On $q$-Deformations of the Heun Equation

On $\boldsymbol{q}$-Deformations of the Heun Equation††This paper is a
contribution to the Special Issue on Elliptic Hypergeometric Functions and
Their Applications. The full collection is available at
https://www.emis.de/journals/SIGMA/EHF2017.html

Kouichi TAKEMURA

K. Takemura

Department of Mathematics, Faculty of Science and Engineering, Chuo
University,
1-13-27 Kasuga, Bunkyo-ku Tokyo 112-8551, Japan<EMAIL_ADDRESS>
Received January 18, 2018, in final form May 29, 2018; Published online June
18, 2018

The $q$-Heun equation and its variants arise as degenerations of
Ruijsenaars–van Diejen operators with one particle. We investigate local
properties of these equations. In particular we characterize the variants of
the $q$-Heun equation by using analysis of regular singularities. We also
consider the quasi-exact solvability of the $q$-Heun equation and its
variants. Namely we investigate finite-dimensional subspaces which are
invariant under the action of the $q$-Heun operator or variants of the
$q$-Heun operator.

Heun equation; $q$-deformation; regular singularity; quasi-exact solvability;
degeneration

39A13; 33E10

##  1 Introduction

Accessory parameters of an ordinary linear differential equation are the
parameters which are not governed by local data (or local exponents) of the
differential equation, and absence or existence of accessory parameters
affects the strategy of analyzing the differential equation. A typical example
which does not have an accessory parameter is the hypergeometric equation of
Gauss. On the other hand, the Heun equation is an example which has an
accessory parameter. Heun’s differential equation is a standard form of the
second order linear differential equation with four regular singularities on
the Riemann sphere, and it is written as

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

with the condition $\gamma+\delta+\epsilon=\alpha+\beta+1$. The parameter $B$
is an accessory parameter, which is independent of the local exponents.

In [10], $q$-difference deformations of Heun’s differential equation were
obtained by degenerations of Ruijsenaars–van Diejen operators [2, 7], which
were also obtained in connection with $q$-Painlevé equations [5, 12]. One of
the $q$-deformed difference equations is written as

$\displaystyle\big{(}x-q^{h_{1}+1/2}t_{1}\big{)}\big{(}x-q^{h_{2}+1/2}t_{2}\big{)}g(x/q)+q^{\alpha_{1}+\alpha_{2}}\big{(}x-q^{l_{1}-1/2}t_{1}\big{)}\big{(}x-q^{l_{2}-1/2}t_{2}\big{)}g(qx)$
$\displaystyle\qquad{}-\big{\\{}\big{(}q^{\alpha_{1}}+q^{\alpha_{2}}\big{)}x^{2}+Ex+q^{(h_{1}+h_{2}+l_{1}+l_{2}+\alpha_{1}+\alpha_{2})/2}\big{(}q^{\beta/2}+q^{-\beta/2}\big{)}t_{1}t_{2}\big{\\}}g(x)=0.$
(1.2)

It is characterized as the linear homogeneous $q$-difference equation

$\displaystyle p^{\langle 0\rangle}(x)g(x/q)+p^{\langle
1\rangle}(x)g(x)+p^{\langle 2\rangle}(x)g(qx)=0,$ (1.3)

where the polynomials $p^{\langle i\rangle}(x)$, $i=0,1,2$, are quadratic
$p^{\langle i\rangle}(x)=p^{\langle i\rangle}_{0}+p^{\langle
i\rangle}_{1}x+p^{\langle i\rangle}_{2}x^{2}$ with the condition $p^{\langle
0\rangle}_{0}\neq 0$, $p^{\langle 0\rangle}_{2}\neq 0$, $p^{\langle
2\rangle}_{0}\neq 0$ and $p^{\langle 2\rangle}_{2}\neq 0$. It was discovered
by Hahn [4] in 1971, and we call equation (1.3) (or equation (1.2)) the
$q$-Heun equation. Note that Hahn recognized that the parameter $p^{\langle
1\rangle}_{1}$ in equation (1.3) (or the parameter $E$ in equation (1.2)) can
be regarded as an accessory parameter.

The $q$-difference equation such that the polynomials $p^{\langle
i\rangle}(x)$, $i=0,1,2$, are linear $p^{\langle i\rangle}(x)=p^{\langle
i\rangle}_{0}+p^{\langle i\rangle}_{1}x$ with the condition $p^{\langle
0\rangle}_{0}\neq 0$, $p^{\langle 0\rangle}_{1}\neq 0$, $p^{\langle
2\rangle}_{0}\neq 0$ and $p^{\langle 2\rangle}_{1}\neq 0$ is better known as
the $q$-difference hypergeometric equation ($q$-hypergeometric equation), and
a standard form is given as

$\displaystyle(x-q)f(x/q)-((a+b)x-q-c)f(x)+(abx-c)f(qx)=0.$ (1.4)

Then the basic hypergeometric series ($q$-hypergeometric series)

$\displaystyle{}_{2}\phi_{1}(a,b;c;x)=\sum_{n=0}^{\infty}\frac{(a;q)_{n}(b;q)_{n}}{(q;q)_{n}(c;q)_{n}}x^{n},\qquad(\lambda,q)_{n}=\prod_{i=0}^{n-1}\big{(}1-\lambda
q^{i}\big{)}$

is a solution to equation (1.4). Note that the $q$-hypergeometric equation
appears as a special case of the $q$-Heun equation. Namely, if the parameters
in equation (1.2) satisfy $l_{2}=h_{2}+1$ and
$E=-\big{(}q^{\alpha_{1}}+q^{\alpha_{2}}\big{)}q^{h_{2}+1/2}t_{2}-q^{(h_{1}-h_{2}+l_{1}+l_{2}+\alpha_{1}+\alpha_{2}-1)/2}\big{(}q^{\beta/2}+q^{-\beta/2}\big{)}t_{1}$,
then the $q$-difference equation has the common factor
$\big{(}x-q^{h_{2}+1/2}t_{2}\big{)}$ and we obtain the $q$-hypergeometric
difference equation.

Recall that the $q$-Heun equation was introduced in [10] as an eigenfunction
of the fourth degeneration of the Ruijsenaars–van Diejen operator of one
variable

$\displaystyle A^{\langle
4\rangle}=x^{-1}\big{(}x-q^{h_{1}+1/2}t_{1}\big{)}\big{(}x-q^{h_{2}+1/2}t_{2}\big{)}T_{q^{-1}}$
$\displaystyle\hphantom{A^{\langle
4\rangle}=}{}+q^{\alpha_{1}+\alpha_{2}}x^{-1}\big{(}x-q^{l_{1}-1/2}t_{1}\big{)}\big{(}x-q^{l_{2}-1/2}t_{2}\big{)}T_{q}$
$\displaystyle\hphantom{A^{\langle
4\rangle}=}{}-\big{\\{}\big{(}q^{\alpha_{1}}+q^{\alpha_{2}}\big{)}x+q^{(h_{1}+h_{2}+l_{1}+l_{2}+\alpha_{1}+\alpha_{2})/2}\big{(}q^{\beta/2}+q^{-\beta/2}\big{)}t_{1}t_{2}x^{-1}\big{\\}}$
(1.5)

with eigenvalue $E$, where $T_{q^{-1}}g(x)=g(x/q)$ and $T_{q}g(x)=g(qx)$.
Namely equation (1.2) is written as

$\displaystyle A^{\langle 4\rangle}g(x)=Eg(x).$

The derivation of the $q$-Heun equation in [10] was motivated by the
relationship between the elliptic $4$-parameter Heun differential operator and
the 8-parameter elliptic difference operator (Ruijsenaars–van Diejen
operator). The (non-degenerate) Ruijsenaars–van Diejen operator [2, 7] of one
variable is given by

$\displaystyle A^{\langle
0\rangle}f(z)=V(z)f(z-ia_{-})+V(-z)f(z+ia_{-})+U(z)f(z),$

where

$\displaystyle
V(z)=\frac{\prod\limits_{n=1}^{8}R_{+}(z-h_{n}-ia_{-}/2)}{R_{+}(2z+ia_{+}/2)R_{+}(2z-ia_{-}+ia_{+}/2)},$

$R_{+}(z)=\prod\limits_{k=1}^{\infty}\big{(}1-e^{-(2k-1)\pi a_{+}}e^{2\pi
iz}\big{)}\big{(}1-e^{-(2k-1)\pi a_{+}}e^{-2\pi iz}\big{)}$ is a modified
version of the theta function, and $U(z)$ is an elliptic function whose
definition is omitted here (see [10]). The operator $A^{\langle 0\rangle}$ has
8 parameters $h_{1},\dots,h_{8}$ apart from the parameter $a_{+}$ in the
elliptic function and the difference interval $ia_{-}$. The multi-variable
Ruijsenaars–van Diejen system can be regarded as a relativistic deformation of
the quantum system called the Inozemtsev system [2, 7]. By the non-
relativistic limit $a_{-}\to 0$ of the one-variable Ruijsenaars–van Diejen
operator, we obtain the Heun operator in term of the elliptic function written
as

$\displaystyle H=-\frac{{\rm d}^{2}}{{\rm
d}x^{2}}+\sum_{i=0}^{3}l_{i}(l_{i}+1)\wp(x+\omega_{i}),$

where $\wp(x)$ is the Weierstrass elliptic function with basic periods
$(2\omega_{1},2\omega_{3})$ and $\\{\omega_{0}(=0),\omega_{1}$,
$\omega_{2}(=-\omega_{1}-\omega_{3}),\omega_{3}\\}$ is a set of half periods
(see [7] for details). Here the 8 parameters in the relativistic system are
reduced to the 4 parameters $l_{0}$, $l_{1}$, $l_{2}$, $l_{3}$ in the non-
relativistic limit. It is known that the Heun equation given in equation (1.1)
is equivalent to the equation $Hf(x)=Ef(x)$, $E\in\mathbb{C}$ (see [9] for
details). Hence the Ruijsenaars–van Diejen system is a difference analogue of
the elliptic form of the Heun equation. A motivation of the paper [10] was a
construction of the difference analogue of the Heun equation without the
elliptic expression, and this was realized by considering degenerations four
times. After writing the first version of the manuscript of [10], the author
was informed of Hahn’s paper [4] by Professor Ohyama.

In [10], other degenerate operators were also obtained. The third and the
second degenerate Ruijsenaars–van Diejen operators are written as

$\displaystyle A^{\langle
3\rangle}=x^{-1}\prod_{n=1}^{3}\big{(}x-q^{h_{n}+1/2}t_{n}\big{)}T_{q^{-1}}+x^{-1}\prod_{n=1}^{3}(x-q^{l_{n}-1/2}t_{n})T_{q}-\big{(}q^{1/2}+q^{-1/2}\big{)}x^{2}$
$\displaystyle\hphantom{A^{\langle
3\rangle}=}{}+\sum_{n=1}^{3}\big{(}q^{h_{n}}+q^{l_{n}}\big{)}t_{n}x+q^{(l_{1}+l_{2}+l_{3}+h_{1}+h_{2}+h_{3})/2}\big{(}q^{\beta/2}+q^{-\beta/2}\big{)}t_{1}t_{2}t_{3}x^{-1},$
(1.6) $\displaystyle A^{\langle
2\rangle}=x^{-2}\prod_{n=1}^{4}\big{(}x-q^{h_{n}+1/2}t_{n}\big{)}T_{q^{-1}}+x^{-2}\prod_{n=1}^{4}\big{(}x-q^{l_{n}-1/2}t_{n}\big{)}T_{q}$
$\displaystyle\hphantom{A^{\langle
2\rangle}=}{}-\big{(}q^{1/2}+q^{-1/2}\big{)}x^{2}+\sum_{n=1}^{4}\big{(}q^{h_{n}}+q^{l_{n}}\big{)}t_{n}x$
$\displaystyle\hphantom{A^{\langle
2\rangle}=}{}+\prod_{n=1}^{4}q^{(h_{n}+l_{n})/2}t_{n}\left[-\big{(}q^{1/2}+q^{-1/2}\big{)}x^{-2}+\sum_{n=1}^{4}\left(\frac{1}{q^{h_{n}}t_{n}}+\frac{1}{q^{l_{n}}t_{n}}\right)x^{-1}\right].$
(1.7)

The variants of the $q$-Heun equation were introduced as $A^{\langle
3\rangle}g(x)=Eg(x)$ and $A^{\langle 2\rangle}g(x)=Eg(x)$, $E\in\mathbb{C}$.

In this paper, we investigate some properties of the $q$-Heun equation and its
variants. One of the results is a characterization of variants of the $q$-Heun
equation. Recall that the equation $A^{\langle 4\rangle}g(x)=Eg(x)$ (i.e.,
equation (1.2)) is written as equation (1.3) whose coefficients are generic
quadratic polynomials.

The equation $A^{\langle 3\rangle}g(x)=Eg(x)$ (resp. $A^{\langle
2\rangle}g(x)=Eg(x)$) is written as

$\displaystyle a(x)g(x/q)+b(x)g(x)+c(x)g(qx)=0,$ (1.8)

where $a(x)$, $b(x)$, $c(x)$ are polynomials such that
$\deg_{x}a(x)=\deg_{x}c(x)=3$, $a(0)\neq 0\neq c(0)$ and $\deg_{x}b(x)\leq 3$
(resp. $\deg_{x}a(x)=\deg_{x}c(x)=4$, $a(0)\neq 0\neq c(0)$ and
$\deg_{x}b(x)\leq 4$), although there are relations among the coefficients of
the polynomials $a(x)$, $b(x)$, $c(x)$. In this paper we describe these
relations using a theory of the regular singularity of the $q$-difference
equation. The definition of the regular singularity is analogous to the
differential equation (see Section 2). Let us consider equation (1.8) such
that $a(x)$, $b(x)$ and $c(x)$ are polynomials, $\deg_{x}a(x)=\deg_{x}c(x)=3$,
$\deg_{x}b(x)\leq 3$ and $a(0)\neq 0\neq c(0)$. Without loss of generality, we
assume that the polynomials $a(x)$ and $c(x)$ are monic (see the argument
around equation (3.13)). We factorize the polynomials as
$a(x)=\big{(}x-q^{h_{1}+1/2}t_{1}\big{)}\big{(}x-q^{h_{2}+1/2}t_{2}\big{)}\big{(}x-q^{h_{3}+1/2}t_{3}\big{)}$
and
$c(x)=\big{(}x-q^{l_{1}-1/2}t_{1}\big{)}\big{(}x-q^{l_{2}-1/2}t_{2}\big{)}\big{(}x-q^{l_{3}-1/2}t_{3}\big{)}$.
Write the polynomial $b(x)$ as

$\displaystyle b(x)=b_{3}x^{3}+b_{2}x^{2}+b_{1}x+b_{0}.$

In the equation $A^{\langle 3\rangle}g(x)=Eg(x)$ in the form of equation
(1.8), the constants $b_{3}$, $b_{2}$, $b_{1}$ and $b_{0}$ are specified. Here
we are going to find a characterization of the values $b_{3}$, $b_{2}$ and
$b_{0}$ by imposing conditions of local behavior of equation (1.8). Namely, if
the exponents of equation (1.8) about $x=\infty$ are $-1/2$, $1/2$ and the
singularity $x=\infty$ is apparent (non-logarithmic), then the values $b_{3}$
and $b_{2}$ are determined as

$\displaystyle b_{3}=-\big{(}q^{1/2}+q^{-1/2}\big{)},\qquad
b_{2}=\sum_{n=1}^{3}\big{(}q^{h_{n}}+q^{l_{n}}\big{)}t_{n}.$

The value $b_{0}$ is determined as
$b_{0}=q^{(l_{1}+l_{2}+l_{3}+h_{1}+h_{2}+h_{3})/2}\big{(}q^{\beta/2}+q^{-\beta/2}\big{)}t_{1}t_{2}t_{3}$
by imposing a condition of the local exponents about $x=0$. See Theorem 3.1
for the precise description. Note that the value $b_{1}=E$ is independent of
the local conditions about $x=0,\infty$, and it is reasonable to regard $E$ as
an accessory parameter. An analogous result for the equation $A^{\langle
2\rangle}g(x)=Eg(x)$ is described in Theorem 3.2.

