\documentclass[reqno]{amsart}
\usepackage{hyperref}

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2016 (2016), No. 262, pp. 1--18.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu}
\thanks{\copyright 2016 Texas State University.}
\vspace{8mm}}

\begin{document}
\title[\hfilneg EJDE-2016/262\hfil Painlev\'e equations I and II]
{Meromorphic solutions to difference Painlev\'e equations I and II}

\author[Z.-T. Wen \hfil EJDE-2016/262\hfilneg]
{Zhi-Tao Wen}

\address{Zhi-Tao Wen \newline
Taiyuan University of Technology,
Department of Mathematics,
No. 79 Yingze West Street, 030024 Taiyuan, China}
\email{zhitaowen@gmail.com}

\thanks{Submitted April 28, 2016. Published September 28, 2016.}
\subjclass[2010]{39A10, 30D35, 39A12}
\keywords{Painlev\'e difference equation;  rational solution; 
 Nevanlinna theory;
\hfill\break\indent difference Riccati equation;
 singularity confinement}

\begin{abstract}
 In this article, we study the solutions of difference Painlev\'e equations
 I and II. to do this, we first give some properties of solutions to
 difference Riccati equations.
 We show that there are no rational solutions for the difference Painlev\'e
 equation I. We also study rational solutions for the difference Painlev\'e
 equation II. Later, we present a new way to seek solutions of the difference
 Painlev\'e equation II by using the difference Riccati equation.
 At last, we construct new solutions of the difference
 Painlev\'e equation II through a known solution.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{remark}[theorem]{Remark}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{example}[theorem]{Example}
\allowdisplaybreaks

\section{Introduction}

An ordinary differential equation is said to possess the Painlev\'e property
if all of its solutions are single-values about all movable singularities, see
\cite{Gromak-Laine-Shimomura}. Over a century ago
Painlev\'e \cite{painleve:00,painleve:02}, Fuchs \cite{fuchs:05} and
Gambier \cite{gambier:10} completed a substantial classification work,
which comprised of sieving through a large class of second-order
differential equations by making use of a criterion proposed by
 Picard \cite{picard:89}, now known as the Painlev\'e property.
Painlev\'e and his colleagues discovered six new equations,
later named as Painlev\'e equations, which were not solvable in terms
of known functions.

Painlev\'e equations are the most complicated systems that one can solve
(in a nontrivial way). Anything simpler becomes trivially
integrable, anything more complicated becomes hopelessly non-integrable.
Therefore, Painlev\'e equations are the borderline between
trivial integrability and non-integrable, a fact that bestows upon them
a lot of of interesting properties.

There are several candidates for the discrete analogue of the Painlev\'e property.
The singularity confinement test by Grammaticos, Ramani and Papageorgiou
\cite{grammaticosrp:91} has been very successful in finding new discrete
equations of Painlev\'e type \cite{ramanigh:91}. Hietarinta and Viallet
suggested to amend singularity confinement by demanding additionally that
the algebraic complexity of the solutions remains relatively low as the
iteration progresses \cite{hietarintav:98}. The first approach using a
low growth criterion of iterates as a sign of integrability is
due to Veselov \cite{veselov:92} who showed that generic exponential
growth rate of the degree of iterates corresponds to the non-existence of
certain types of first integrals. A method due to Roberts and Vivaldi relies
on orbit dynamics over finite fields \cite{robertsv:03},
while Halburd's Diophantine integrability \cite{halburd:05} makes use of
 polynomial growth of heights of iterates over number fields.

The analogues of Painlev\'{e} property for difference equations in the
complex plane have been discussed, for example, in \cite{Ablowitz-Halburd-Herbst},
Ablowitz et.al. consider the solutions of difference Painlev\'{e} equations
by Nevanlinna theory, which is a landmark in the application
of Nevanlinna theory in the study of difference equations. They have observed
that all of the relevant difference equations have obvious analytic versions,
and hence can be studied by using the methods of complex analysis, and in
particular Nevanlinna theory.

