformation for the variant of the $q$-Heun
equation of degree four. The operator $A^{\langle
2\rangle}(x;h_{1},h_{2},h_{3},h_{4},l_{1},l_{2},l_{3},l_{4},\alpha)$ in Eq.
(2.15) was used to write down the variant of the $q$-Heun equation of degree
four. In Theorem 2.3, we obtained the identity

(4.35) $\displaystyle A^{\langle
2\rangle}(x;h_{1},h_{2},h_{3},h_{4},l_{1},l_{2},l_{3},l_{4},\alpha)x^{-\alpha}s^{\chi+1-\tilde{\alpha}}P_{\mu,\mu_{0}}(x,s)$
$\displaystyle=q^{\nu}A^{\langle
2\rangle}(s;\tilde{h}_{1},\tilde{h}_{2},\tilde{h}_{3},\tilde{h}_{4},\tilde{l}_{1},\tilde{l}_{2},\tilde{l}_{3},\tilde{l}_{4},\tilde{\alpha})\\}x^{-\alpha}s^{\chi+1-\tilde{\alpha}}P_{\mu,\mu_{0}}(x,s),$

where the parameters satisfy Eq. (2.22). We now apply Theorem 3.1 for the
variant of the $q$-Heun equation of degree four. Let $E$ and $\tilde{E}$ be
constants such that $E=q^{\nu}\tilde{E}$ and write

(4.36) $\displaystyle a(x)T_{x}^{-1}+b(x)+c(x)T_{x}=A^{\langle
2\rangle}(x;h_{1},h_{2},h_{3},h_{4},l_{1},l_{2},l_{3},l_{4},\alpha)-E,$
$\displaystyle\tilde{a}(s)T_{s}^{-1}+\tilde{b}(s)+\tilde{c}(s)T_{s}=q^{\nu}\\{A^{\langle
2\rangle}(s;\tilde{h}_{1},\tilde{h}_{2},\tilde{h}_{3},\tilde{h}_{4},\tilde{l}_{1},\tilde{l}_{2},\tilde{l}_{3},\tilde{l}_{4},\tilde{\alpha})-\tilde{E}\\}.$

Eq. (4.35) gives a realization of Eq. (3.2) by setting
$\Phi(x,s)=x^{-\alpha}s^{\chi+1-\tilde{\alpha}}P_{\mu,\mu_{0}}(x,s)$. In this
situation, the coefficients of the equation

(4.37)
$\\{q\tilde{a}(qs)T_{s}+\tilde{b}(s)+q^{-1}\tilde{c}(s/q)T_{s}^{-1}\\}h(s)=0$

are described as

(4.38) $\displaystyle
q\tilde{a}(qs)=q^{\nu+3}s^{-2}(s-q^{\tilde{h}_{1}-1/2}t_{1})(s-q^{\tilde{h}_{2}-1/2}t_{2})(s-q^{\tilde{h}_{3}-1/2}t_{3})(s-q^{\tilde{h}_{4}-1/2}t_{4}),$
$\displaystyle
q^{-1}\tilde{c}(s/q)=q^{\nu+2\tilde{\alpha}-2}s^{-2}(s-q^{\tilde{l}_{1}+1/2}t_{1})(s-q^{\tilde{l}_{2}+1/2}t_{2})(s-q^{\tilde{l}_{3}+1/2}t_{3})(s-q^{\tilde{l}_{4}+1/2}t_{4}),$
$\displaystyle\tilde{b}(s)=q^{\nu+\tilde{\alpha}+1/2}\Big{[}-(q^{1/2}+q^{-1/2})s^{2}+\sum_{n=1}^{4}(q^{\tilde{h}_{n}}+q^{\tilde{l}_{n}})t_{n}s$
$\displaystyle\quad+\prod_{n=1}^{4}q^{(\tilde{h}_{n}+\tilde{l}_{n})/2}t_{n}\cdot\Big{\\{}-(q^{1/2}+q^{-1/2})s^{-2}+\sum_{n=1}^{4}\Big{(}\frac{1}{q^{\tilde{h}_{n}}t_{n}}+\frac{1}{q^{\tilde{l}_{n}}t_{n}}\Big{)}s^{-1}\Big{\\}}\Big{]}-q^{\nu}\tilde{E}.$

Eq. (4.37) is equivalent to the equation $A^{\langle
2\rangle}(x;h^{\prime}_{1},h^{\prime}_{2},h^{\prime}_{3},h^{\prime}_{4},l^{\prime}_{1},l^{\prime}_{2},l^{\prime}_{3},l^{\prime}_{4},\alpha^{\prime})h(s)=E^{\prime}h(s)$,
where

(4.39) $\displaystyle
l^{\prime}_{i}=\tilde{h}_{i},\;h^{\prime}_{i}=\tilde{l}_{i},\;(i=1,2,3,4),\;\alpha^{\prime}=2-\tilde{\alpha},\;E^{\prime}=q^{2-2\tilde{\alpha}}\tilde{E}.$

Note that the exponents of Eq. (4.37) about $s=\infty$ are
$\\{\alpha^{\prime},\alpha^{\prime}+1\\}$ and the exponents about $s=0$ are
$\\{\lambda^{\prime},\lambda^{\prime}+1\\}$, where
$\lambda^{\prime}=(h^{\prime}_{1}+h^{\prime}_{2}+h^{\prime}_{3}+h^{\prime}_{4}-l^{\prime}_{1}-l^{\prime}_{2}-l^{\prime}_{3}-l^{\prime}_{4}-2\alpha^{\prime}+2)/2$.
The singular points $s=0$ and $s=\infty$ are non-logarithmic. Then we obtain
the following theorem from Theorem 3.1.