By a suitable limit in the equation $A^{\langle 3\rangle}g(x)=Eg(x)$, we
obtain a differential equation which is equivalent to Heun’s differential
equation by a linear fractional transformation (see Section 3). Then the
singularity $x=\infty$ of the equation $A^{\langle 3\rangle}g(x)=Eg(x)$
corresponds to a regular (non-singular) point of the differential equation in
the limit, and we may say that the equation $(A^{\langle 3\rangle}-E)g(x)=0$
is a $q$-analogue of the Fuchsian equation with four regular singularities
$\\{0,t_{1},t_{2},t_{3}\\}$, $t_{1},t_{2},t_{3}\in\mathbb{C}_{\neq 0}$. For
details see Section 3.2. Similarly, the equation $\big{(}A^{\langle
2\rangle}-E\big{)}g(x)=0$ is a $q$-analogue of the Fuchsian equation with four
regular singularities $\\{t_{1},t_{2},t_{3},t_{4}\\}$,
$t_{1},t_{2},t_{3},t_{4}\in\mathbb{C}_{\neq 0}$ (see Section 3.3).

In this paper we also investigate special solutions to the $q$-Heun equation
and its variants. For this purpose, we consider quasi-exact solvability [11]
of the equations. Namely, we investigate finite-dimensional invariant spaces
which are invariant under the action of the degenerate Ruijsenaars–van Diejen
operator $A^{\langle 4\rangle}$, $A^{\langle 3\rangle}$ or $A^{\langle
2\rangle}$. Recall that quasi-exact solvability was applied to investigate the
Inozemtsev system of type $BC_{N}$ [3, 8], which is a generalization of the
Heun equation to multiple variables. For the Ruijsenaars–van Diejen system
(the Ruijsenaars system of type $BC_{N}$), Komori [6] obtained finite-
dimensional subspaces spanned by theta functions which are invariant under the
action of the Ruijsenaars–van Diejen system. The author believes that our
invariant subspaces of the operator $A^{\langle 4\rangle}$, $A^{\langle
3\rangle}$ or $A^{\langle 2\rangle}$ are degenerations of those of Komori,
although they are obtained independently and straightforwardly. We hope to
develop further our understandings of these invariant subspaces and to find a
path for analysis of the (non-degenerate) Ruijsenaars–van Diejen system in the
near future.

Here we present a simple example of an invariant subspace for the operator
$A^{\langle 4\rangle}$ in equation (1.5). Set
$\lambda_{1}=(h_{1}+h_{2}-l_{1}-l_{2}-\alpha_{1}-\alpha_{2}-\beta)/2+1$ and
assume that $\lambda_{1}=-\alpha_{1}$. Then the operator $A^{\langle
4\rangle}$ preserves the one-dimensional space spanned by the function
$x^{\lambda_{1}}$, and we have

$\displaystyle A^{\langle
4\rangle}x^{\lambda_{1}}=-\big{\\{}\big{(}q^{h_{1}+1/2}t_{1}+q^{h_{2}+1/2}t_{2}\big{)}q^{\alpha_{1}}+\big{(}q^{l_{1}-1/2}t_{1}+q^{l_{2}-1/2}t_{2}\big{)}q^{\alpha_{2}}\big{\\}}x^{\lambda_{1}}.$
(1.9)

Therefore the function $x^{\lambda_{1}}$ is an eigenfunction of the operator
$A^{\langle 4\rangle}$ with eigenvalue $-\big{\\{}\big{(}q^{h_{1}+1/2}t_{1}$
$+q^{h_{2}+1/2}t_{2}\big{)}q^{\alpha_{1}}+\big{(}q^{l_{1}-1/2}t_{1}+q^{l_{2}-1/2}t_{2}\big{)}q^{\alpha_{2}}\big{\\}}$
under the assumption $h_{1}+h_{2}-l_{1}-l_{2}+\alpha_{1}-\alpha_{2}$
$-\beta+2=0$.

The reminder of the paper is organized as follows. In Section 2, we explain
local properties of solutions to second-order $q$-difference equation about
$x=0$ and $x=\infty$. In Section 3, we investigate local behaviors of
solutions about $x=0$ and $x=\infty$ to the $q$-Heun equation and its
variants. We obtain theorems for characterization of $q$-difference equations
to variants of the $q$-Heun equation. We also investigate the limit $q\to 1$
and observe relationships with Heun’s differential equation. In Section 4, we
investigate invariant subspaces related with the $q$-Heun equation and its
variants.

In this paper, we assume that $q$ is a positive real number which is not equal
to $1$, and all exponents are real numbers.

##  2 Regular singularity to the difference equation

We investigate the linear difference equation

$\displaystyle u(x)g(x/q)+v(x)g(x)+w(x)g(qx)=0$ (2.1)

such that $u(x)$, $v(x)$, $w(x)$ are Laurent polynomials which are written as

$\displaystyle u(x)=\sum_{k=M}^{N}u_{k}x^{k},\qquad
v(x)=\sum_{k=M^{\prime}}^{N^{\prime}}v_{k}x^{k},\qquad
w(x)=\sum_{k=M^{\prime\prime}}^{N^{\prime\prime}}w_{k}x^{k},$ (2.2)

where $u_{M}$, $v_{M^{\prime}}$, $w_{M^{\prime\prime}}$, $u_{N^{\prime}}$,
$v_{N^{\prime}}$, $w_{N^{\prime\prime}}$ are all non-zero.

Then the point $x=0$ (resp. $x=\infty$) of equation (2.1) is a regular
singularity, iff $M=M^{\prime\prime}\leq M^{\prime}$ (resp.
$N=N^{\prime\prime}\geq N^{\prime}$). The power series solutions about the
regular singularity is by now well-understood (see [1] and related papers).

Assume that the point $x=0$ of equation (2.1) is a regular singularity. Then
$M=M^{\prime\prime}\leq M^{\prime}$. We investigate local solutions about
$x=0$ of the form

$\displaystyle g(x)=x^{\lambda}\sum_{\ell=0}^{\infty}c_{\ell}x^{\ell},\qquad
c_{0}\neq 0.$

If we substitute this into equation (2.1) and set $k+\ell=n+M$, then we get

$\displaystyle\sum_{n=0}^{\infty}\sum_{\ell=0}^{n}\big{(}q^{-\lambda-\ell}u_{n+M-\ell}+v_{n+M-\ell}+q^{\lambda+\ell}w_{n+M-\ell}\big{)}c_{\ell}x^{n+M+\lambda}=0.$

Hence

$\displaystyle\sum_{\ell=0}^{n}\big{(}q^{-\lambda-\ell}u_{n+M-\ell}+v_{n+M-\ell}+q^{\lambda+\ell}w_{n+M-\ell}\big{)}c_{\ell}=0,$

where $n$ is a non-negative integer. If $n=0$, then it follows from $c_{0}\neq
0$ that

$\displaystyle q^{-\lambda}u_{M}+v_{M}+q^{\lambda}w_{M}=0,$

which we call the characteristic equation about the regular singularity $x=0$.
The exponents about the regular singularity $x=0$ are the values $\lambda$
which satisfy the characteristic equation. If the quadratic equation
$u_{M}+v_{M}t+w_{M}t^{2}=0$ has two positive roots, then we have two real-
valued exponents. Let $\lambda^{\prime}$ be an exponent. If
$\lambda^{\prime}+n$ ($n\in{\mathbb{Z}}_{\geq 1}$) is not an exponent, then
the coefficient $c_{n}$ is determined by

$\displaystyle\big{(}q^{-\lambda^{\prime}-n}u_{M}+v_{M}+q^{\lambda^{\prime}+n}w_{M}\big{)}c_{n}=-\sum_{\ell=0}^{n-1}\big{(}q^{-\lambda^{\prime}-\ell}u_{n+M-\ell}+v_{n+M-\ell}+q^{\lambda^{\prime}+\ell}w_{n+M-\ell}\big{)}c_{\ell}.$

If $\lambda^{\prime}+n$ is an exponent for some positive integer $n$, then we
need the equation

$\displaystyle\sum_{\ell=0}^{n-1}\big{(}q^{-\lambda^{\prime}-\ell}u_{n+M-\ell}+v_{n+M-\ell}+q^{\lambda^{\prime}+\ell}w_{n+M-\ell}\big{)}c_{\ell}=0$
(2.3)

in order to have series solutions. Otherwise we need logarithmic terms for the
solutions. If the difference of the exponents is a non-zero integer and
equation (2.3) is satisfied, then the singularity $x=0$ is called apparent (or
non-logarithmic).

Next, we assume that the point $x=\infty$ of equation (2.1) is a regular
singularity, i.e., $N=N^{\prime\prime}\geq N^{\prime}$ in equation (2.2), and
investigate local solutions about $x=\infty$ of the form

$\displaystyle
h(x)=(1/x)^{\lambda}\sum_{\ell=0}^{\infty}\tilde{c}_{\ell}(1/x)^{\ell},\qquad\tilde{c}_{0}\neq
0.$

Then we have

$\displaystyle\sum_{\ell=0}^{n}\big{(}q^{\lambda+\ell}u_{N+\ell-n}+v_{N+\ell-n}+q^{-\lambda-\ell}w_{N+\ell-n}\big{)}\tilde{c}_{\ell}=0,$

where $n$ is a non-negative integer. By setting $n=0$, we obtain the
characteristic equation at the regular singularity $x=\infty$ as follows

$\displaystyle q^{\lambda}u_{N}+v_{N}+q^{-\lambda}w_{N}=0.$ (2.4)

The exponents about the regular singularity $x=\infty$ are the values
$\lambda$ which satisfy the characteristic equation, i.e., equation (2.4).
Assume that $\lambda^{\prime}$ is an exponent at $x=\infty$ and
$\lambda^{\prime}+n$ is also an exponent at $x=\infty$ for some positive
integer $n$. Then the singularity is called apparent, iff

$\displaystyle\sum_{\ell=0}^{n-1}\big{(}q^{\lambda+\ell}u_{N+\ell-n}+v_{N+\ell-n}+q^{-\lambda-\ell}w_{N+\ell-n}\big{)}\tilde{c}_{\ell}=0.$

##  3 Local behaviors of solutions to the $\boldsymbol{q}$-Heun equation
and its variants

We investigate local behaviors of solutions about $x=0$ and $x=\infty$ to the
$q$-Heun equation and its variants. We obtain theorems that the local
behaviors of solutions determine the coefficients of the $q$-difference
equation for the cases of the variants of the $q$-Heun equation.

### 3.1

Let $A^{\langle 4\rangle}$ be the operator in equation (1.5). Then the
$q$-Heun equation is written as

$\displaystyle\big{(}A^{\langle 4\rangle}-E\big{)}g(x)=0,$ (3.1)

where $E$ is a constant. The singularities $x=0$ and $x=\infty$ of the
$q$-Heun equation are regular, and the constants $M$ and $N$ in Section 2 are
given by $M=-1$ and $N=1$. The operator $A^{\langle 4\rangle}$ acts on
$x^{\mu}$ as

$\displaystyle A^{\langle 4\rangle}x^{\mu}=d^{\langle
4\rangle,+}(\mu)x^{\mu+1}+d^{\langle 4\rangle,0}(\mu)x^{\mu}+d^{\langle
4\rangle,-}(\mu)x^{\mu-1},$

where

$\displaystyle d^{\langle
4\rangle,+}(\mu)=q^{\alpha_{1}+\alpha_{2}}q^{\mu}-\big{(}q^{\alpha_{1}}+q^{\alpha_{2}}\big{)}+q^{-\mu},$
$\displaystyle d^{\langle
4\rangle,0}(\mu)=-\big{(}q^{h_{1}+1/2}t_{1}+q^{h_{2}+1/2}t_{2}\big{)}q^{-\mu}-\big{(}q^{l_{1}-1/2}t_{1}+q^{l_{2}-1/2}t_{2}\big{)}q^{\alpha_{1}+\alpha_{2}+\mu},$
(3.2) $\displaystyle d^{\langle
4\rangle,-}(\mu)=\big{\\{}q^{h_{1}+h_{2}+1}q^{-\mu}\\!-q^{(h_{1}+h_{2}+l_{1}+l_{2}+\alpha_{1}+\alpha_{2})/2}\big{(}q^{\beta/2}+q^{-\beta/2}\big{)}+q^{l_{1}+l_{2}+\alpha_{1}+\alpha_{2}-1}q^{\mu}\big{\\}}t_{1}t_{2}.$

We investigate the singularity $x=0$. Set

$\displaystyle g(x)=x^{\lambda}\sum_{n=0}^{\infty}c_{n}x^{n},\qquad c_{0}\neq
0,$ (3.3)

and substitute this into equation (3.1). Then we have

$\displaystyle c_{0}d^{\langle 4\rangle,-}(\lambda)=0,\qquad c_{1}d^{\langle
4\rangle,-}(\lambda+1)+c_{0}\big{(}d^{\langle
4\rangle,0}(\lambda)-E\big{)}=0,$ $\displaystyle c_{n}d^{\langle
4\rangle,-}(\lambda+n)+c_{n-1}\big{(}d^{\langle
4\rangle,0}(\lambda+n-1)-E\big{)}+c_{n-2}d^{\langle
4\rangle,+}(\lambda+n-2)=0,\qquad n\geq 2.$