######  Theorem 4.4.

Assume that the function $h(s)$ satisfies

(4.40) $\displaystyle A^{\langle
2\rangle}(x;h^{\prime}_{1},h^{\prime}_{2},h^{\prime}_{3},h^{\prime}_{4},l^{\prime}_{1},l^{\prime}_{2},l^{\prime}_{3},l^{\prime}_{4},\alpha^{\prime})h(s)=E^{\prime}h(s)$

and

(4.41)
$\displaystyle\lim_{L\to+\infty}\frac{h(s)}{s^{(h^{\prime}_{1}+h^{\prime}_{2}+h^{\prime}_{3}+h^{\prime}_{4}-l^{\prime}_{1}-l^{\prime}_{2}-l^{\prime}_{3}-l^{\prime}_{4}-2\alpha^{\prime}+4)/2}}\big{|}_{s=q^{L}\xi}=0,\;\lim_{K\to-\infty}\frac{h(s)}{s^{-\alpha^{\prime}-1}}\big{|}_{s=q^{K}\xi}=0$

for some constant $E^{\prime}$. Then the Jackson integral

(4.42) $\displaystyle
g(x)=x^{-\alpha}\int^{\xi\infty}_{0}s^{-(h^{\prime}_{1}+h^{\prime}_{2}+h^{\prime}_{3}+h^{\prime}_{4}-l^{\prime}_{1}-l^{\prime}_{2}-l^{\prime}_{3}-l^{\prime}_{4}-2\alpha^{\prime}+2)/2}h(s)P^{(1)}_{\mu,\mu_{0}}(x,s)\,d_{q}s$

converges and it satisfies

(4.43) $\displaystyle A^{\langle
2\rangle}(x;h_{1},h_{2},h_{3},h_{4},l_{1},l_{2},l_{3},l_{4},\alpha)g(x)=Eg(x),$

where

(4.44)
$\displaystyle\chi=(l^{\prime}_{1}+l^{\prime}_{2}+l^{\prime}_{3}+l^{\prime}_{4}-h^{\prime}_{1}-h^{\prime}_{2}-h^{\prime}_{3}-h^{\prime}_{4})/2,\;E=q^{2\mu_{0}+\alpha-\alpha^{\prime}+\chi}E^{\prime},$
$\displaystyle\mu=\mu_{0}+1+\chi,\;l_{i}=l^{\prime}_{i}+\mu_{0},\;h_{i}=h^{\prime}_{i}+\mu_{0}+\chi,\;(i=1,2,3,4).$

##  5\. Special solutions of the $q$-Heun equation

We investigate special solutions of the $q$-Heun equation. Recall that the
$q$-Heun equation is written as

(5.1) $A^{\langle
4\rangle}(x;h_{1},h_{2},l_{1},l_{2},\alpha_{1},\alpha_{2},\beta)g(x)=Eg(x),$

where $E$ is an arbitrary complex number and $A^{\langle
4\rangle}(x;h_{1},h_{2},l_{1},l_{2},\alpha_{1},\alpha_{2},\beta)$ is written
as

(5.2) $\displaystyle A^{\langle
4\rangle}(x;h_{1},h_{2},l_{1},l_{2},\alpha_{1},\alpha_{2},\beta)$
$\displaystyle=x^{-1}(x-q^{h_{1}+1/2}t_{1})(x-q^{h_{2}+1/2}t_{2})T_{x}^{-1}+q^{\alpha_{1}+\alpha_{2}}x^{-1}(x-q^{l_{1}-1/2}t_{1})(x-q^{l_{2}-1/2}t_{2})T_{x}$
$\displaystyle-\\{(q^{\alpha_{1}}+q^{\alpha_{2}})x+q^{(h_{1}+h_{2}+l_{1}+l_{2}+\alpha_{1}+\alpha_{2})/2}(q^{\beta/2}+q^{-\beta/2})t_{1}t_{2}x^{-1}\\}.$

### 5.1.

To investigate monomial solutions of the $q$-Heun equation, we look for
monomial eigenfunctions of the operator $A^{\langle
4\rangle}(x;h_{1},h_{2},l_{1},l_{2},\alpha_{1},\alpha_{2},\beta)$.

######  Proposition 5.1.

Set $(i,i^{\prime})=(1,2)$ or $(2,1)$. If
$\pm\beta=h_{1}+h_{2}-l_{1}-l_{2}+\alpha_{i}-\alpha_{i^{\prime}}+2$, then

(5.3) $\displaystyle A^{\langle
4\rangle}(x;h_{1},h_{2},l_{1},l_{2},\alpha_{1},\alpha_{2},\beta)x^{-\alpha_{i}}$
$\displaystyle=-(q^{\alpha_{i}+h_{1}+1/2}t_{1}+q^{\alpha_{i}+h_{2}+1/2}t_{2}+q^{\alpha_{i^{\prime}}+l_{1}-1/2}t_{1}+q^{\alpha_{i^{\prime}}+l_{2}-1/2}t_{2})x^{-\alpha_{i}}.$

###### Proof.

Let us find eigenfunctions of the operator $A^{\langle 4\rangle}$ in the form
$x^{\nu}$. We have

(5.4) $\displaystyle A^{\langle
4\rangle}(x;h_{1},h_{2},l_{1},l_{2},\alpha_{1},\alpha_{2},\beta)x^{\nu}$
$\displaystyle=(q^{-\nu}+q^{\alpha_{1}+\alpha_{2}+\nu}-q^{\alpha_{1}}-q^{\alpha_{2}})x^{\nu+1}$
$\displaystyle\quad+(q^{h_{1}+h_{2}+1-\nu}+q^{\alpha_{1}+\alpha_{2}+l_{1}+l_{2}-1+\nu}$
$\displaystyle\qquad-q^{(h_{1}+h_{2}+l_{1}+l_{2}+\alpha_{1}+\alpha_{2}+\beta)/2}-q^{(h_{1}+h_{2}+l_{1}+l_{2}+\alpha_{1}+\alpha_{2}-\beta)/2})t_{1}t_{2}x^{\nu-1}$
$\displaystyle\quad-(q^{-\nu+h_{1}+1/2}t_{1}+q^{-\nu+h_{2}+1/2}t_{2}+q^{\alpha_{1}+\alpha_{2}+\nu+l_{1}-1/2}t_{1}+q^{\alpha_{1}+\alpha_{2}+\nu+l_{2}-1/2}t_{2})x^{\nu}.$

If $\nu=-\alpha_{1}$ or $\nu=-\alpha_{2}$, then the coefficient of $x^{\nu+1}$
vanishes. In the case $\nu=-\alpha_{1}$ (resp. $\nu=-\alpha_{2}$), the
condition $\pm\beta=h_{1}+h_{2}-l_{1}-l_{2}+\alpha_{1}-\alpha_{2}+2$ (resp.
$\pm\beta=h_{1}+h_{2}-l_{1}-l_{2}-\alpha_{1}+\alpha_{2}+2$) is sufficient for
elimination of the term $x^{\nu-1}$. Then we obtain the expression in the
proposition. ∎

Note that we have

(5.5) $\displaystyle
x^{-1}[(x-q^{h_{1}+1/2}t_{1})(x-q^{h_{2}+1/2}t_{2})-q^{\alpha_{i^{\prime}}}(x-q^{l_{1}-1/2}t_{1})(x-q^{l_{2}-1/2}t_{2})T_{x}][T_{x}^{-1}-q^{\alpha_{i}}]$
$\displaystyle=A^{\langle
4\rangle}(x;h_{1},h_{2},l_{1},l_{2},\alpha_{1},\alpha_{2},\beta)$
$\displaystyle\qquad+q^{\alpha_{i}+h_{1}+1/2}t_{1}+q^{\alpha_{i}+h_{2}+1/2}t_{2}+q^{\alpha_{i^{\prime}}+l_{1}-1/2}t_{1}+q^{\alpha_{i^{\prime}}+l_{2}-1/2}t_{2},$

if $\pm\beta=h_{1}+h_{2}-l_{1}-l_{2}+\alpha_{i}-\alpha_{i^{\prime}}+2$, and
the function $x^{-\alpha_{i}}$ satisfies
$[T_{x}^{-1}-q^{\alpha_{i}}]x^{-\alpha_{i}}=0$.

On a gauge transformation, the following proposition is shown readily.

######  Proposition 5.2.

If the function $f(x)$ satisfies

$\displaystyle
x^{-1}(x-q^{l_{1}+1/2}t_{1})(x-q^{h_{2}+1/2}t_{2})f(x/q)+q^{\alpha_{1}+\alpha_{2}}x^{-1}(x-q^{h_{1}-1/2}t_{1})(x-q^{l_{2}-1/2}t_{2})f(qx)$
$\displaystyle-\\{(q^{\alpha_{1}}+q^{\alpha_{2}})x+q^{(h_{1}+h_{2}+l_{1}+l_{2}+\alpha_{1}+\alpha_{2})/2}(q^{\beta/2}+q^{-\beta/2})t_{1}t_{2}x^{-1}\\}f(x)=Ef(x)$

and

$\displaystyle
g_{1}(x)=\frac{(q^{h_{1}+1/2}t_{1}/x;q)_{\infty}}{(q^{l_{1}+1/2}t_{1}/x;q)_{\infty}}f(x),\;g_{2}(x)=x^{h_{1}-l_{1}}\frac{(x/(q^{l_{1}-1/2}t_{1});q)_{\infty}}{(x/(q^{h_{1}-1/2}t_{1});q)_{\infty}}f(x),$

then the functions $g_{1}(x)$ and $g_{2}(x)$ satisfy

$\displaystyle
x^{-1}(x-q^{h_{1}+1/2}t_{1})(x-q^{h_{2}+1/2}t_{2})g(x/q)+q^{\alpha_{1}+\alpha_{2}}x^{-1}(x-q^{l_{1}-1/2}t_{1})(x-q^{l_{2}-1/2}t_{2})g(qx)$
$\displaystyle-\\{(q^{\alpha_{1}}+q^{\alpha_{2}})x+q^{(h_{1}+h_{2}+l_{1}+l_{2}+\alpha_{1}+\alpha_{2})/2}(q^{\beta/2}+q^{-\beta/2})t_{1}t_{2}x^{-1}\\}g(x)=Eg(x).$

By applying gauge transformations to Proposition 5.1, we have the following
solutions of the $q$-Heun equation.

######  Proposition 5.3.

Set $(i,i^{\prime})=(1,2)$ or $(2,1)$.
(i) If $\pm\beta=-h_{1}+h_{2}+l_{1}-l_{2}+\alpha_{i}-\alpha_{i^{\prime}}+2$,
then

(5.6) $\displaystyle A^{\langle
4\rangle}(x;h_{1},h_{2},l_{1},l_{2},\alpha_{1},\alpha_{2},\beta)x^{-\alpha_{i}}\frac{(q^{h_{1}+1/2}t_{1}/x;q)_{\infty}}{(q^{l_{1}+1/2}t_{1}/x;q)_{\infty}}$
$\displaystyle=-(q^{\alpha_{i}+l_{1}+1/2}t_{1}+q^{\alpha_{i}+h_{2}+1/2}t_{2}+q^{\alpha_{i^{\prime}}+h_{1}-1/2}t_{1}+q^{\alpha_{i^{\prime}}+l_{2}-1/2}t_{2})$
$\displaystyle\qquad\cdot
x^{-\alpha_{i}}\frac{(q^{h_{1}+1/2}t_{1}/x;q)_{\infty}}{(q^{l_{1}+1/2}t_{1}/x;q)_{\infty}},$
$\displaystyle A^{\langle
4\rangle}(x;h_{1},h_{2},l_{1},l_{2},\alpha_{1},\alpha_{2},\beta)x^{-\alpha_{i}+h_{1}-l_{1}}\frac{(x/(q^{l_{1}-1/2}t_{1});q)_{\infty}}{(x/(q^{h_{1}-1/2}t_{1});q)_{\infty}}$
$\displaystyle=-(q^{\alpha_{i}+l_{1}+1/2}t_{1}+q^{\alpha_{i}+h_{2}+1/2}t_{2}+q^{\alpha_{i^{\prime}}+h_{1}-1/2}t_{1}+q^{\alpha_{i^{\prime}}+l_{2}-1/2}t_{2})$
$\displaystyle\qquad\cdot
x^{-\alpha_{i}+h_{1}-l_{1}}\frac{(x/(q^{l_{1}-1/2}t_{1});q)_{\infty}}{(x/(q^{h_{1}-1/2}t_{1});q)_{\infty}}.$

(ii) If $\pm\beta=-h_{1}-h_{2}+l_{1}+l_{2}+\alpha_{i}-\alpha_{i^{\prime}}+2$,
then