The characteristic equation about $x=0$ is written as $d^{\langle
4\rangle,-}(\lambda)=0$, i.e.,

$\displaystyle
q^{h_{1}+h_{2}+1}q^{-\lambda}-q^{(h_{1}+h_{2}+l_{1}+l_{2}+\alpha_{1}+\alpha_{2})/2}\big{(}q^{\beta/2}+q^{-\beta/2}\big{)}+q^{l_{1}+l_{2}+\alpha_{1}+\alpha_{2}-1}q^{\lambda}=0.$

Hence the values

$\displaystyle\lambda_{1}=(h_{1}+h_{2}-l_{1}-l_{2}-\alpha_{1}-\alpha_{2}-\beta+2)/2,$
$\displaystyle\lambda_{2}=(h_{1}+h_{2}-l_{1}-l_{2}-\alpha_{1}-\alpha_{2}+\beta+2)/2$
(3.4)

are exponents about $x=0$. If $\lambda_{2}-\lambda_{1}=\beta$ is not a
positive integer, then we have $d^{\langle 4\rangle,-}(\lambda_{1}+n)\neq 0$
for $n\in\mathbb{Z}_{>0}$ and the coefficients $c_{n}$ for
$\lambda=\lambda_{1}$ and $n\geq 1$ are determined recursively. If
$\lambda_{1}-\lambda_{2}\not\in\mathbb{Z}$, then we have solutions to equation
(3.1) written as

$\displaystyle
x^{\lambda_{1}}\left\\{1+\sum_{n=1}^{\infty}c_{n}x^{n}\right\\},\qquad
x^{\lambda_{2}}\left\\{1+\sum_{n=1}^{\infty}c^{\prime}_{n}x^{n}\right\\}.$

We investigate equation (3.1) about $x=\infty$. Set

$\displaystyle
g(x)=(1/x)^{\lambda}\sum_{n=0}^{\infty}\tilde{c}_{n}(1/x)^{n}=\sum_{n=0}^{\infty}\tilde{c}_{n}x^{-\lambda-n},\qquad\tilde{c}_{0}\neq
0.$ (3.5)

By substituting this into equation (3.1), we have

$\displaystyle\tilde{c}_{0}d^{\langle
4\rangle,+}(-\lambda)=0,\qquad\tilde{c}_{1}d^{\langle
4\rangle,+}(-\lambda-1)+\tilde{c}_{0}\big{(}d^{\langle
4\rangle,0}(-\lambda)-E\big{)}=0,$ $\displaystyle\tilde{c}_{n}d^{\langle
4\rangle,+}(-\lambda-n)+\tilde{c}_{n-1}\big{(}d^{\langle
4\rangle,0}(-\lambda-n+1)-E\big{)}+\tilde{c}_{n-2}d^{\langle
4\rangle,+}(-\lambda-n+2)=0.$

The characteristic equation at $x=\infty$ is written as $d^{\langle
4\rangle,+}(-\lambda)=0$, i.e.,

$\displaystyle
q^{\alpha_{1}+\alpha_{2}}q^{-\lambda}-\big{(}q^{\alpha_{1}}+q^{\alpha_{2}}\big{)}+q^{\lambda}=0.$

Hence the values $\alpha_{1}$ and $\alpha_{2}$ are exponents about $x=\infty$.
If $\alpha_{2}-\alpha_{1}$ is not a positive integer, then we have $d^{\langle
4\rangle,+}(-\alpha_{1}-n)\neq 0$ for $n\in\mathbb{Z}_{>0}$ and the
coefficients $\tilde{c}_{n}$, $n\geq 1$, are determined recursively. If
$\alpha_{1}-\alpha_{2}\not\in\mathbb{Z}$, then we have solutions to equation
(3.1) written as

$\displaystyle
x^{-\alpha_{1}}\left\\{1+\sum_{n=1}^{\infty}\tilde{c}_{n}x^{-n}\right\\},\qquad
x^{-\alpha_{2}}\left\\{1+\sum_{n=1}^{\infty}\tilde{c}^{\prime}_{n}x^{-n}\right\\}.$

Equation (3.1) is written as

$\displaystyle a(x)g(x/q)+b(x)g(x)+c(x)g(qx)=0,$

where

$\displaystyle
a(x)=\big{(}x-q^{h_{1}+1/2}t_{1}\big{)}\big{(}x-q^{h_{2}+1/2}t_{2}\big{)},\qquad
b(x)=b_{2}x^{2}+Ex+b_{0},$ $\displaystyle
c(x)=q^{\alpha_{1}+\alpha_{2}}\big{(}x-q^{l_{1}-1/2}t_{1}\big{)}\big{(}x-q^{l_{2}-1/2}t_{2}\big{)},$

and

$\displaystyle b_{2}=-\big{(}q^{\alpha_{1}}+q^{\alpha_{2}}\big{)},\qquad
b_{0}=-q^{(h_{1}+h_{2}+l_{1}+l_{2}+\alpha_{1}+\alpha_{2})/2}\big{(}q^{\beta/2}+q^{-\beta/2}\big{)}t_{1}t_{2}.$
(3.6)

If the exponents of equation (3.1) about $x=0$ (resp. $x=\infty$) are
$\lambda_{1}$ and $\lambda_{2}$ in equation (3.4) (resp. $\alpha_{1}$ and
$\alpha_{2}$), then $b_{0}$ (resp. $b_{2}$) is given by equation (3.6). The
parameter $E$ is independent of the exponents about $x=0,\infty$, and it is
reasonable to regard $E$ as an accessory parameter.

It was shown in [10] that the $q$-Heun equation reduces to Heun’s differential
equation in the $q\to 1$ limit.

### 3.2

Let $A^{\langle 3\rangle}$ be the operator in equation (1.6). Then the
equation

$\displaystyle\big{(}A^{\langle 3\rangle}-E\big{)}g(x)=0$ (3.7)

for a fixed constant $E$ is a variant of the $q$-Heun equation. The
singularities $x=0$ and $x=\infty$ of equation (3.7) are regular, and the
constants $M$ and $N$ in Section 2 are given by $M=-1$ and $N=2$. The operator
$A^{\langle 3\rangle}$ acts on $x^{\mu}$ as

$\displaystyle A^{\langle 3\rangle}x^{\mu}=d^{\langle
3\rangle,++}(\mu)x^{\mu+2}+d^{\langle 3\rangle,+}(\mu)x^{\mu+1}+d^{\langle
3\rangle,0}(\mu)x^{\mu}+d^{\langle 3\rangle,-}(\mu)x^{\mu-1},$ (3.8)

where

$\displaystyle d^{\langle
3\rangle,++}(\mu)=q^{-\mu}+q^{\mu}-q^{1/2}-q^{-1/2},$ $\displaystyle
d^{\langle
3\rangle,+}(\mu)=\sum_{i=1}^{3}\big{(}q^{h_{i}}+q^{l_{i}}-q^{h_{i}+1/2-\mu}-q^{l_{i}-1/2+\mu}\big{)}t_{i},$
$\displaystyle d^{\langle 3\rangle,0}(\mu)=\sum_{1\leq i<j\leq
3}\big{(}q^{h_{i}+h_{j}+1-\mu}+q^{l_{i}+l_{j}-1+\mu}\big{)}t_{i}t_{j},$
$\displaystyle d^{\langle
3\rangle,-}(\mu)=\big{\\{}{-}q^{h_{1}+h_{2}+h_{3}+3/2-\mu}-q^{l_{1}+l_{2}+l_{3}-3/2+\mu}$
$\displaystyle\hphantom{d^{\langle
3\rangle,-}(\mu)=}{}+q^{(l_{1}+l_{2}+l_{3}+h_{1}+h_{2}+h_{3})/2}\big{(}q^{\beta/2}+q^{-\beta/2}\big{)}\big{\\}}t_{1}t_{2}t_{3}.$
(3.9)

We investigate the singularity $x=0$ of equation (3.7). We substitute equation
(3.3) into equation (3.7). Then we have

$\displaystyle d^{\langle 3\rangle,-}(\lambda)=0,\qquad c_{1}d^{\langle
3\rangle,-}(\lambda+1)=-c_{0}\big{(}d^{\langle 3\rangle,0}(\lambda)-E\big{)},$
$\displaystyle c_{2}d^{\langle
3\rangle,-}(\lambda+2)=-\big{\\{}c_{1}\big{(}d^{\langle
3\rangle,0}(\lambda+1)-E\big{)}+c_{0}d^{\langle
3\rangle,+}(\lambda)\big{\\}},$ $\displaystyle c_{n}d^{\langle
3\rangle,-}(\lambda+n)=-\big{\\{}c_{n-1}\big{(}d^{\langle
3\rangle,0}(\lambda+n-1)-E\big{)}+c_{n-2}d^{\langle 3\rangle,+}(\lambda+n-2)$
$\displaystyle\hphantom{c_{n}d^{\langle
3\rangle,-}(\lambda+n)=}{}+c_{n-3}d^{\langle
3\rangle,++}(\lambda+n-3)\big{\\}},\qquad n\geq 3.$

The characteristic equation at $x=0$ is written as $d^{\langle
3\rangle,-}(\lambda)=0$, i.e.,

$\displaystyle-q^{h_{1}+h_{2}+h_{3}+3/2-\lambda}-q^{l_{1}+l_{2}+l_{3}-3/2+\lambda}+q^{(l_{1}+l_{2}+l_{3}+h_{1}+h_{2}+h_{3})/2}\big{(}q^{\beta/2}+q^{-\beta/2}\big{)}=0.$

Hence the values

$\displaystyle\lambda_{1}=(h_{1}+h_{2}+h_{3}-\beta-l_{1}-l_{2}-l_{3}+3)/2,$
$\displaystyle\lambda_{2}=(h_{1}+h_{2}+h_{3}+\beta-l_{1}-l_{2}-l_{3}+3)/2$

are exponents about $x=0$.

We investigate the singularity $x=\infty$ of equation (3.7). We substitute
equation (3.5) into equation (3.7). Then we have

$\displaystyle d^{\langle
3\rangle,++}(-\lambda)=0,\qquad\tilde{c}_{1}d^{\langle
3\rangle,++}(-\lambda-1)=-\tilde{c}_{0}d^{\langle 3\rangle,+}(-\lambda),$
$\displaystyle\tilde{c}_{2}d^{\langle
3\rangle,++}(-\lambda-2)=-\big{\\{}\tilde{c}_{1}d^{\langle
3\rangle,+}(-\lambda-1)+\tilde{c}_{0}(d^{\langle
3\rangle,0}(-\lambda)-E)\big{\\}},$ $\displaystyle\tilde{c}_{n}d^{\langle
3\rangle,++}(-\lambda-n)=-\big{\\{}\tilde{c}_{n-1}d^{\langle
3\rangle,+}(-\lambda-n+1)+\tilde{c}_{n-2}\big{(}d^{\langle
3\rangle,0}(-\lambda-n+2)-E\big{)}$
$\displaystyle\hphantom{\tilde{c}_{n}d^{\langle
3\rangle,++}(-\lambda-n)=}{}+\tilde{c}_{n-3}d^{\langle
3\rangle,+}(-\lambda-n+3)\big{\\}},\qquad n\geq 3.$

The singularity $x=\infty$ is regular and the characteristic equation at
$x=\infty$ is written as $d^{\langle 3\rangle,++}(-\lambda)=0$, i.e.,

$\displaystyle q^{\lambda}+q^{-\lambda}-q^{1/2}-q^{-1/2}=0.$

Hence the values $\lambda_{1}=-1/2$ and $\lambda_{2}=1/2$ are exponents about
$x=\infty$, and the difference of the exponents is $1$. In the case
$\lambda_{1}=-1/2$, we have $d^{\langle 3\rangle,++}(-\lambda_{1}-1)=0$ and it
follows from the relation $d^{\langle 3\rangle,+}(-\lambda_{1})=0$ (see
equation (3.9)) that the regular singularity $x=\infty$ is apparent. We have
two independent solutions written as

$\displaystyle
x^{1/2}\left\\{1+\sum_{n=2}^{\infty}\tilde{c}_{n}x^{-n}\right\\},\qquad
x^{-1/2}\left\\{1+\sum_{n=1}^{\infty}{\tilde{c}}^{\prime}_{n}x^{-n}\right\\}.$

Recall that equation (3.7) is written as

$\displaystyle a(x)g(x/q)+b(x)g(x)+c(x)g(qx)=0,$ (3.10)

where

$\displaystyle
a(x)=\big{(}x-q^{h_{1}+1/2}t_{1}\big{)}\big{(}x-q^{h_{2}+1/2}t_{2}\big{)}\big{(}x-q^{h_{3}+1/2}t_{3}\big{)},$
$\displaystyle b(x)=b_{3}x^{3}+b_{2}x^{2}+Ex+b_{0},$ $\displaystyle
c(x)=\big{(}x-q^{l_{1}-1/2}t_{1}\big{)}\big{(}x-q^{l_{2}-1/2}t_{2}\big{)}\big{(}x-q^{l_{3}-1/2}t_{3}\big{)},$
(3.11)

and

$\displaystyle b_{3}=-\big{(}q^{1/2}+q^{-1/2}\big{)},\qquad
b_{2}=\sum_{n=1}^{3}\big{(}q^{h_{n}}+q^{l_{n}}\big{)}t_{n},$ $\displaystyle
b_{0}=q^{(l_{1}+l_{2}+l_{3}+h_{1}+h_{2}+h_{3})/2}\big{(}q^{\beta/2}+q^{-\beta/2}\big{)}t_{1}t_{2}t_{3}.$
(3.12)

Note that the parameter $E$ is independent of the exponents about $x=0,\infty$
and apparency of the regular singularity $x=\infty$. Therefore it is
reasonable to regard $E$ as an accessory parameter.

Next we consider a characterization of a variant of the $q$-Heun equation in
equation (3.7) by restricting the polynomials $a(x)$, $b(x)$ and $c(x)$ in the
difference equation (3.10) which satisfy $\deg_{x}a(x)=\deg_{x}c(x)=3$,
$\deg_{x}b(x)\leq 3$ and $a(0)\neq 0\neq c(0)$. If the function $g(x)$
satisfies equation (3.10) and $g(x)=x^{\nu}h(x)$, then the function $h(x)$
satisfies

$\displaystyle q^{-\nu}a(x)h(x/q)+b(x)h(x)+q^{\nu}c(x)h(qx)=0.$ (3.13)

Therefore we may assume that the polynomials $a(x)$ and $c(x)$ are monic by a
gauge transformation. We denote the zeros of the polynomials $a(x)$ and $c(x)$
by equation (3.11). Then the coefficients $b_{3}$, $b_{2}$ and $b_{0}$ are
determined by imposing conditions of local behaviors about $x=0$ and
$x=\infty$. Namely we have the following theorem.