(5.7) $\displaystyle A^{\langle
4\rangle}(x;h_{1},h_{2},l_{1},l_{2},\alpha_{1},\alpha_{2},\beta)x^{-\alpha_{i}}\frac{(q^{h_{1}+1/2}t_{1}/x,q^{h_{2}+1/2}t_{2}/x;q)_{\infty}}{(q^{l_{1}+1/2}t_{1}/x,q^{l_{2}+1/2}t_{2}/x;q)_{\infty}}$
$\displaystyle=-(q^{\alpha_{i}+l_{1}+1/2}t_{1}+q^{\alpha_{i}+l_{2}+1/2}t_{2}+q^{\alpha_{i^{\prime}}+h_{1}-1/2}t_{1}+q^{\alpha_{i^{\prime}}+h_{2}-1/2}t_{2})$
$\displaystyle\qquad\cdot
x^{-\alpha_{i}}\frac{(q^{h_{1}+1/2}t_{1}/x,q^{h_{2}+1/2}t_{2}/x;q)_{\infty}}{(q^{l_{1}+1/2}t_{1}/x,q^{l_{2}+1/2}t_{2}/x;q)_{\infty}},$
$\displaystyle A^{\langle
4\rangle}(x;h_{1},h_{2},l_{1},l_{2},\alpha_{1},\alpha_{2},\beta)x^{-\alpha_{i}+h_{1}+h_{2}-l_{1}-l_{2}}\frac{(x/(q^{l_{1}-1/2}t_{1}),x/(q^{l_{2}-1/2}t_{2});q)_{\infty}}{(x/(q^{h_{1}-1/2}t_{1}),x/(q^{h_{2}-1/2}t_{2});q)_{\infty}}$
$\displaystyle=-(q^{\alpha_{i}+l_{1}+1/2}t_{1}+q^{\alpha_{i}+l_{2}+1/2}t_{2}+q^{\alpha_{i^{\prime}}+h_{1}-1/2}t_{1}+q^{\alpha_{i^{\prime}}+h_{2}-1/2}t_{2})$
$\displaystyle\qquad\cdot
x^{-\alpha_{i}+h_{1}+h_{2}-l_{1}-l_{2}}\frac{(x/(q^{l_{1}-1/2}t_{1}),x/(q^{l_{2}-1/2}t_{2});q)_{\infty}}{(x/(q^{h_{1}-1/2}t_{1}),x/(q^{h_{2}-1/2}t_{2});q)_{\infty}}.$

Note that, if
$\pm\beta=-h_{1}+h_{2}+l_{1}-l_{2}+\alpha_{i}-\alpha_{i^{\prime}}+2$, then

(5.8) $\displaystyle
x^{-1}[(x-q^{h_{2}+1/2}t_{2})-q^{\alpha_{i^{\prime}}-1}(x-q^{l_{2}-1/2}t_{2})T_{x}][(x-q^{h_{1}+1/2}t_{1})T_{x}^{-1}-q^{\alpha_{i}}(x-q^{l_{1}+1/2}t_{1})]$
$\displaystyle=A^{\langle
4\rangle}(x;h_{1},h_{2},l_{1},l_{2},\alpha_{1},\alpha_{2},\beta)$
$\displaystyle\qquad+q^{\alpha_{i}+l_{1}+1/2}t_{1}+q^{\alpha_{i}+h_{2}+1/2}t_{2}+q^{\alpha_{i^{\prime}}+h_{1}-1/2}t_{1}+q^{\alpha_{i^{\prime}}+l_{2}-1/2}t_{2},$

and, if $\pm\beta=-h_{1}-h_{2}+l_{1}+l_{2}+\alpha_{i}-\alpha_{i^{\prime}}+2$,
then

(5.9) $\displaystyle
x^{-1}[1-q^{\alpha_{i^{\prime}}-2}T_{x}][(x-q^{h_{1}+1/2}t_{1})(x-q^{h_{2}+1/2}t_{2})T_{x}^{-1}-q^{\alpha_{i}}(x-q^{l_{1}+1/2}t_{1})(x-q^{l_{2}+1/2}t_{2})]$
$\displaystyle=A^{\langle
4\rangle}(x;h_{1},h_{2},l_{1},l_{2},\alpha_{1},\alpha_{2},\beta)$
$\displaystyle\qquad+q^{\alpha_{i}+l_{1}+1/2}t_{1}+q^{\alpha_{i}+l_{2}+1/2}t_{2}+q^{\alpha_{i^{\prime}}+h_{1}-1/2}t_{1}+q^{\alpha_{i^{\prime}}+h_{2}-1/2}t_{2}.$

Next, we apply the $q$-integral transformations in Theorem 4.1 to some
solutions of the $q$-Heun equation given in Propositions 5.3 and 5.1.

### 5.2.

Assume that the parameters in $A^{\langle
4\rangle}(x;h^{\prime}_{1},h^{\prime}_{2},l^{\prime}_{1},l^{\prime}_{2},\alpha^{\prime}_{1},\alpha^{\prime}_{2},\beta^{\prime})$
satisfy

(5.10)
$\beta^{\prime}=-h^{\prime}_{1}+h^{\prime}_{2}+l^{\prime}_{1}-l^{\prime}_{2}-\alpha^{\prime}_{1}+\alpha^{\prime}_{2}+2.$

Then it follows from Proposition 5.3 (i) that

(5.11) $\displaystyle A^{\langle
4\rangle}(x;h^{\prime}_{1},h^{\prime}_{2},l^{\prime}_{1},l^{\prime}_{2},\alpha^{\prime}_{1},\alpha^{\prime}_{2},\beta^{\prime})x^{-\alpha^{\prime}_{2}+h^{\prime}_{1}-l^{\prime}_{1}}\frac{(x/(q^{l^{\prime}_{1}-1/2}t_{1});q)_{\infty}}{(x/(q^{h^{\prime}_{1}-1/2}t_{1});q)_{\infty}}$
$\displaystyle=-(q^{\alpha^{\prime}_{2}+l^{\prime}_{1}+1/2}t_{1}+q^{\alpha^{\prime}_{2}+h^{\prime}_{2}+1/2}t_{2}+q^{\alpha^{\prime}_{1}+h^{\prime}_{1}-1/2}t_{1}+q^{\alpha^{\prime}_{1}+l^{\prime}_{2}-1/2}t_{2})$
$\displaystyle\qquad\cdot
x^{-\alpha^{\prime}_{2}+h^{\prime}_{1}-l^{\prime}_{1}}\frac{(x/(q^{l^{\prime}_{1}-1/2}t_{1});q)_{\infty}}{(x/(q^{h^{\prime}_{1}-1/2}t_{1});q)_{\infty}}.$

We apply Theorem 4.1 in the case

(5.12) $\displaystyle
E^{\prime}=-(q^{\alpha^{\prime}_{2}+l^{\prime}_{1}+1/2}t_{1}+q^{\alpha^{\prime}_{2}+h^{\prime}_{2}+1/2}t_{2}+q^{\alpha^{\prime}_{1}+h^{\prime}_{1}-1/2}t_{1}+q^{\alpha^{\prime}_{1}+l^{\prime}_{2}-1/2}t_{2}),$
$\displaystyle
h(s)=s^{-\alpha^{\prime}_{2}+h^{\prime}_{1}-l^{\prime}_{1}}\frac{(s/(q^{l^{\prime}_{1}-1/2}t_{1});q)_{\infty}}{(s/(q^{h^{\prime}_{1}-1/2}t_{1});q)_{\infty}},\;\mu_{0}=0.$

It follows from Eq. (5.10) that the parameters in Theorem 4.1 satisfy

(5.13)
$\displaystyle\chi=-h^{\prime}_{2}+l^{\prime}_{2}-1,\;\mu=-h^{\prime}_{2}+l^{\prime}_{2},\;\alpha_{2}=\alpha_{1}-\alpha^{\prime}_{1}+\alpha^{\prime}_{2}+h^{\prime}_{2}-l^{\prime}_{2}+1,$
$\displaystyle\beta=h^{\prime}_{1}-l^{\prime}_{1}+\alpha^{\prime}_{1}-\alpha^{\prime}_{2}-1,\;l_{1}=l^{\prime}_{1},\;l_{2}=l^{\prime}_{2},\;h_{1}=h^{\prime}_{1}-h^{\prime}_{2}+l^{\prime}_{2}-1,\;h_{2}=l^{\prime}_{2}-1,$
$\displaystyle
E=-q^{\alpha_{1}}(q^{\alpha^{\prime}_{2}-\alpha^{\prime}_{1}+l^{\prime}_{1}+1/2}t_{1}+q^{\alpha^{\prime}_{2}-\alpha^{\prime}_{1}+h^{\prime}_{2}+1/2}t_{2}+q^{h^{\prime}_{1}-1/2}t_{1}+q^{l^{\prime}_{2}-1/2}t_{2}).$

In particular, we have $h_{2}=l_{2}-1$. The Jackson integral $g(x)$ in Eq.
(4.8) is written as

(5.14) $\displaystyle g(x)$
$\displaystyle=x^{-\alpha_{1}}\int^{\xi\infty}_{0}s^{-\beta^{\prime}}\frac{(s/(q^{l^{\prime}_{1}-1/2}t_{1});q)_{\infty}}{(s/(q^{h^{\prime}_{1}-1/2}t_{1});q)_{\infty}}\frac{(q^{-h^{\prime}_{2}+l^{\prime}_{2}}s/x;q)_{\infty}}{(s/x;q)_{\infty}}\,d_{q}s$
$\displaystyle=(1-q)\xi^{-\beta^{\prime}+1}x^{-\alpha_{1}}\frac{(q^{-l^{\prime}_{1}+1/2}\xi/t_{1},q^{-h^{\prime}_{2}+l^{\prime}_{2}}\xi/x;q)_{\infty}}{(q^{-h^{\prime}_{1}+1/2}\xi/t_{1},\xi/x;q)_{\infty}}$
$\displaystyle\qquad\cdot\sum^{\infty}_{n=-\infty}\frac{(q^{-h^{\prime}_{1}+1/2}\xi/t_{1},\xi/x;q)_{n}}{(q^{-l^{\prime}_{1}+1/2}\xi/t_{1},q^{-h^{\prime}_{2}+l^{\prime}_{2}}\xi/x;q)_{n}}q^{(-\beta^{\prime}+1)n}.$

If the parameters satisfy $\beta^{\prime}<0$ and
$\alpha^{\prime}_{1}<\alpha^{\prime}_{2}$, then the constants $C_{1}$ and
$C_{2}$ in Theorem 4.1 are equal to $0$ respectively, the Jackson integral
$g(x)$ in Eq. (5.14) converges and it satisfies

(5.15) $\displaystyle A^{\langle
4\rangle}(x;h_{1},h_{2},l_{1},l_{2},\alpha_{1},\alpha_{2},\beta)g(x)=Eg(x),$

where the parameters satisfy Eq. (5.13). We have

(5.16) $\displaystyle A^{\langle
4\rangle}(x;h_{1},h_{2},l_{1},l_{2},\alpha_{1},\alpha_{2},\beta)+q^{\alpha_{1}}(q^{\alpha^{\prime}_{2}-\alpha^{\prime}_{1}+l^{\prime}_{1}+1/2}t_{1}+q^{\alpha^{\prime}_{2}-\alpha^{\prime}_{1}+h^{\prime}_{2}+1/2}t_{2}+q^{h^{\prime}_{1}-1/2}t_{1}+q^{l^{\prime}_{2}-1/2}t_{2})$
$\displaystyle=x^{-1}(x-q^{l^{\prime}_{2}-1/2}t_{2})[(x-q^{h^{\prime}_{1}-h^{\prime}_{2}+l^{\prime}_{2}-1/2}t_{1})T_{x}^{-1}+q^{2\alpha_{1}-\alpha^{\prime}_{1}+\alpha^{\prime}_{2}+h^{\prime}_{2}-l^{\prime}_{2}+1}(x-q^{l^{\prime}_{1}-1/2}t_{1})T_{x}$
$\displaystyle\qquad-q^{\alpha_{1}}\\{(1+q^{-\alpha^{\prime}_{1}+\alpha^{\prime}_{2}+h^{\prime}_{2}-l^{\prime}_{2}+1})x-(q^{h^{\prime}_{1}-1/2}+q^{-\alpha^{\prime}_{1}+\alpha^{\prime}_{2}+l^{\prime}_{1}+1/2})t_{1}\\}],$

and we essentially obtain the $q$-hypergeometric equation. If
$\xi=q^{l^{\prime}_{1}+1/2}t_{1}$, then