######  Theorem 3.1.

Assume that the functions $a(x)$, $b(x)$ and $c(x)$ take the form of equation
(3.11). If the difference of the exponents about $x=0$ is $\beta(\neq 0)$, the
difference of the exponents about $x=\infty$ is $1$, and the regular
singularity $x=\infty$ is apparent, then we obtain equation (3.12) and recover
a variant of the $q$-Heun equation in equation (3.7).

###### Proof.

Let $\lambda$ and $\lambda+\beta$ be the exponents about $x=0$. Then we have

$\displaystyle-q^{h_{1}+h_{2}+h_{3}+3/2-\lambda}t_{1}t_{2}t_{3}-q^{l_{1}+l_{2}+l_{3}-3/2+\lambda}t_{1}t_{2}t_{3}+b_{0}=0,$
$\displaystyle-q^{h_{1}+h_{2}+h_{3}+3/2-\lambda-\beta}t_{1}t_{2}t_{3}-q^{l_{1}+l_{2}+l_{3}-3/2+\lambda+\beta}t_{1}t_{2}t_{3}+b_{0}=0.$

Hence we have
$\big{(}q^{h_{1}+h_{2}+h_{3}+3/2-\lambda}-q^{l_{1}+l_{2}+l_{3}-3/2+\lambda+\beta}\big{)}\big{(}1-q^{-\beta}\big{)}=0$,
$\lambda=(l_{1}+l_{2}+l_{3}+h_{1}+h_{2}+h_{3}+3-\beta)/2$ and
$b_{0}=q^{(l_{1}+l_{2}+l_{3}+h_{1}+h_{2}+h_{3})/2}\big{(}q^{\beta/2}+q^{-\beta/2}\big{)}t_{1}t_{2}t_{3}$.
Similarly it follows from the condition of the exponents about $x=\infty$ that
$b_{3}=-\big{(}q^{1/2}+q^{-1/2}\big{)}$ and the exponents are $1/2$ and
$-1/2$. The condition that the singularity $x=\infty$ is apparent is written
as
$-q^{-1/2}\big{(}q^{h_{1}+1/2}t_{1}+q^{h_{2}+1/2}t_{2}+q^{h_{3}+1/2}t_{3}\big{)}+b_{2}-q^{1/2}\big{(}q^{l_{1}-1/2}t_{1}+q^{l_{2}-1/2}t_{2}+q^{l_{3}-1/2}t_{3}\big{)}=0$.
Therefore we obtain equation (3.12). ∎

We are going to obtain a differential equation from the difference equation
(3.7) by taking a suitable limit. We rewrite equation (3.7) as

$\displaystyle\big{(}x-t_{1}q^{h_{1}+1/2}\big{)}\big{(}x-t_{2}q^{h_{2}+1/2}\big{)}\big{(}x-t_{3}q^{h_{3}+1/2}\big{)}g(x/q)$
$\displaystyle\qquad{}+\big{(}x-t_{1}q^{l_{1}-1/2}\big{)}\big{(}x-t_{2}q^{l_{2}-1/2}\big{)}\big{(}x-t_{3}q^{l_{3}-1/2}\big{)}g(xq)$
$\displaystyle\qquad{}+\bigg{[}{-}\big{(}q^{1/2}+q^{-1/2}\big{)}x^{3}+\sum_{n=1}^{3}t_{n}\big{(}q^{h_{n}}+q^{l_{n}}\big{)}x^{2}-\big{\\{}2(t_{1}t_{2}+t_{2}t_{3}+t_{3}t_{1})+(q-1)E_{1}$
$\displaystyle\qquad{}+(q-1)^{2}\tilde{E}\big{\\}}x+t_{1}t_{2}t_{3}q^{(l_{1}+l_{2}+l_{3}+h_{1}+h_{2}+h_{3})/2}\big{(}q^{\beta/2}+q^{-\beta/2}\big{)}\bigg{]}g(x)=0,$
(3.14)

where

$\displaystyle
E_{1}=(h_{1}+h_{2}+l_{1}+l_{2})t_{1}t_{2}+(h_{2}+h_{3}+l_{2}+l_{3})t_{2}t_{3}+(h_{3}+h_{1}+l_{3}+l_{1})t_{3}t_{1}.$

Set $q=1+\varepsilon$. We divide equation (3.14) by $\varepsilon^{2}$. By
Taylor’s expansion

$\displaystyle
g(x/q)=g(x)+\big{(}{-}\varepsilon+\varepsilon^{2}\big{)}xg^{\prime}(x)+\varepsilon^{2}x^{2}g^{\prime\prime}(x)/2+O\big{(}\varepsilon^{3}\big{)},$
$\displaystyle g(qx)=g(x)+\varepsilon
xg^{\prime}(x)+\varepsilon^{2}x^{2}g^{\prime\prime}(x)/2+O\big{(}\varepsilon^{3}\big{)},$

we find the following limit as $\varepsilon\to 0$:

$\displaystyle x^{2}(x-t_{1})(x-t_{2})(x-t_{3})g^{\prime\prime}(x)$
$\displaystyle\qquad{}+x^{2}(x-t_{1})(x-t_{2})(x-t_{3})\left[\frac{h_{1}-l_{1}+1}{x-t_{1}}+\frac{h_{2}-l_{2}+1}{x-t_{2}}+\frac{h_{3}-l_{3}+1}{x-t_{3}}-\frac{2\tilde{l}}{x}\right]g^{\prime}(x)$
$\displaystyle\qquad{}-\big{[}x^{3}/4+\\{(2h_{1}-2l_{1}+1)t_{1}+(2h_{2}-2l_{2}+1)t_{2}+(2h_{3}-2l_{3}+1)t_{3}\\}x^{2}/4$
$\displaystyle\qquad{}+\tilde{B}x+t_{1}t_{2}t_{3}\big{(}\tilde{l}+1/2+\beta/2\big{)}\big{(}\tilde{l}+1/2-\beta/2\big{)}\big{]}g(x)=0,$
(3.15)

where

$\displaystyle\tilde{l}=(h_{1}+h_{2}+h_{3}-l_{1}-l_{2}-l_{3}+2)/2,\qquad\tilde{B}=\tilde{E}+\mbox{const}.$

This is a Fuchsian differential equation with five singularities
$\\{0,t_{1},t_{2},t_{3},\infty\\}$ and the local exponents are given by the
following Riemann scheme

$\displaystyle\begin{pmatrix}x=0&x=t_{1}&x=t_{2}&x=t_{3}&x=\infty\\\
\tilde{l}+\beta/2+1/2&0&0&0&-1/2\\\
\tilde{l}-\beta/2+1/2&l_{1}-h_{1}&l_{2}-h_{2}&l_{3}-h_{3}&1/2\end{pmatrix}.$

By the linear fractional transformation $z=t_{2}(1-t_{1}/x)/(t_{2}-t_{1})$ and
the gauge transformation $g(x)=x^{1/2}\tilde{g}(z)$, the function
$y=\tilde{g}(z)$ satisfies Heun’s differential equation

$\displaystyle\frac{{\rm d}^{2}y}{{\rm
d}z^{2}}+\left(\frac{\gamma}{z}+\frac{\delta}{z-1}+\frac{\epsilon}{z-t}\right)\frac{{\rm
d}y}{{\rm d}z}+\frac{\alpha^{\prime}\beta^{\prime}z-q}{z(z-1)(z-t)}y=0,$

where $t=t_{2}(t_{1}-t_{3})/\\{t_{3}(t_{1}-t_{2})\\}$, $\gamma=1+h_{1}-l_{1}$,
$\delta=1+h_{2}-l_{2}$, $\epsilon=1+h_{3}-l_{3}$,
$\\{\alpha^{\prime},\beta^{\prime}\\}=\\{\tilde{l}-\beta/2,\tilde{l}+\beta/2\\}$,
$q=\tilde{E}/(t_{3}(t_{2}-t_{1}))+\mbox{const}$. Hence the point $x=\infty$ in
equation (3.15) is essentially non-singular (apparent), and we may say that
the equation $(A^{\langle 3\rangle}-E)g(x)=0$ is a $q$-analogue of the
Fuchsian equation with four regular singularities $\\{0,t_{1},t_{2},t_{3}\\}$,
$t_{1},t_{2},t_{3}\in\mathbb{C}_{\neq 0}$.

### 3.3

Let $A^{\langle 2\rangle}$ be the operator in equation (1.7). Then the
equation

$\displaystyle\big{(}A^{\langle 2\rangle}-E\big{)}g(x)=0.$ (3.16)

for a fixed constant $E$ is a variant of the $q$-Heun equation. The
singularities $x=0$ and $x=\infty$ of equation (3.16) are regular, and the
constants $M$ and $N$ in Section 2 are given by $M=-2$ and $N=2$. The operator
$A^{\langle 2\rangle}$ acts on $x^{\mu}$ as

$\displaystyle A^{\langle 2\rangle}x^{\mu}=d^{\langle
2\rangle,++}(\mu)x^{\mu+2}+d^{\langle 2\rangle,+}(\mu)x^{\mu+1}+d^{\langle
2\rangle,0}(\mu)x^{\mu}$ $\displaystyle\hphantom{A^{\langle
2\rangle}x^{\mu}=}{}+d^{\langle 2\rangle,-}(\mu)x^{\mu-1}+d^{\langle
2\rangle,--}(\mu)x^{\mu-2},$ (3.17)

where

$\displaystyle d^{\langle
2\rangle,++}(\mu)=q^{-\mu}+q^{\mu}-q^{1/2}-q^{-1/2},$ $\displaystyle
d^{\langle
2\rangle,+}(\mu)=\sum_{i=1}^{4}\big{(}q^{h_{i}}+q^{l_{i}}-q^{h_{i}+1/2-\mu}-q^{l_{i}-1/2+\mu}\big{)}t_{i},$
$\displaystyle d^{\langle 2\rangle,0}(\mu)=\sum_{1\leq i<j\leq
4}\big{(}q^{h_{i}+h_{j}+1-\mu}+q^{l_{i}+l_{j}-1+\mu}\big{)}t_{i}t_{j},$
$\displaystyle d^{\langle
2\rangle,-}(\mu)=t_{1}t_{2}t_{3}t_{4}\sum_{i=1}^{4}\big{\\{}q^{(h_{1}+h_{2}+h_{3}+h_{4}+l_{1}+l_{2}+l_{3}+l_{4})/2}\big{(}q^{-h_{i}}+q^{-l_{i}}\big{)}$
$\displaystyle\hphantom{d^{\langle
2\rangle,-}(\mu)=}{}-q^{h_{1}+h_{2}+h_{3}+h_{4}+3/2-\mu}q^{-h_{i}}-q^{l_{1}+l_{2}+l_{3}+l_{4}-3/2+\mu}q^{-l_{i}}\big{\\}}t_{i}^{-1},$
$\displaystyle d^{\langle
2\rangle,--}(\mu)=\big{\\{}q^{h_{1}+h_{2}+h_{3}+h_{4}+2-\mu}+q^{l_{1}+l_{2}+l_{3}+l_{4}-2+\mu}$
$\displaystyle\hphantom{d^{\langle
2\rangle,--}(\mu)=}{}-q^{(l_{1}+l_{2}+l_{3}+l_{4}+h_{1}+h_{2}+h_{3}+l_{4})/2}\big{(}q^{1/2}+q^{-1/2}\big{)}\big{\\}}t_{1}t_{2}t_{3}t_{4}.$

We investigate the singularity $x=0$ of equation (3.16). We substitute
equation (3.3) into equation (3.16). Then we have

$\displaystyle d^{\langle 2\rangle,--}(\lambda)=0,\qquad c_{1}d^{\langle
2\rangle,--}(\lambda+1)=-c_{0}d^{\langle 2\rangle,-}(\lambda),$ $\displaystyle
c_{n}d^{\langle 3\rangle,--}(\lambda+n)=-\big{\\{}c_{n-1}d^{\langle
3\rangle,-}(\lambda+n-1)+c_{n-2}\big{(}d^{\langle
3\rangle,0}(\lambda+n-2)-E\big{)}$ $\displaystyle\hphantom{c_{n}d^{\langle
3\rangle,--}(\lambda+n)=}{}+c_{n-3}d^{\langle
3\rangle,+}(\lambda+n-3)+c_{n-4}d^{\langle
3\rangle,++}(\lambda+n-4)\big{\\}},\qquad n\geq 2,$

where we set $c_{-1}=c_{-2}=0$. The singularity $x=0$ is regular and the
characteristic equation at $x=0$ is written as $d^{\langle
2\rangle,--}(\lambda)=0$, i.e.,

$\displaystyle
q^{h_{1}+h_{2}+h_{3}+h_{4}+2-\lambda}+q^{l_{1}+l_{2}+l_{3}+l_{4}-2+\lambda}-q^{(l_{1}+l_{2}+l_{3}+l_{4}+h_{1}+h_{2}+h_{3}+l_{4})/2}\big{(}q^{1/2}+q^{-1/2}\big{)}=0.$

Hence the values

$\displaystyle\lambda_{1}=(h_{1}+h_{2}+h_{3}+h_{4}-l_{1}-l_{2}-l_{3}-l_{4}+3)/2,\qquad\lambda_{2}=\lambda_{1}+1$

are exponents about $x=0$, i.e., $d^{\langle
2\rangle,--}(\lambda_{1})=d^{\langle 2\rangle,--}(\lambda_{2})=0$. It follows
from $d^{\langle 2\rangle,-}(\lambda_{1})=0$ that the singularity $x=0$ is
apparent, and we have two independent solutions written as

$\displaystyle
x^{\lambda_{1}}\left\\{1+\sum_{n=2}^{\infty}c_{n}x^{n}\right\\},\qquad
x^{\lambda_{1}+1}\left\\{1+\sum_{n=1}^{\infty}c^{\prime}_{n}x^{n}\right\\}.$

We investigate the singularity $x=\infty$ of equation (3.16). We substitute
equation (3.5) into equation (3.16). Then we have

$\displaystyle d^{\langle
2\rangle,++}(-\lambda)=0,\qquad\tilde{c}_{1}d^{\langle
2\rangle,++}(-\lambda-1)=-\tilde{c}_{0}d^{\langle 2\rangle,+}(-\lambda),$
$\displaystyle\tilde{c}_{n}d^{\langle
2\rangle,++}(-\lambda-n)=-\big{\\{}\tilde{c}_{n-1}d^{\langle
2\rangle,+}(-\lambda-n+1)+\tilde{c}_{n-2}\big{(}d^{\langle
2\rangle,0}(-\lambda-n+2)-E\big{)}$
$\displaystyle\hphantom{\tilde{c}_{n}d^{\langle
2\rangle,++}(-\lambda-n)=}{}+\tilde{c}_{n-3}d^{\langle
2\rangle,+}(-\lambda-n+3)+\tilde{c}_{n-4}d^{\langle
2\rangle,++}(-\lambda-n+4)\big{\\}},\qquad n\geq 2,$

where we set $\tilde{c}_{-1}=\tilde{c}_{-2}=0$. The singularity $x=\infty$ is
regular and the characteristic equation at $x=\infty$ is written as
$d^{\langle 2\rangle,++}(-\lambda)=0$, i.e.,

$\displaystyle q^{\lambda}+q^{-\lambda}-q^{1/2}-q^{-1/2}=0.$

Hence the values $\lambda_{1}=-1/2$ and $\lambda_{2}=1/2$ are exponents about
$x=\infty$. It is also shown that the singularity $x=\infty$ is apparent.

We have observed that the regular singularities $x=0$ and $x=\infty$ of
equation (3.16) are apparent and the differences of the exponents are $1$.
Recall that equation (3.16) is written as

$\displaystyle a(x)g(x/q)+b(x)g(x)+c(x)g(qx)=0,$

where

$\displaystyle
a(x)=\big{(}x-q^{h_{1}+1/2}t_{1}\big{)}\big{(}x-q^{h_{2}+1/2}t_{2}\big{)}\big{(}x-q^{h_{3}+1/2}t_{3}\big{)}\big{(}x-q^{h_{4}+1/2}t_{4}\big{)},$
$\displaystyle b(x)=b_{4}x^{4}+b_{3}x^{3}+Ex^{2}+b_{1}x+b_{0},$ $\displaystyle
c(x)=\big{(}x-q^{l_{1}-1/2}t_{1}\big{)}\big{(}x-q^{l_{2}-1/2}t_{2}\big{)}\big{(}x-q^{l_{3}-1/2}t_{3}\big{)}\big{(}x-q^{l_{4}-1/2}t_{4}\big{)},$
(3.18)

and

$\displaystyle b_{4}=-\big{(}q^{1/2}+q^{-1/2}\big{)},\qquad
b_{3}=\sum_{n=1}^{4}\big{(}q^{h_{n}}+q^{l_{n}}\big{)}t_{n},$ $\displaystyle
b_{1}=q^{(l_{1}+l_{2}+l_{3}+l_{4}+h_{1}+h_{2}+h_{3}+h_{4})/2}t_{1}t_{2}t_{3}t_{4}\sum_{n=1}^{4}\big{(}q^{-h_{n}}+q^{-l_{n}}\big{)}t_{n}^{-1},$
$\displaystyle
b_{0}=-q^{(l_{1}+l_{2}+l_{3}+l_{4}+h_{1}+h_{2}+h_{3}+h_{4})/2}\big{(}q^{1/2}+q^{-1/2}\big{)}t_{1}t_{2}t_{3}t_{4}.$
(3.19)

Note that the parameter $E$ is independent of the exponents about $x=0,\infty$
and apparency of the regular singularities $x=0,\infty$. Therefore it is
reasonable to regard $E$ as an accessory parameter.

Next we consider a characterization of the coefficients $b_{4}$, $b_{3}$,
$b_{1}$ and $b_{0}$ as an analogue of Theorem 3.1. Namely we have the
following theorem, which is proved similarly to Theorem 3.1.

######  Theorem 3.2.

Assume that the functions $a(x)$, $b(x)$ and $c(x)$ take the form of equation
(3.18). If the difference of the exponents about $x=0$ is $1$, the difference
of the exponents about $x=\infty$ is $1$, and the regular singularities
$x=0,\infty$ are apparent, then we obtain equation (3.19) and recover a
variant of the $q$-Heun equation in equation (3.16).

We are going to obtain a differential equation from the difference equation
(3.16) by taking a suitable limit. We rewrite equation (3.16) as

$\displaystyle\big{(}x-t_{1}q^{h_{1}+1/2}\big{)}\big{(}x-t_{2}q^{h_{2}+1/2}\big{)}\big{(}x-t_{3}q^{h_{3}+1/2}\big{)}\big{(}x-t_{4}q^{h_{4}+1/2}\big{)}g(x/q)$
$\displaystyle\qquad{}+\big{(}x-t_{1}q^{l_{1}-1/2}\big{)}\big{(}x-t_{2}q^{l_{2}-1/2}\big{)}\big{(}x-t_{3}q^{l_{3}-1/2}\big{)}\big{(}x-t_{4}q^{l_{4}-1/2}\big{)}g(xq)$
$\displaystyle\qquad{}+\bigg{[}{}-\big{(}q^{1/2}+q^{-1/2}\big{)}x^{4}+\sum_{n=1}^{4}t_{n}\big{(}q^{h_{n}}+q^{l_{n}}\big{)}x^{3}$
$\displaystyle\qquad{}-\big{(}2(t_{1}t_{2}+t_{1}t_{3}+t_{1}t_{4}+t_{2}t_{3}+t_{2}t_{4}+t_{3}t_{4})+(q-1)E_{1}+(q-1)^{2}\tilde{E}\big{)}x^{2}$
$\displaystyle\qquad{}+t_{1}t_{2}t_{3}t_{4}q^{(l_{1}+l_{2}+l_{3}+l_{4}+h_{1}+h_{2}+h_{3}+h_{4})/2}\bigg{\\{}\sum_{n=1}^{4}\big{(}1/\big{(}t_{n}q^{h_{n}}\big{)}+1/\big{(}t_{n}q^{l_{n}}\big{)}\big{)}x$
$\displaystyle\qquad{}-\big{(}q^{1/2}+q^{-1/2}\big{)}\bigg{\\}}\bigg{]}g(x)=0,$

where

$\displaystyle E_{1}=\sum_{1\leq m<n\leq
4}(h_{m}+h_{n}+l_{m}+l_{n})t_{m}t_{n}.$

Set $q=1+\varepsilon$. Then we obtain the following differential equation
after taking the limit $\varepsilon\to 0$.

$\displaystyle
x^{2}(x-t_{1})(x-t_{2})(x-t_{3})(x-t_{4})\left[g^{\prime\prime}(x)+\left\\{-\frac{2\tilde{l}}{x}+\sum_{i=1}^{4}\frac{h_{i}-l_{i}+1}{x-t_{i}}\right\\}g^{\prime}(x)\right]$
$\displaystyle\qquad{}-\left[\frac{x^{4}}{4}+\left\\{\sum_{i=1}^{4}(2h_{i}-2l_{i}+1)t_{i}\right\\}\frac{x^{3}}{4}+\tilde{B}x^{2}\right.$
$\displaystyle\left.\qquad{}+t_{1}t_{2}t_{3}t_{4}\tilde{l}\left\\{\sum_{i=1}^{4}\frac{\tilde{l}-h_{i}+l_{i}}{t_{i}}\right\\}x+t_{1}t_{2}t_{3}t_{4}\tilde{l}\big{(}\tilde{l}+1\big{)}\right]g(x)=0,$
(3.20)

where

$\displaystyle\tilde{l}=(h_{1}+h_{2}+h_{3}+h_{4}-l_{1}-l_{2}-l_{3}-l_{4}+3)/2,\qquad\tilde{B}=\tilde{E}+\mbox{const}.$

This is a Fuchsian differential equation with six singularities
$\\{0,t_{1},t_{2},t_{3},t_{4},\infty\\}$ and the local exponents are given by
the following Riemann scheme

$\displaystyle\begin{pmatrix}x=0&x=t_{1}&x=t_{2}&x=t_{3}&x=t_{4}&x=\infty\\\
\tilde{l}&0&0&0&0&-1/2\\\
\tilde{l}+1&l_{1}-h_{1}&l_{2}-h_{2}&l_{3}-h_{3}&l_{4}-h_{4}&1/2\end{pmatrix}.$

By the linear fractional transformation
$z=(x-t_{2})(t_{3}-t_{1})/\\{(x-t_{1})(t_{3}-t_{2})\\}$ and the gauge
transformation $g(x)=x^{\tilde{l}}(x-t_{1})^{-\tilde{l}+1/2}\tilde{g}(z)$, the
function $y=\tilde{g}(z)$ satisfies Heun’s differential equation

$\displaystyle\frac{{\rm d}^{2}y}{{\rm
d}z^{2}}+\left(\frac{\gamma}{z}+\frac{\delta}{z-1}+\frac{\epsilon}{z-t}\right)\frac{{\rm
d}y}{{\rm d}z}+\frac{\alpha^{\prime}\beta^{\prime}z-q}{z(z-1)(z-t)}y=0,$

where $t=(t_{4}-t_{2})(t_{3}-t_{1})/\\{(t_{4}-t_{1})(t_{3}-t_{2})\\}$,
$\gamma=1+h_{2}-l_{2}$, $\delta=1+h_{3}-l_{3}$, $\epsilon=1+h_{4}-l_{4}$,
$\\{\alpha^{\prime},\beta^{\prime}\\}=\\{\tilde{l}-1/2,\tilde{l}-1/2+l_{1}-h_{1}\\}$,
$q=\tilde{E}/((t_{1}-t_{4})(t_{3}-t_{2}))+\mbox{const}$. Hence the points
$x=0,\infty$ in equation (3.20) are essentially non-singular (apparent), and
we may say that the equation $\big{(}A^{\langle 2\rangle}-E\big{)}g(x)=0$ is a
$q$-analogue of the Fuchsian equation with four regular singularities
$\\{t_{1},t_{2},t_{3},t_{4}\\}$, $t_{1},t_{2},t_{3},t_{4}\in\mathbb{C}_{\neq
0}$.

##  4 Invariant subspaces related with the $\boldsymbol{q}$-Heun equation
and its variants

In order to find special solutions to the $q$-Heun equation or its variants,
we consider quasi-exact solvability [11], i.e., we investigate subspaces which
are invariant under the action of the operators $A^{\langle 4\rangle}$,
$A^{\langle 3\rangle}$ or $A^{\langle 2\rangle}$.

We look for subspaces which are invariant under the action of the operator
$A^{\langle 4\rangle}$ given in equation (1.5).

######  Proposition 4.1.

Set

$\displaystyle\lambda_{1}=(h_{1}+h_{2}-l_{1}-l_{2}-\alpha_{1}-\alpha_{2}-\beta+2)/2,$
$\displaystyle\lambda_{2}=(h_{1}+h_{2}-l_{1}-l_{2}-\alpha_{1}-\alpha_{2}+\beta+2)/2.$
(4.1)

Let $\lambda\in\\{\lambda_{1},\lambda_{2}\\}$,
$\alpha\in\\{\alpha_{1},\alpha_{2}\\}$ and assume that $n:=-\lambda-\alpha$ is
a non-negative integer. Let $V^{\langle 4\rangle}$ be the space spanned by the
monomials $x^{\lambda+k}$, $k=0,\dots,n$, i.e.,

$\displaystyle V^{\langle
4\rangle}=\big{\\{}c_{0}x^{\lambda}+c_{1}x^{\lambda+1}+\dots+c_{n}x^{\lambda+n}\,|\,c_{0},c_{1},\dots,c_{n}\in\mathbb{C}\big{\\}}.$

Then the operator $A^{\langle 4\rangle}$ preserves the space $V^{\langle
4\rangle}$.

###### Proof.

The operator $A^{\langle 4\rangle}$ acts on $x^{\lambda+k}$ as

$\displaystyle A^{\langle 4\rangle}x^{\lambda+k}=d^{\langle
4\rangle,+}(\lambda+k)x^{\lambda+k+1}+d^{\langle
4\rangle,0}(\lambda+k)x^{\lambda+k}+d^{\langle
4\rangle,-}(\lambda+k)x^{\lambda+k-1}.$

See equation (3.2) for the values $d^{\langle 4\rangle,+}(\lambda+k)$,
$d^{\langle 4\rangle,0}(\lambda+k)$ and $d^{\langle 4\rangle,-}(\lambda+k)$.
Therefore we have $A^{\langle 4\rangle}x^{\lambda+k}\in V^{\langle 4\rangle}$
for $k=1,\dots,n-1$. When $k=0$, we have $d^{\langle 4\rangle,-}(\lambda)=0$
and $A^{\langle 4\rangle}x^{\lambda}\in V^{\langle 4\rangle}$. When $k=n$, we
have $d^{\langle 4\rangle,+}(\lambda+n)=d^{\langle 4\rangle,+}(-\alpha)=0$ and
$A^{\langle 4\rangle}x^{\lambda+n}\in V^{\langle 4\rangle}$. ∎

Note that the values $\lambda_{1}$ and $\lambda_{2}$ in equation (4.1) are
exponents of the $q$-Heun equation (equation (3.1)) about $x=0$, and the
values $\alpha_{1}$ and $\alpha_{2}$ are exponents about $x=\infty$.

We look for subspaces which are invariant under the action of the operator
$A^{\langle 3\rangle}$ in equation (1.6).

######  Proposition 4.2.

Set

$\displaystyle\lambda_{1}=(h_{1}+h_{2}+h_{3}-l_{1}-l_{2}-l_{3}-\beta+3)/2,$
$\displaystyle\lambda_{2}=(h_{1}+h_{2}+h_{3}-l_{1}-l_{2}-l_{3}+\beta+3)/2.$
(4.2)

Let $\lambda\in\\{\lambda_{1},\lambda_{2}\\}$ and assume that
$n:=-\lambda+1/2$ is a non-negative integer. Let $V^{\langle 3\rangle}$ be the
space spanned by the monomials $x^{\lambda+k}$, $k=0,\dots,n$, i.e.,

$\displaystyle V^{\langle
3\rangle}=\big{\\{}c_{0}x^{\lambda}+c_{1}x^{\lambda+1}+\dots+c_{n}x^{1/2}\,|\,c_{0},c_{1},\dots,c_{n}\in\mathbb{C}\big{\\}}.$

Then the operator $A^{\langle 3\rangle}$ preserves the space $V^{\langle
3\rangle}$.