(5.17) $\displaystyle g(x)$
$\displaystyle=(1-q)(q^{l^{\prime}_{1}+1/2}t_{1})^{-\beta^{\prime}+1}x^{-\alpha_{1}}\frac{(q,q^{-h^{\prime}_{2}+l^{\prime}_{2}+l^{\prime}_{1}+1/2}t_{1}/x;q)_{\infty}}{(q^{-h^{\prime}_{1}+l^{\prime}_{1}+1},q^{l^{\prime}_{1}+1/2}t_{1}/x;q)_{\infty}}$
(5.20) $\displaystyle\qquad\cdot\
_{2}\phi_{1}\Bigl{(}\begin{array}[]{c}q^{l^{\prime}_{1}+1/2}t_{1}/x,q^{-h^{\prime}_{1}+l^{\prime}_{1}+1}\\\
q^{-h^{\prime}_{2}+l^{\prime}_{2}+l^{\prime}_{1}+1/2}t_{1}/x\end{array};q,q^{-\beta^{\prime}+1}\Bigr{)}.$

If $\xi=q^{h^{\prime}_{2}-l^{\prime}_{2}+1}x$, then

(5.21) $\displaystyle g(x)$
$\displaystyle=(1-q)(q^{h^{\prime}_{2}-l^{\prime}_{2}+1}x)^{-\beta^{\prime}+1}x^{-\alpha_{1}}\frac{(q^{h^{\prime}_{2}-l^{\prime}_{1}-l^{\prime}_{2}+3/2}x/t_{1},q;q)_{\infty}}{(q^{-h^{\prime}_{1}+h^{\prime}_{2}-l^{\prime}_{2}+3/2}x/t_{1},q^{h^{\prime}_{2}-l^{\prime}_{2}+1};q)_{\infty}}$
(5.24) $\displaystyle\qquad\cdot\
_{2}\phi_{1}\Bigl{(}\begin{array}[]{c}q^{-h^{\prime}_{1}+h^{\prime}_{2}-l^{\prime}_{2}+3/2}x/t_{1},q^{h^{\prime}_{2}-l^{\prime}_{2}+1}\\\
q^{h^{\prime}_{2}-l^{\prime}_{1}-l^{\prime}_{2}+3/2}x/t_{1}\end{array};q,q^{-\beta^{\prime}+1}\Bigr{)}.$

These results agree with the $q$-integral representations of the
$q$-hypergeometric equation obtained in [1].

### 5.3.

Assume that the parameters in $A^{\langle
4\rangle}(x;h^{\prime}_{1},h^{\prime}_{2},l^{\prime}_{1},l^{\prime}_{2},\alpha^{\prime}_{1},\alpha^{\prime}_{2},\beta^{\prime})$
satisfy

(5.25)
$\beta^{\prime}=-h^{\prime}_{1}-h^{\prime}_{2}+l^{\prime}_{1}+l^{\prime}_{2}-\alpha^{\prime}_{1}+\alpha^{\prime}_{2}+2.$

Then it follows from Proposition 5.3 (ii) that

(5.26) $\displaystyle A^{\langle
4\rangle}(x;h^{\prime}_{1},h^{\prime}_{2},l^{\prime}_{1},l^{\prime}_{2},\alpha^{\prime}_{1},\alpha^{\prime}_{2},\beta^{\prime})x^{-\alpha^{\prime}_{2}+h^{\prime}_{1}+h^{\prime}_{2}-l^{\prime}_{1}-l^{\prime}_{2}}\frac{(x/(q^{l^{\prime}_{1}-1/2}t_{1},x/(q^{l^{\prime}_{2}-1/2}t_{2});q)_{\infty}}{(x/(q^{h^{\prime}_{1}-1/2}t_{1}),x/(q^{h^{\prime}_{2}-1/2}t_{2});q)_{\infty}}$
$\displaystyle=-(q^{\alpha^{\prime}_{2}+l^{\prime}_{1}+1/2}t_{1}+q^{\alpha^{\prime}_{2}+l^{\prime}_{2}+1/2}t_{2}+q^{\alpha^{\prime}_{1}+h^{\prime}_{1}-1/2}t_{1}+q^{\alpha^{\prime}_{1}+h^{\prime}_{2}-1/2}t_{2})$
$\displaystyle\qquad\cdot
x^{-\alpha^{\prime}_{2}+h^{\prime}_{1}+h^{\prime}_{2}-l^{\prime}_{1}-l^{\prime}_{2}}\frac{(x/(q^{l^{\prime}_{1}-1/2}t_{1}),x/(q^{l^{\prime}_{2}-1/2}t_{2});q)_{\infty}}{(x/(q^{h^{\prime}_{1}-1/2}t_{1}),x/(q^{h^{\prime}_{2}-1/2}t_{2});q)_{\infty}}.$

We apply Theorem 4.1 in the case

(5.27) $\displaystyle
E^{\prime}=-(q^{\alpha^{\prime}_{2}+l^{\prime}_{1}+1/2}t_{1}+q^{\alpha^{\prime}_{2}+l^{\prime}_{2}+1/2}t_{2}+q^{\alpha^{\prime}_{1}+h^{\prime}_{1}-1/2}t_{1}+q^{\alpha^{\prime}_{1}+h^{\prime}_{2}-1/2}t_{2}),$
$\displaystyle
h(s)=s^{-\alpha^{\prime}_{2}+h^{\prime}_{1}+h^{\prime}_{2}-l^{\prime}_{1}-l^{\prime}_{2}}\frac{(s/(q^{l^{\prime}_{1}-1/2}t_{1}),s/(q^{l^{\prime}_{2}-1/2}t_{2});q)_{\infty}}{(s/(q^{h^{\prime}_{1}-1/2}t_{1}),s/(q^{h^{\prime}_{2}-1/2}t_{2});q)_{\infty}},\;\mu_{0}=0.$