###### Proof.

It follows from equation (3.8) that

$\displaystyle A^{\langle 3\rangle}x^{\lambda+k}=d^{\langle
3\rangle,++}(\lambda+k)x^{\lambda+k+2}+d^{\langle
3\rangle,+}(\lambda+k)x^{\lambda+k+1}$ $\displaystyle\hphantom{A^{\langle
3\rangle}x^{\lambda+k}=}{}+d^{\langle
3\rangle,0}(\lambda+k)x^{\lambda+k}+d^{\langle
3\rangle,-}(\lambda+k)x^{\lambda+k-1}.$

Since $d^{\langle 3\rangle,-}(\lambda)=d^{\langle
3\rangle,++}(-1/2)=d^{\langle 3\rangle,++}(1/2)=d^{\langle
3\rangle,+}(1/2)=0$, we obtain the proposition. ∎

Note that the values $\lambda_{1}$ and $\lambda_{2}$ in equation (4.2) are
exponents of equation (3.7) about $x=0$, and the values $-1/2$ and $1/2$ are
exponents about $x=\infty$.

Invariant subspaces of the operator $A^{\langle 2\rangle}$ in equation (1.7)
are described as follows:

######  Proposition 4.3.

Set

$\displaystyle\lambda=(h_{1}+h_{2}+h_{3}+h_{4}-l_{1}-l_{2}-l_{3}-l_{4}+3)/2$

and assume that $n:=-\lambda+1/2$ is a non-negative integer. Let $V^{\langle
2\rangle}$ be the space spanned by the monomials $x^{\lambda+k}$,
$k=0,\dots,n$, i.e.,

$\displaystyle V^{\langle
2\rangle}=\big{\\{}c_{0}x^{\lambda}+c_{1}x^{\lambda+1}+\dots+c_{n}x^{1/2}\,|\,c_{0},c_{1},\dots,c_{n}\in\mathbb{C}\big{\\}}.$

Then the operator $A^{\langle 2\rangle}$ preserves the space $V^{\langle
2\rangle}$.

###### Proof.

It follows from equation (3.17) that

$\displaystyle A^{\langle 2\rangle}x^{\lambda+k}=d^{\langle
2\rangle,++}(\lambda+k)x^{\lambda+k+2}+d^{\langle
2\rangle,+}(\lambda+k)x^{\lambda+k+1}$ $\displaystyle\hphantom{A^{\langle
2\rangle}x^{\lambda+k}=}{}+d^{\langle
2\rangle,0}(\lambda+k)x^{\lambda+k}+d^{\langle
2\rangle,-}(\lambda+k)x^{\lambda+k-1}+d^{\langle
2\rangle,--}(\lambda+k)x^{\lambda+k-2}.$

Since $d^{\langle 2\rangle,--}(\lambda)=d^{\langle
2\rangle,-}(\lambda)=d^{\langle 2\rangle,--}(\lambda+1)=d^{\langle
2\rangle,++}(-1/2)=d^{\langle 2\rangle,++}(1/2)=d^{\langle
2\rangle,+}(1/2)=0$, we obtain the proposition. ∎

If the dimension of the invariant subspace is one, then the eigenvalue and the
eigenfunction of the subspace are calculated explicitly, and they are
described as follows:

######  Proposition 4.4.

Let $\lambda_{1}$, $\lambda_{2}$ be the values in equation (4.1),
$\lambda\in\\{\lambda_{1},\lambda_{2}\\}$,
$\alpha\in\\{\alpha_{1},\alpha_{2}\\}$ and assume that $\lambda=-\alpha$. Then
the operator $A^{\langle 4\rangle}$ preserves the one-dimensional space
spanned by the function $x^{\lambda}$, and we have

$\displaystyle A^{\langle 4\rangle}x^{\lambda}=d^{\langle
4\rangle,0}(-\alpha)x^{\lambda},$ $\displaystyle d^{\langle
4\rangle,0}(-\alpha)=-\big{(}q^{h_{1}+1/2}t_{1}+q^{h_{2}+1/2}t_{2}\big{)}q^{\alpha}-\big{(}q^{l_{1}-1/2}t_{1}+q^{l_{2}-1/2}t_{2}\big{)}q^{\alpha_{1}+\alpha_{2}-\alpha}.$

If $\lambda=\lambda_{1}$, $\alpha=\alpha_{1}$ and $-\lambda_{1}-\alpha_{1}=0$,
then we recover the eigenfunction of $A^{\langle 4\rangle}$ in the
introduction (see equation (1.9)).

######  Proposition 4.5.

Assume that $\beta=h_{1}+h_{2}+h_{3}-l_{1}-l_{2}-l_{3}+2$ or
$\beta=-(h_{1}+h_{2}+h_{3}-l_{1}-l_{2}-l_{3}+2)$. Then the operator
$A^{\langle 3\rangle}$ preserves the one-dimensional space spanned by the
function $x^{1/2}$, and we have

$\displaystyle A^{\langle 3\rangle}x^{1/2}=\bigg{[}\sum_{1\leq i<j\leq
3}\big{(}q^{h_{i}+h_{j}+1/2}+q^{l_{i}+l_{j}-1/2}\big{)}t_{i}t_{j}\bigg{]}x^{1/2}.$

######  Proposition 4.6.

Assume that $h_{1}+h_{2}+h_{3}+h_{4}-l_{1}-l_{2}-l_{3}-l_{4}+2=0$. Then the
operator $A^{\langle 2\rangle}$ preserves the one-dimensional space spanned by
the function $x^{1/2}$, and we have

$\displaystyle A^{\langle 2\rangle}x^{1/2}=\bigg{[}\sum_{1\leq i<j\leq
4}\big{(}q^{h_{i}+h_{j}+1/2}+q^{l_{i}+l_{j}-1/2}\big{)}t_{i}t_{j}\bigg{]}x^{1/2}.$

### Acknowledgements

The author is grateful to Simon Ruijsenaars for valuable comments and fruitful
discussions. He thanks to the referees and the editor for valuable comments.
He is supported by JSPS KAKENHI Grant Number JP26400122.

## References

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# OpenICS: Open Image Compressive Sensing Toolbox and Benchmark

Jonathan Zhao*, Matthew Westerham*, Mark Lakatos-Toth*, Zhikang Zhang*, Avi
Moskoff, Fengbo Ren
* equally contribute Parallel Systems and Computing Laboratory
Arizona State University
Tempe, USA
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We present OpenICS, an image compressive sensing toolbox that includes
multiple image compressive sensing and reconstruction algorithms proposed in
the past decade. Due to the lack of standardization in the implementation and
evaluation of the proposed algorithms, the application of image compressive
sensing in the real-world is limited. We believe this toolbox is the first
framework that provides a unified and standardized implementation of multiple
image compressive sensing algorithms. In addition, we also conduct a
benchmarking study on the methods included in this framework from two aspects:
reconstruction accuracy and reconstruction efficiency. We wish this toolbox
and benchmark can serve the growing research community of compressive sensing
and the industry applying image compressive sensing to new problems as well as
developing new methods more efficiently. Code and models are available at
https://github.com/PSCLab-ASU/OpenICS. The project is still under maintenance,
and we will keep this document updated.

###### Index Terms:

compressive sensing, computer vision, machine learning, signal processing

##  I Introduction

Compressive sensing is a signal sensing technique that performs the sensing
and compression of signals simultaneously to reduce the sensing cost without
losing information. Over the past decade, there is a wide variety of image
compressive sensing reconstruction methods proposed. However, due to the lack
of standardization in the implementation and evaluation of the proposed
algorithms, the application of image compressive sensing in the real-world is
still limited. Towards the goal of efficient deployment and evaluation of
image compressive sensing, we build OpenICS which is an image compressive
sensing toolbox containing multiple image compressive sensing reconstruction
methods implemented in a unified interface and structure.

Major features of OpenICS are 1. Unified interface. We rewrite the code of
multiple image compressive sensing algorithms to build a unified interface for
all the methods. This unified design greatly improves the usability and
availability of our toolbox. 2. Modular design. Each method locates in a
separate folder, and there is no cross-dependency between different methods.
This modular design improves the reusability of our toolbox. 3. Out-of-the-box
usage. Our toolbox contains the most representative methods of image
compressive sensing methods. See Section 2 for the full list.

In addition to the toolbox we implemented, we also propose a benchmark to
evaluate all the methods included in the toolbox from two aspects:
reconstruction accuracy and reconstruction speed. The benchmark results
include the performance of all the methods evaluated on six different datasets
and five different compression ratios. We believe our benchmark is the most
complete benchmark in the domain of image compressive sensing so far in terms
of the variety of datasets and the range of compression ratios.

Our contribution are summarized as follows:

1\. We provide a toolbox in the domain of image compressive sensing that
consists of multiple most representative algorithms in this domain. The
toolbox has a unified interface, modular design, and it can be used out of
box.

2\. We propose a benchmark in the domain of image compressive sensing and use
it to evaluate the methods included in our toolbox. It is by far the most
complete benchmark in the domain of image compressive sensing in terms of the
variety of datasets and the range of compression ratios.

##  II Methods Included

OpenICS contains implementations of multiple image CS reconstruction methods.
Based on whether the method is data-dependent, we divide implemented methods
into two categories: Model-based methods and data-driven methods.

###  II-A Model-based Methods

Model-based methods use pre-defined models based on prior knowledge of the
signals to perform the reconstruction. The included model-based methods are
listed and summarized in table I.

L1[1]: The first reconstruction methods in the domain of compressive
sensing(Only the total-variation-based methods are currently implemented).

NLR-CS[2]: A reconstruction method based on non-local low-rank regularization.

TVAL-3[3]: An efficient image reconstruction method based on total variation
minimization.

D-AMP[4]: An reconstruction method based on model-based image denoising
algorithms.

###  II-B Data-driven Methods

Data-driven methods do not rely on pre-defined models of signals. Instead,
they use neural networks to model the images and perform the reconstruction
tasks. The included data-driven methods are listed below.

ReconNet[5]: An end-to-end reconstruction network based on convolutional
neural networks.

LDAMP[6]: An end-to-end reconstruction network built from the unrolled
iterative image denoising process by replacing the model-based image denoisers
with neural-network-based denoisers.

ISTA-Net[7]: An end-to-end reconstruction network built by unrolling the
conventional iterative shrinkage-thresholding algorithm.

LAPRAN[8]: An end-to-end reconstruction network based on deep laplacian
pyramid neural networks.

CSGM[9]: An iterative reconstruction method based on generative adversial
neural network.

CSGAN[10]: A variant of CSGM method enhanced by meta-learning to improve
reconstruction speed.

Methods | Data dependent | Running process | Platform
---|---|---|---
L1 | No | Iterative | CPU
TVAL-3 | No | Iterative | CPU
NLR-CS | No | Iterative | CPU
D-AMP | No | Iterative | CPU
ReconNet | Yes | End-to-end | GPU
ISTA-Net | Yes | End-to-end | GPU
LDAMP | Yes | End-to-end | GPU
CSGM | Yes | Iterative | GPU
LAPRAN | Yes | End-to-end | GPU
CSGAN | Yes | Iterative | GPU
TABLE I: List of methods included in OpenICS

##  III Architecture

###  III-A Toolbox Structure

There are two programing languages used to implement all the methods. L1, NLR-
CS, TVAL-3, D-AMP are implemented in Matlab. ReconNet, ISTA-Net, LAPRAN are
implemented in Python with Pytorch[11]. CSGM, CSGAN, LDAMP are implemented in
Python with Tensorflow[12].

We provide a unified interface to run all the methods. Specifically, the
common parameters of all methods are listed as follows:

  1. 1.

dataset: the name of dataset to be used

  2. 2.

input_channel: number of channels training/testing images have

  3. 3.

input_width: width of training/testing images

  4. 4.

input_height: height of training/testing images

  5. 5.

m: number of measurements/outputs of sensing matrix

  6. 6.

n: number of inputs to sensing matrix

Besides, there are method-specific parameters that are included in a
container-like object called ”specifics”. In python, it is a dictionary with
its keys as parameter names and its values as actual parameters. In Matlab, it
is a structure array with its field names as parameter names, and its field
values are the parameters.

We also provide the functionality to directly call certain methods from the
main interface. The parameters of the main interface are listed below:

  1. 1.

sensing: method of sensing

  2. 2.

reconstruction: method of reconstruction

  3. 3.

stage: training or testing(model-based methods do not have this parameter)

  4. 4.

default: will use default parameters if it’s true. Will override other
parameters set manually.

  5. 5.

dataset: same as method’s corresponding parameter.

  6. 6.

input_channel: same as method’s corresponding parameter.

  7. 7.

input_width: same as method’s corresponding parameter.

  8. 8.

input_height: same as method’s corresponding parameter.

  9. 9.

m: same as method’s corresponding parameter.

  10. 10.

n: same as method’s corresponding parameter.

  11. 11.

specifics: specific parameter settings of chosen reconstruction method. Will
be passed to the actual method.

Given an image to be sensed and reconstructed, for model-based methods, it can
be directly reconstructed by calling the main function of specific methods.
For data-driven methods, the networks of specific methods have to be trained
first. We provide pre-trained networks of each method at five compression
ratios(2,4,8,16,32) on six datasets. We also provide the functionality of
training new networks on new datasets and compression ratios from scratch.

More details regarding how to use our code are listed in the main page of our
Github repository(https://github.com/PSCLab-ASU/OpenICS).

##  IV Benchmarks

###  IV-A Benchmark Design

Dataset. We use six widely used datasets to evaluate all the methods in
benchmark. They are MNIST[13], CIFAR10[14], CIFAR10(grayscaled), CELEBA[15],
Bigset, Bigset(grayscaled). Bigset stands for a manually composed dataset. It
was initially used in [16, 17, 18] in the domain of single image super-
resolution. Later it was used in LAPRAN[8] for image compressive sensing. The
training set of Bigset was composed of 91 images from [19] and 200 images from
the BSD[20] dataset. The 291 images are augmented (rotation and flip) and cut
into 228688 patches as training samples. The testing set of Bigset consists of
image patches from Set5[21] and Set14[22] with same patch size. For MNIST,
CIFAR10, CIFAR10(gray), the image size of samples is 32x32. For CELEBA,
Bigset(gray) and Bigset, the image size of samples is 64x64.

Compression ratios. We take five different compression ratios to evaluate each
method: 2,4,8,16,32. The compression is always performed channel-wise, i.e.,
for colored images(with RGB color channels), we perform the compression over
each channel seperately. The measurements of all three channels are then
grouped together for subsequent reconstruction. The training procedure of each
data-driven method is almost the same as the original training guideline
provided by the original authors. The discrepancy is detailed in our github
repository.