It follows from Eq. (5.25) that the parameters in Theorem 4.1 satisfy

(5.28)
$\displaystyle\chi=-\beta^{\prime}-1=(-l_{1}-l_{2}+h_{1}+h_{2}+\alpha_{1}-\alpha_{2}-1)/2,\;\mu=\chi+1,$
$\displaystyle\beta=1,\;\alpha_{2}=\alpha_{1}-\alpha^{\prime}_{1}+\alpha^{\prime}_{2}-\chi,\;l_{1}=l^{\prime}_{1},\;l_{2}=l^{\prime}_{2},\;h_{1}=h^{\prime}_{1}+\chi,\;h_{2}=h^{\prime}_{2}+\chi,$
$\displaystyle
E=-q^{\alpha_{1}}(q^{\alpha^{\prime}_{2}-\alpha^{\prime}_{1}+l^{\prime}_{1}+1/2}t_{1}+q^{\alpha^{\prime}_{2}-\alpha^{\prime}_{1}+l^{\prime}_{2}+1/2}t_{2}+q^{h^{\prime}_{1}-1/2}t_{1}+q^{h^{\prime}_{2}-1/2}t_{2}).$

The Jackson integral $g(x)$ in Eq. (4.8) is written as

(5.29) $\displaystyle g(x)$
$\displaystyle=x^{-\alpha_{1}}\int^{\xi\infty}_{0}\frac{(s/(q^{l^{\prime}_{1}-1/2}t_{1}),s/(q^{l^{\prime}_{2}-1/2}t_{2});q)_{\infty}}{(s/(q^{h^{\prime}_{1}-1/2}t_{1}),q^{h^{\prime}_{2}-1/2}t_{2});q)_{\infty}}\frac{(q^{\mu}s/x;q)_{\infty}}{(s/x;q)_{\infty}}\,d_{q}s$
$\displaystyle=(1-q)\xi
x^{-\alpha_{1}}\frac{(q^{-l_{1}+1/2}\xi/t_{1},q^{-l_{2}+1/2}\xi/t_{2},q^{\chi+1}\xi/x;q)_{\infty}}{(q^{-h_{1}+\chi+1/2}\xi/t_{1},q^{-h_{2}+\chi+1/2}\xi/t_{2},\xi/x;q)_{\infty}}$
$\displaystyle\qquad\cdot\sum^{\infty}_{n=-\infty}\frac{(q^{-h_{1}+\chi+1/2}\xi/t_{1},q^{-h_{2}+\chi+1/2}\xi/t_{2},\xi/x;q)_{n}}{(q^{-l_{1}+1/2}\xi/t_{1},q^{-l_{2}+1/2}\xi/t_{2},q^{\chi+1}\xi/x;q)_{n}}q^{n}.$

If the parameters satisfy $\alpha^{\prime}_{1}<\alpha^{\prime}_{2}$, then the
constants $C_{1}$ and $C_{2}$ in Theorem 4.1 satisfy $C_{1}=1$ and $C_{2}=0$,
the Jackson integral $g(x)$ in Eq. (5.29) converges and it satisfies

(5.30) $\displaystyle A^{\langle
4\rangle}(x;h_{1},h_{2},l_{1},l_{2},\alpha_{1},\alpha_{2},\beta)g(x)=Eg(x)-(1-q)x^{-\alpha_{1}}q^{\alpha_{1}+h_{1}+h_{2}-\chi}(q^{-\chi-1}-1)t_{1}t_{2}.$

Note that the homogeneous version of Eq. (5.30) is written in the variant of
the $q$-hypergeometric equation of degree two (see [5] for the definition),
and our results agree with the results obtained in [1].

### 5.4.

Assume that the parameters in $A^{\langle
4\rangle}(x;h^{\prime}_{1},h^{\prime}_{2},l^{\prime}_{1},l^{\prime}_{2},\alpha^{\prime}_{1},\alpha^{\prime}_{2},\beta^{\prime})$
satisfy

(5.31)
$\beta^{\prime}=h^{\prime}_{1}+h^{\prime}_{2}-l^{\prime}_{1}-l^{\prime}_{2}-\alpha^{\prime}_{1}+\alpha^{\prime}_{2}+2.$

Then it follows from Proposition 5.1 that

(5.32) $\displaystyle A^{\langle
4\rangle}(x;h^{\prime}_{1},h^{\prime}_{2},l^{\prime}_{1},l^{\prime}_{2},\alpha^{\prime}_{1},\alpha^{\prime}_{2},\beta^{\prime})x^{-\alpha^{\prime}_{2}}$
$\displaystyle=-(q^{\alpha^{\prime}_{2}+h^{\prime}_{1}+1/2}t_{1}+q^{\alpha^{\prime}_{2}+h^{\prime}_{2}+1/2}t_{2}+q^{\alpha^{\prime}_{1}+l^{\prime}_{1}-1/2}t_{1}+q^{\alpha^{\prime}_{1}+l^{\prime}_{2}-1/2}t_{2})x^{-\alpha^{\prime}_{2}}.$

We apply Theorem 4.1 in the case

(5.33) $\displaystyle
E^{\prime}=-(q^{\alpha^{\prime}_{2}+h^{\prime}_{1}+1/2}t_{1}+q^{\alpha^{\prime}_{2}+h^{\prime}_{2}+1/2}t_{2}+q^{\alpha^{\prime}_{1}+l^{\prime}_{1}-1/2}t_{1}+q^{\alpha^{\prime}_{1}+l^{\prime}_{2}-1/2}t_{2}),$
$\displaystyle h(s)=s^{-\alpha^{\prime}_{2}},\;\mu_{0}=0.$

It follows from Eq. (5.31) that the parameters in Theorem 4.1 satisfy

(5.34)
$\displaystyle\chi=-\beta^{\prime}-\alpha^{\prime}_{1}+\alpha^{\prime}_{2}+1,\;\mu=\chi+1,\;\alpha_{2}=\alpha_{1}+\beta^{\prime}-1,$
$\displaystyle\beta=\alpha^{\prime}_{1}-\alpha^{\prime}_{2}-1,\;l_{1}=l^{\prime}_{1},\;l_{2}=l^{\prime}_{2},\;h_{1}=h^{\prime}_{1}+\chi,\;h_{2}=h^{\prime}_{2}+\chi,$
$\displaystyle
E=-q^{\alpha_{1}}(q^{\alpha^{\prime}_{2}-\alpha^{\prime}_{1}+h^{\prime}_{1}+1/2}t_{1}+q^{\alpha^{\prime}_{2}-\alpha^{\prime}_{1}+h^{\prime}_{2}+1/2}t_{2}+q^{\alpha^{\prime}_{1}+l^{\prime}_{1}-1/2}t_{1}+q^{\alpha^{\prime}_{1}+l^{\prime}_{2}-1/2}t_{2}).$

In particular, we have
$\beta+h_{1}+h_{2}-l_{1}-l_{2}-\alpha_{1}+\alpha_{2}+2=0$. The Jackson
integral $g(x)$ in Eq. (4.8) is written as

(5.35) $\displaystyle g(x)$
$\displaystyle=x^{-\alpha_{1}}\int^{\xi\infty}_{0}s^{-\beta^{\prime}}\frac{(q^{-\beta^{\prime}-\alpha^{\prime}_{1}+\alpha^{\prime}_{2}+2}s/x;q)_{\infty}}{(s/x;q)_{\infty}}\,d_{q}s$
$\displaystyle=(1-q)\xi^{-\beta^{\prime}+1}x^{-\alpha_{1}}\frac{(q^{-\beta^{\prime}-\alpha^{\prime}_{1}+\alpha^{\prime}_{2}+2}\xi/x;q)_{\infty}}{(\xi/x;q)_{\infty}}\sum^{\infty}_{n=-\infty}\frac{(\xi/x;q)_{n}}{(q^{-\beta^{\prime}-\alpha^{\prime}_{1}+\alpha^{\prime}_{2}+2}\xi/x;q)_{n}}q^{(-\beta^{\prime}+1)n}.$

If the parameters satisfy $\beta^{\prime}<0$ and
$\alpha^{\prime}_{1}<\alpha^{\prime}_{2}$, then the constants $C_{1}$ and
$C_{2}$ in Theorem 4.1 are equal to $0$ respectively, the Jackson integral
$g(x)$ in Eq. (5.35) converges and it satisfies

(5.36) $\displaystyle A^{\langle
4\rangle}(x;h_{1},h_{2},l_{1},l_{2},\alpha_{1},\alpha_{2},\beta)g(x)=Eg(x),$

where the parameters satisfy Eq. (5.34).

Recall that the Ramanujan’s sum for $\\!{}_{1}\psi_{1}(a;b;q,z)$ (the
bilateral summation formula) is written as

(5.37)
$\displaystyle\sum_{n=-\infty}^{\infty}\frac{(a;q)_{n}}{(b;q)_{n}}z^{n}=\frac{(q,b/a,az,q/(az);q)_{\infty}}{(b,q/a,z,b/(az);q)_{\infty}}$

for $|b/a|<|z|<1$ and $|q|<1$. It follows from Eq. (5.37) in the case
$a=\xi/x$,
$b=q^{-\beta^{\prime}-\alpha^{\prime}_{1}+\alpha^{\prime}_{2}+2}\xi/x$,
$z=q^{-\beta^{\prime}+1}$ that the function $g(x)$ in Eq. (5.35) is written as

(5.38) $\displaystyle
g(x)=x^{-\alpha_{1}}\frac{(q^{-\beta^{\prime}+1}\xi/x,x/(q^{-\beta^{\prime}}\xi),q^{-\beta^{\prime}-\alpha^{\prime}_{1}+\alpha^{\prime}_{2}+2},q;q)_{\infty}}{(\xi/x,qx/\xi,q^{-\beta^{\prime}+1},q^{-\alpha^{\prime}_{1}+\alpha^{\prime}_{2}+1};q)_{\infty}}.$

Since
$\vartheta_{q}(qax)/\vartheta_{q}(qbx)=(b/a)\vartheta_{q}(ax)/\vartheta_{q}(bx)$,
the function $g(x)$ in Eq. (5.38) behaves the same as $x^{-\alpha_{2}}$ on the
transformation $x\mapsto qx$. Therefore, we may conclude that the $q$-integral
transformation on this case produces another monomial solution from a monomial
solution.

##  6\. Concluding remarks

In this paper, we found kernel function identities for the $q$-Heun equation
and the variants of the $q$-Heun equation of degree three and four. We applied
them to obtain $q$-integral transformations of solutions to the $q$-Heun
equation and its variants. Moreover, we investigated special solutions of the
$q$-Heun equation from the perspective of the $q$-integral transformation.

We give some comments on issues related to the results in this paper.

The variants of the $q$-Heun equation was found by considering degenerations
of the Ruijsenaars-van Diejen system, and kernel functions of the Ruijsenaars-
van Diejen system had been found in [14, 15, 10]. On the other hand, we found
kernel functions of the $q$-Heun equation and its variants directly in this
paper. It is expected to obtain degenerations of kernel functions from the
non-degenerate Ruijsenaars-van Diejen system, although calculation would be
extremely complicated without nice ideas. Extension of our identities of
kernel functions to the multivariable cases is also expected.

We investigated $q$-integral transformations of monomial-type solutions of the
$q$-Heun equation in section 5. Polynomial solutions of the $q$-Heun equation
and its variants were investigated in [24, 9], and it is expected to study
$q$-integral transformations related to the polynomial-type solutions. Results
in [22] for Heun’s differential equation might be helpful on this direction.

In [18], the $q$-middle convolution by Sakai and Yamaguchi [16] was applied to
the linear $q$-difference equation related with the $q$-Painlevé equation
whose symmetry is of type $D^{(1)}_{5}$, and the $q$-integral transformation
of the $q$-Heun equation was obtained as a corollary. The variants of the
$q$-Heun equation of degree three and four is related with the $q$-Painlevé
equation of type $E^{(1)}_{6}$ and $E^{(1)}_{7}$ through the space of initial
conditions [17]. In this paper, we obtained $q$-integral transformations of
solutions to the $q$-Heun equation and its variants by using the identities of
the kernel functions. It seems that relationship between the variants of the
$q$-Heun equation of degree three and four with the $q$-Painlevé equation of
type $E^{(1)}_{6}$ and $E^{(1)}_{7}$ through the $q$-middle convolution is not
fully understood. The paper [3] by Fujii and Nobukawa might be related to this
problem.

## Acknowledgements

The author is grateful to Yumi Arai for discussion. He is supported by JSPS
KAKENHI Grant Number JP22K03368.

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