Metrics. We evaluate all the methods from two aspects: reconstruction accuracy
and reconstruction speed. The reconstruction accuracy is quantified with two
metrics: PSNR(0-48) and SSIM(0-1) between reconstructed images and original
images in the testing set. Higher values indicate higher accuracy. For each
experiment we conduct, the reported results are the averaged values of both
metrics over all the samples of the corresponding testing set. The
reconstruction speed is quantified with the number of images reconstructed per
second. This value is averaged over all the samples in the testing set as
well.

Benchmark calculation. After obtaining the all raw benchmark results of each
method(total of $6\text{\ datasets}\times 5\text{\ compression ratios}\times
3\text{\ metrics}=90\text{\ raw results}$), we use the following equation to
calculate the final benchmark score:

$score=\sum_{i=1}^{90}w_{dataset}*w_{cr}*w_{metric}*\bar{v}_{i}$ (1)

$\bar{v}_{i}$ is the ith normalized raw experiment result. Due to the
different value ranges of each metric, we have to normalize the raw values to
0-100 range to avoid the dominance of one metric over the others. The function
used to normalize PSNR values is $\bar{v}=10^{\frac{v}{48}-1}*100$. The
function used to normalize SSIM values is $\bar{v}=10^{v-1}*100$. The function
used to normalize reconstruction speed values is
$\bar{v}=\frac{100}{1+1/log(1+v)}$.

$w_{dataset}$ is the weight of the corresponding dataset of $v_{i}$. Different
weights are assigned to different datasets according to their relative
complexity in reconstruction compared with other datasets. The relative
complexity is determined based on the results reported in literatures[1, 3, 2,
4, 5, 7, 6, 8, 9, 10] in the domain of image compressive sensing.

$w_{cr}$ is the weight of the corresponding compression ratio of $v_{i}$.
Since images compressed at higher compression ratios are more difficult to
reconstruct than images compressed at lower compression ratios, we assign
higher weights to higher compression ratios.

$w_{metric}$ is the weight of corresponding metric of $v_{i}$. we assign
different weights to different metrics as
$\text{PSNR}:\text{SSIM}:\text{Speed}=1:1:2$. As such, there is no bias
between reconstruction accuracy and reconstruction speed. One can specify own
weights to different metrics to make the score reflects one’s own preferences.

The actual weights are listed in table II,table III and table IV.

Dataset | Weight
---|---
MNIST | 1/21
CelebA | 4/21
CIFAR10 | 3/21
CIFAR10 Gray | 2/21
Bigset | 6/21
Bigset Gray | 5/21
TABLE II: Weights of Datasets Compression ratio | Weight
---|---
2 | 1/31
4 | 2/31
8 | 4/31
16 | 8/31
32 | 16/31
TABLE III: Weights of compression ratios Metric | Weight
---|---
PSNR | 1/4
SSIM | 1/4
Reconstruction speed | 1/2
TABLE IV: Weights of metrics

###  IV-B Benchmark Results

The raw benchmark results are listed in Table VI,VII,VIII,IX,X,XI,XII,XIII,XIV
and XV in the appendix. The benchmark score of each method is shown in Table V
and Fig 1.

Method | Speed | Accuracy | Score
---|---|---|---
LDAMP | 30.25 | 17.21 | 47.46
ISTA-Net | 30.02 | 20.69 | 50.71
CSGAN | 32.58 | 19.03 | 51.61
LAPRAN | 34.69 | 23.60 | 58.30
CSGM | 4.75 | 13.07 | 17.82
ReconNet | 37.00 | 19.15 | 56.15
TVAL-3 | 18.43 | 18.92 | 37.35
L1 | 3.78 | 19.69 | 23.46
D-AMP | 2.35 | 21.83 | 24.19
NLR-CS | 1.69 | 20.35 | 22.04
TABLE V: The benchmark scores Figure 1: The benchamark scores

LAPRAN has the highest benchmark score due to its prominent performance in
accuracy and speed. LDAMP has the highest performance in accuracy but bad
performance in speed due to its heavyweight design of network structure(more
than 200 neural layers). ReconNet has the highest performance in
reconstruction speed due to its lightweight design in structure(only seven
layers), but its performance in accuracy is limited as well. In general,
model-based methods have lower performance on both accuracy and speed than
data-driven methods due to their static, pre-defined signal prior and
iterative running process. CSGM is a special case in data-driven methods. The
unsatisfying performance in accuracy is due to the GAN model it uses, which is
DCGAN[23] proposed in 2015. Over the past few years, there have been more
successful GAN models proposed, such as StyleGAN[24] that has much higher
performance in modeling signals from data, which may improve the performance
of CSGM if it is used. For all the model-based methods, NLR-CS and D-AMP have
higher performance in reconstruction accuracy but lower performance in
reconstruction speed compared with the other two methods.

To conclude, in general, data-driven methods achieve the highest performance
in terms of accuracy and performance. With enough training data and hardware
platforms that have sufficient computation capacity, one should always choose
end-to-end data-driven methods. If there is no sufficient data, one should
choose model-based methods that have the highest reconstruction accuracy. If
the reconstruction speed is a critical factor to consider as well, TVAL-3 has
a significantly higher reconstruction speed than other model-based methods and
comparable reconstruction accuracy to other methods.

## Acknowledgment

This work is supported by the Research Experiences for Undergraduates (REU)
funding of an NSF grant (IIS/CPS-1652038) and the Fulton Undergraduate
Research Initiative (FURI) program at Arizona State University. Part of the
NVIDIA GPUs used for this work was donated by NVIDIA Corporation. The CPU
servers used for this work were donated by Intel Corporation.

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##  V Appendix

| Compression ratio | PSNR | SSIM | Reconstruction per second
---|---|---|---|---
MNIST | 2 | 47.9998 | 0.9997 | 27.5482
MNIST | 4 | 47.5808 | 0.9994 | 34.6021
MNIST | 8 | 39.8992 | 0.9965 | 34.9650
MNIST | 16 | 30.8985 | 0.9751 | 34.7222
MNIST | 32 | 16.2898 | 0.5901 | 34.3643
CelebA | 2 | 41.1183 | 0.9911 | 23.4192
CelebA | 4 | 32.0614 | 0.9459 | 27.5482
CelebA | 8 | 27.5465 | 0.8723 | 30.9598
CelebA | 16 | 23.592 | 0.7415 | 33.5570
CelebA | 32 | 17.5607 | 0.3732 | 32.5733
CIFAR10 | 2 | 35.1695 | 0.9748 | 35.2113
CIFAR10 | 4 | 28.5017 | 0.9052 | 35.4610
CIFAR10 | 8 | 22.7743 | 0.7285 | 33.5570
CIFAR10 | 16 | 18.8613 | 0.498 | 34.6021
CIFAR10 | 32 | 13.7782 | 0.1406 | 34.1297
CIFAR10(Gray) | 2 | 34.3648 | 0.9713 | 32.5733
CIFAR10(Gray) | 4 | 27.8835 | 0.8945 | 33.0033
CIFAR10(Gray) | 8 | 23.2726 | 0.7499 | 34.4828
CIFAR10(Gray) | 16 | 16.6404 | 0.3471 | 35.2113
CIFAR10(Gray) | 32 | 12.2314 | 0.0701 | 35.0877
Bigset | 2 | 38.4191 | 0.9432 | 23.9234
Bigset | 4 | 34.3927 | 0.8936 | 29.1545
Bigset | 8 | 30.9195 | 0.8096 | 29.2398
Bigset | 16 | 27.608 | 0.7055 | 34.3643
Bigset | 32 | 17.3593 | 0.1853 | 33.2226
Bigset(Gray) | 2 | 38.5702 | 0.9508 | 22.8311
Bigset(Gray) | 4 | 34.5885 | 0.8956 | 27.3224
Bigset(Gray) | 8 | 31.0164 | 0.808 | 30.1205
Bigset(Gray) | 16 | 28.3581 | 0.7207 | 33.4448
Bigset(Gray) | 32 | 17.283 | 0.178 | 35.5872
TABLE VI: Benchmark results of LDAMP Dataset | Compression ratio | PSNR | SSIM | Reconstruction per second
---|---|---|---|---
MNIST | 2 | 47.99 | 0.9999 | 93.4579
MNIST | 4 | 44.27 | 0.9988 | 55.2486
MNIST | 8 | 35.12 | 0.9907 | 75.7576
MNIST | 16 | 27.31 | 0.9532 | 75.7576
MNIST | 32 | 19.76 | 0.7747 | 82.6446
CelebA | 2 | 37.43 | 0.9798 | 11.9190
CelebA | 4 | 31.14 | 0.9297 | 14.3062
CelebA | 8 | 27.07 | 0.8499 | 4.8170
CelebA | 16 | 23.77 | 0.7406 | 7.3475
CelebA | 32 | 21.13 | 0.6295 | 17.5439
CIFAR10 | 2 | 34.12 | 0.9703 | 30.1205
CIFAR10 | 4 | 27.66 | 0.8932 | 33.2226
CIFAR10 | 8 | 23.41 | 0.7632 | 31.4465
CIFAR10 | 16 | 20.25 | 0.5979 | 25.9067
CIFAR10 | 32 | 17.95 | 0.435 | 26.3852
CIFAR10(Gray) | 2 | 33.63 | 0.9679 | 138.8889
CIFAR10(Gray) | 4 | 27.46 | 0.8886 | 68.0272
CIFAR10(Gray) | 8 | 23.15 | 0.7501 | 75.1880
CIFAR10(Gray) | 16 | 20.25 | 0.5911 | 81.3008
CIFAR10(Gray) | 32 | 18.13 | 0.4406 | 86.2069
Bigset | 2 | 37.28 | 0.9393 | 15.8479
Bigset | 4 | 33 | 0.8686 | 19.5313
Bigset | 8 | 29.88 | 0.7823 | 22.0751
Bigset | 16 | 27.03 | 0.682 | 25.6410
Bigset | 32 | 24.67 | 0.5864 | 29.7619
Bigset(Gray) | 2 | 38.49 | 0.95 | 57.8035
Bigset(Gray) | 4 | 34.06 | 0.8874 | 76.9231
Bigset(Gray) | 8 | 30.69 | 0.8007 | 82.6446
Bigset(Gray) | 16 | 27.66 | 0.7035 | 74.0741
Bigset(Gray) | 32 | 25.16 | 0.6074 | 65.3595
TABLE VII: Benchmark results of ISTA-Net Dataset | Compression ratio | PSNR | SSIM | Reconstruction per second
---|---|---|---|---
MNIST | 2 | 29.2126 | 0.9640 | 118.4133
MNIST | 4 | 30.5120 | 0.9563 | 120.9862
MNIST | 8 | 29.1593 | 0.9558 | 123.8261
MNIST | 16 | 25.2426 | 0.9275 | 122.9381
MNIST | 32 | 21.3493 | 0.8455 | 121.1784
CelebA | 2 | 20.2156 | 0.8108 | 71.2754
CelebA | 4 | 17.7137 | 0.6881 | 73.5295
CelebA | 8 | 18.0141 | 0.7477 | 71.7567
CelebA | 16 | 21.0598 | 0.8439 | 73.0777
CelebA | 32 | 21.5447 | 0.8538 | 71.7036
CIFAR10 | 2 | 19.8350 | 0.8110 | 70.4578
CIFAR10 | 4 | 21.6027 | 0.8646 | 69.8564
CIFAR10 | 8 | 21.7896 | 0.8692 | 71.6831
CIFAR10 | 16 | 21.2262 | 0.8506 | 72.7173
CIFAR10 | 32 | 19.7638 | 0.7946 | 71.1875
CIFAR10(Gray) | 2 | 23.4110 | 0.7581 | 75.6761
CIFAR10(Gray) | 4 | 22.8502 | 0.7318 | 75.1091
CIFAR10(Gray) | 8 | 21.7888 | 0.6794 | 74.7101
CIFAR10(Gray) | 16 | 19.7298 | 0.5493 | 74.7030
CIFAR10(Gray) | 32 | 18.1149 | 0.4288 | 74.9074
Bigset | 2 | 18.1686 | 0.5370 | 72.5171
Bigset | 4 | 16.3798 | 0.3874 | 80.3709
Bigset | 8 | 17.6498 | 0.4811 | 72.8396
Bigset | 16 | 21.4860 | 0.6384 | 72.8716
Bigset | 32 | 21.0166 | 0.6326 | 66.1600
Bigset(Gray) | 2 | 18.2207 | 0.2187 | 70.1563
Bigset(Gray) | 4 | 21.5229 | 0.3874 | 68.3210
Bigset(Gray) | 8 | 23.2503 | 0.5378 | 71.6273
Bigset(Gray) | 16 | 23.5321 | 0.5401 | 72.4786
Bigset(Gray) | 32 | 22.8348 | 0.5089 | 71.2181
TABLE VIII: Benchmark results of CSGAN Dataset | Compression ratio | PSNR | SSIM | Reconstruction per second
---|---|---|---|---
MNIST | 2 | 32.0483 | 0.9156 | 223.1187
MNIST | 4 | 32.1388 | 0.9869 | 221.9279
MNIST | 8 | 26.5582 | 0.954 | 222.8450
MNIST | 16 | 23.3674 | 0.8691 | 229.3073
MNIST | 32 | 19.7423 | 0.7701 | 219.0102
CelebA | 2 | 29.4438 | 0.975 | 137.9121
CelebA | 4 | 33.2433 | 0.9888 | 150.3084
CelebA | 8 | 28.9183 | 0.9698 | 146.9329
CelebA | 16 | 26.2793 | 0.9483 | 151.5757
CelebA | 32 | 23.1517 | 0.8969 | 151.6358
CIFAR10 | 2 | 31.6029 | 0.9862 | 174.7553
CIFAR10 | 4 | 27.808 | 0.9674 | 225.8276
CIFAR10 | 8 | 25.4534 | 0.9428 | 227.8823
CIFAR10 | 16 | 22.262 | 0.8844 | 230.9234
CIFAR10 | 32 | 19.852 | 0.7874 | 214.7841
CIFAR10(Gray) | 2 | 25.7358 | 0.8669 | 141.5323
CIFAR10(Gray) | 4 | 23.6409 | 0.7804 | 219.8296
CIFAR10(Gray) | 8 | 21.621 | 0.6675 | 229.1173
CIFAR10(Gray) | 16 | 19.6054 | 0.5293 | 223.8697
CIFAR10(Gray) | 32 | 18.0317 | 0.3887 | 232.5259
Bigset | 2 | 30.7518 | 0.9314 | 30.5301
Bigset | 4 | 30.2502 | 0.9214 | 34.1619
Bigset | 8 | 28.3798 | 0.881 | 36.4556
Bigset | 16 | 27.3575 | 0.8539 | 34.5518
Bigset | 32 | 24.035 | 0.7489 | 41.8374
Bigset(Gray) | 2 | 30.5937 | 0.8376 | 35.3011
Bigset(Gray) | 4 | 30.2819 | 0.8125 | 41.4439
Bigset(Gray) | 8 | 27.0031 | 0.7037 | 32.2543
Bigset(Gray) | 16 | 25.1241 | 0.6216 | 41.1410
Bigset(Gray) | 32 | 23.6821 | 0.5611 | 44.2111
TABLE IX: Benchmark results of LAPRAN Dataset | Compression ratio | PSNR | SSIM | Reconstruction per second
---|---|---|---|---
MNIST | 2 | 22.6164 | 0.8978 | 2.0329
MNIST | 4 | 22.4511 | 0.8923 | 2.3868
MNIST | 8 | 21.9578 | 0.8742 | 2.6208
MNIST | 16 | 20.6622 | 0.8215 | 2.6924
MNIST | 32 | 17.7698 | 0.6985 | 2.8841
CelebA | 2 | 21.0459 | 0.6081 | 0.0178
CelebA | 4 | 20.9366 | 0.6034 | 0.0405
CelebA | 8 | 20.7178 | 0.5938 | 0.0421
CelebA | 16 | 20.2657 | 0.5737 | 0.0661
CelebA | 32 | 19.3262 | 0.5306 | 0.1634
CIFAR10 | 2 | 19.3796 | 0.5837 | 0.3100
CIFAR10 | 4 | 19.0948 | 0.5679 | 0.3939
CIFAR10 | 8 | 18.5491 | 0.5369 | 0.4501
CIFAR10 | 16 | 17.4882 | 0.4752 | 0.4861
CIFAR10 | 32 | 15.7465 | 0.3754 | 0.5013
CIFAR10(Gray) | 2 | 18.8775 | 0.5630 | 0.4896
CIFAR10(Gray) | 4 | 18.1176 | 0.5190 | 0.5027
CIFAR10(Gray) | 8 | 16.7601 | 0.4420 | 0.5116
CIFAR10(Gray) | 16 | 14.7029 | 0.3264 | 0.5160
CIFAR10(Gray) | 32 | 12.4420 | 0.2042 | 0.5175
Bigset | 2 | 20.5856 | 0.4296 | 0.0294
Bigset | 4 | 20.5270 | 0.4275 | 0.0546
Bigset | 8 | 20.3866 | 0.4207 | 0.0773
Bigset | 16 | 20.0988 | 0.4054 | 0.1022
Bigset | 32 | 19.5600 | 0.3787 | 0.1551
Bigset(Gray) | 2 | 20.5140 | 0.4439 | 0.1111
Bigset(Gray) | 4 | 20.2344 | 0.4295 | 0.1688
Bigset(Gray) | 8 | 19.7265 | 0.4028 | 0.2307
Bigset(Gray) | 16 | 18.6832 | 0.3527 | 0.2580
Bigset(Gray) | 32 | 16.8286 | 0.2712 | 0.2697
TABLE X: Benchmark results of CSGM Dataset | Compression ratio | PSNR | SSIM | Reconstruction per second
---|---|---|---|---
MNIST | 2 | 38.471 | 0.985 | 723.5890
MNIST | 4 | 32.228 | 0.984 | 874.8906
MNIST | 8 | 27.558 | 0.933 | 848.8964
MNIST | 16 | 23.860 | 0.914 | 688.7052
MNIST | 32 | 20.259 | 0.821 | 712.2507
CelebA | 2 | 33.390 | 0.954 | 604.5949
CelebA | 4 | 28.623 | 0.889 | 621.8905
CelebA | 8 | 25.595 | 0.812 | 611.2469
CelebA | 16 | 23.115 | 0.722 | 791.1392
CelebA | 32 | 21.065 | 0.634 | 623.0530
CIFAR10 | 2 | 30.494 | 0.945 | 552.1811
CIFAR10 | 4 | 25.468 | 0.847 | 744.6016
CIFAR10 | 8 | 22.274 | 0.719 | 807.1025
CIFAR10 | 16 | 19.836 | 0.570 | 708.2153
CIFAR10 | 32 | 17.957 | 0.430 | 777.6050
CIFAR10(Gray) | 2 | 30.662 | 0.946 | 683.0601
CIFAR10(Gray) | 4 | 25.466 | 0.842 | 689.6552
CIFAR10(Gray) | 8 | 22.584 | 0.723 | 802.5682
CIFAR10(Gray) | 16 | 20.080 | 0.571 | 736.9197
CIFAR10(Gray) | 32 | 18.264 | 0.433 | 803.8585
Bigset | 2 | 32.544 | 0.873 | 798.7220
Bigset | 4 | 28.862 | 0.782 | 805.1530
Bigset | 8 | 26.872 | 0.705 | 772.7975
Bigset | 16 | 24.945 | 0.618 | 655.3080
Bigset | 32 | 23.377 | 0.556 | 661.3757
Bigset(Gray) | 2 | 34.356 | 0.911 | 566.5722
Bigset(Gray) | 4 | 31.002 | 0.832 | 796.8127
Bigset(Gray) | 8 | 28.232 | 0.742 | 712.2507
Bigset(Gray) | 16 | 26.024 | 0.653 | 662.6905
Bigset(Gray) | 32 | 24.000 | 0.569 | 696.3788
TABLE XI: Benchmark results of ReconNet Dataset | Compression ratio | PSNR | SSIM | Reconstruction per second
---|---|---|---|---
MNIST | 2 | 47.995 | 1 | 16.3934
MNIST | 4 | 33.233 | 0.879 | 12.3457
MNIST | 8 | 20.587 | 0.542 | 12.3457
MNIST | 16 | 15.291 | 0.299 | 15.8730
MNIST | 32 | 13.076 | 0.163 | 16.1290
CelebA | 2 | 32.335 | 0.959 | 0.7283
CelebA | 4 | 26.592 | 0.889 | 1.2516
CelebA | 8 | 22.863 | 0.801 | 1.2563
CelebA | 16 | 19.919 | 0.703 | 1.4104
CelebA | 32 | 17.345 | 0.599 | 1.4025
CIFAR10 | 2 | 29.584 | 0.936 | 4.6948
CIFAR10 | 4 | 24.001 | 0.822 | 4.6083
CIFAR10 | 8 | 20.621 | 0.69 | 4.8077
CIFAR10 | 16 | 18.286 | 0.573 | 5.2356
CIFAR10 | 32 | 16.401 | 0.476 | 5.3476
CIFAR10(Gray) | 2 | 29.766 | 0.9 | 14.4928
CIFAR10(Gray) | 4 | 24.189 | 0.742 | 13.1579
CIFAR10(Gray) | 8 | 20.778 | 0.577 | 13.6986
CIFAR10(Gray) | 16 | 18.343 | 0.446 | 15.3846
CIFAR10(Gray) | 32 | 16.661 | 0.362 | 16.1290
Bigset | 2 | 35.83 | 0.96 | 0.7962
Bigset | 4 | 31.084 | 0.905 | 1.3106
Bigset | 8 | 27.632 | 0.845 | 1.3298
Bigset | 16 | 24.754 | 0.786 | 1.4286
Bigset | 32 | 22.109 | 0.729 | 1.4225
Bigset(Gray) | 2 | 36.154 | 0.915 | 1.6447
Bigset(Gray) | 4 | 31.368 | 0.814 | 3.5971
Bigset(Gray) | 8 | 27.871 | 0.711 | 3.9063
Bigset(Gray) | 16 | 24.954 | 0.62 | 4.2735
Bigset(Gray) | 32 | 22.054 | 0.545 | 4.0984
TABLE XII: Benchmark results of TVAL-3 Dataset | Compression ratio | PSNR | SSIM | Reconstruction per second
---|---|---|---|---
MNIST | 2 | 47.911 | 0.999 | 0.8569
MNIST | 4 | 31.01 | 0.803 | 0.8244
MNIST | 8 | 19.881 | 0.489 | 0.8244
MNIST | 16 | 14.158 | 0.225 | 1.0132
MNIST | 32 | 11.762 | 0.1 | 1.1779
CelebA | 2 | 32.578 | 0.961 | 0.0307
CelebA | 4 | 27.003 | 0.897 | 0.0487
CelebA | 8 | 23.359 | 0.816 | 0.0516
CelebA | 16 | 20.721 | 0.731 | 0.0531
CelebA | 32 | 18.235 | 0.633 | 0.0471
CIFAR10 | 2 | 30.078 | 0.943 | 0.2993
CIFAR10 | 4 | 24.486 | 0.837 | 0.3171
CIFAR10 | 8 | 21.115 | 0.709 | 0.2793
CIFAR10 | 16 | 18.697 | 0.584 | 0.3167
CIFAR10 | 32 | 16.746 | 0.476 | 0.3738
CIFAR10(Gray) | 2 | 30.231 | 0.91 | 0.7868
CIFAR10(Gray) | 4 | 24.619 | 0.758 | 1.0030
CIFAR10(Gray) | 8 | 21.328 | 0.596 | 0.9891
CIFAR10(Gray) | 16 | 18.822 | 0.452 | 1.0194
CIFAR10(Gray) | 32 | 17.046 | 0.344 | 1.1173
Bigset | 2 | 36.084 | 0.961 | 0.0351
Bigset | 4 | 31.6 | 0.91 | 0.0522
Bigset | 8 | 28.475 | 0.855 | 0.0554
Bigset | 16 | 25.951 | 0.802 | 0.0568
Bigset | 32 | 23.821 | 0.754 | 0.0599
Bigset(Gray) | 2 | 36.425 | 0.919 | 0.0862
Bigset(Gray) | 4 | 31.881 | 0.823 | 0.1557
Bigset(Gray) | 8 | 28.687 | 0.727 | 0.1642
Bigset(Gray) | 16 | 26.128 | 0.64 | 0.1740
Bigset(Gray) | 32 | 24.035 | 0.57 | 0.1671
TABLE XIII: Benchmark results of L1 Dataset | Compression ratio | PSNR | SSIM | Reconstruction per second
---|---|---|---|---
MNIST | 2 | 43.45 | 0.972 | 0.3483
MNIST | 4 | 31.787 | 0.891 | 0.3526
MNIST | 8 | 22.582 | 0.724 | 0.3407
MNIST | 16 | 13.238 | 0.322 | 0.3274
MNIST | 32 | 6.53 | 0.093 | 0.3194
CelebA | 2 | 47.126 | 0.998 | 0.0568
CelebA | 4 | 37.495 | 0.985 | 0.0537
CelebA | 8 | 31.095 | 0.95 | 0.0458
CelebA | 16 | 26.328 | 0.877 | 0.0450
CelebA | 32 | 21.517 | 0.723 | 0.0500
CIFAR10 | 2 | 40.401 | 0.993 | 0.2190
CIFAR10 | 4 | 31.595 | 0.957 | 0.2104
CIFAR10 | 8 | 25.851 | 0.867 | 0.1976
CIFAR10 | 16 | 20.248 | 0.645 | 0.1882
CIFAR10 | 32 | 8.307 | 0.113 | 0.1843
CIFAR10(Gray) | 2 | 31.682 | 0.929 | 0.3606
CIFAR10(Gray) | 4 | 25.616 | 0.789 | 0.3524
CIFAR10(Gray) | 8 | 19.996 | 0.537 | 0.3398
CIFAR10(Gray) | 16 | 16.365 | 0.342 | 0.3418
CIFAR10(Gray) | 32 | 8.016 | 0.074 | 0.3398
Bigset | 2 | 42.51 | 0.992 | 0.0474
Bigset | 4 | 37.565 | 0.977 | 0.0520
Bigset | 8 | 33.95 | 0.949 | 0.0516
Bigset | 16 | 30.973 | 0.908 | 0.0480
Bigset | 32 | 27.3 | 0.832 | 0.0496
Bigset(Gray) | 2 | 38.205 | 0.94 | 0.1224
Bigset(Gray) | 4 | 34.138 | 0.87 | 0.1222
Bigset(Gray) | 8 | 30.774 | 0.786 | 0.1216
Bigset(Gray) | 16 | 27.234 | 0.67 | 0.1221
Bigset(Gray) | 32 | 20.727 | 0.435 | 0.1216
TABLE XIV: Benchmark results of D-AMP Dataset | Compression ratio | PSNR | SSIM | Reconstruction per second
---|---|---|---|---
MNIST | 2 | 40.062 | 0.912 | 0.3148
MNIST | 4 | 29.452 | 0.778 | 0.3264
MNIST | 8 | 18.787 | 0.472 | 0.3270
MNIST | 16 | 15.058 | 0.312 | 0.3247
MNIST | 32 | 12.052 | 0.132 | 0.3258
CelebA | 2 | 35.932 | 0.978 | 0.0238
CelebA | 4 | 29.288 | 0.93 | 0.0233
CelebA | 8 | 23.97 | 0.831 | 0.0231
CelebA | 16 | 21.211 | 0.743 | 0.0233
CelebA | 32 | 17.848 | 0.601 | 0.0232
CIFAR10 | 2 | 30.627 | 0.938 | 0.1019
CIFAR10 | 4 | 24.975 | 0.843 | 0.1055
CIFAR10 | 8 | 21.008 | 0.7 | 0.1055
CIFAR10 | 16 | 18.564 | 0.576 | 0.1063
CIFAR10 | 32 | 16.928 | 0.477 | 0.1073
CIFAR10(Gray) | 2 | 31.14 | 0.912 | 0.3108
CIFAR10(Gray) | 4 | 25.336 | 0.781 | 0.3312
CIFAR10(Gray) | 8 | 21.215 | 0.598 | 0.3295
CIFAR10(Gray) | 16 | 18.81 | 0.463 | 0.3301
CIFAR10(Gray) | 32 | 16.921 | 0.353 | 0.3318
Bigset | 2 | 38.114 | 0.973 | 0.0240
Bigset | 4 | 33.706 | 0.938 | 0.0243
Bigset | 8 | 30.003 | 0.883 | 0.0244
Bigset | 16 | 27.201 | 0.831 | 0.0249
Bigset | 32 | 24.191 | 0.758 | 0.0249
Bigset(Gray) | 2 | 38.697 | 0.942 | 0.0720
Bigset(Gray) | 4 | 34.293 | 0.874 | 0.0728
Bigset(Gray) | 8 | 30.449 | 0.781 | 0.0726
Bigset(Gray) | 16 | 27.393 | 0.692 | 0.0741
Bigset(Gray) | 32 | 24.426 | 0.592 | 0.0744
TABLE XV: Benchmark results of NLR-CS