Halburd and Korhonen \cite{Halburd-Korhonen-London} showed that if the
difference equation
 \begin{equation}\label{diff-Pain-I.eq}
 \overline{w}+\underline{w}=R(z,w),
 \end{equation}
where $R(r,w)$ is rational in $w$ with meromorphic coefficients, has an
admissible meromorphic solution of finite order, then the class
\eqref{diff-Pain-I.eq} can be reduced into a short list of equations,
 where the difference Painlev\'{e} equation I and the difference
Painlev\'{e} equation II are included.

Halburd and Korhonen \cite{Halburd-Korhonen-Phys} showed that if the difference
equation
 \begin{equation}\label{diff-Pain-III.eq}
 \overline{w}\underline{w}=\frac{c_2(w-c_+)(w-c_-)}{(w-a_+)(w-a_-)}
 \end{equation}
has an admissible finite order meromorphic solution such that the order of
its poles is bounded, then the equation \eqref{diff-Pain-I.eq} can be
transformed by M\"{o}bius tranformation in $w$
 to a short list of equations, where the difference Painlev\'{e} III is included,
unless $w$ is the solution of difference Riccati equation.
Ronkainen \cite{Ronkainen} removed the assumption on the boundedness of
the order of the poles, difference painlev\'{e} equation III can be obtained from
equation \eqref{diff-Pain-III.eq} when the coefficients are special solutions
of equations. Similarly, difference Painlev\'{e} equation V is also
considered by Ronkainen \cite{Ronkainen}.

The aim of this article is to discuss the solutions of difference
 Painlev\'e equations I and II.
The remainder of this paper is organized as follows.
Difference version of Nevanlinna theory for studying the difference Painlev\'e
equations is shown in Section \ref{lemma.sec}.
We give some properties of the solutions of the difference Riccati equation
in Section \ref{Ri.sec}.
In section \ref{P.sec}, we show the relations of the solutions between the
difference Riccati equation and
difference Painlev\'e equations I and II, respectively.
We conclude that there are no rational solutions of the difference
Painlev\'e equation I. We also discuss the rational solutions
of the difference Painlev\'e equation II in four cases according to the
coefficients of the equations. At the end of section \ref{P.sec}, we give a
way to seek a new solution of the difference Painlev\'e equation II.

\section{Nevalinna theory and difference equations}\label{lemma.sec}

Difference analogue lemma on the logarithmic derivative of meromorphic functions
of finite order was found
by Halburd and Korhonen \cite{Halburd-Korhonen-Ann,Halburd-Korhonen-2006},
and by Chiang and Feng \cite{Chiang-Feng}, independently.
This result was recently extended to meromorphic functions of hype order
strictly less than one in \cite{halburdkt:09}, where it was also shown that
this growth condition cannot be essentially relaxed further.
Let us state latest version of difference analogue lemma on the
logarithmic derivative of meromorphic functions of finite order as follows.

\begin{lemma}[{\cite[Theorem 5.1]{halburdkt:09}}] \label{diff-deri.lemma}
Let $w$ be a non-constant meromorphic function and $c\in\mathbb{C}$.
If $w$ is of finite order, then
 $$
 m\Big(r,\frac{w(z+c)}{w(z)}\Big)=\Big(\frac{\log r}{r}{T(r,w)}\Big)
 $$
for all $r$ outside of a set $E$ satisfying
 $$
 \limsup_{r\to\infty}\frac{\int_{E\cap[1,r)}\,dt/t}{\log r}=0,
 $$
i.e., outside of a set $E$ of zero logarithmic density.
If $\rho_2(w)=\rho_2<1$ and $\varepsilon>0$, then
 $$
 m\Big(r,\frac{w(z+c)}{w(z)}\Big)=o\Big(\frac{T(r,w)}{r^{1-\rho_2-\varepsilon}}
\Big)
 $$
for all $r$ outside of a set of finite logarithmic measure.
\end{lemma}


The following Lemma is improved version of \cite[Lemma 2.2]{Halburd-Korhonen-Ann}.

\begin{lemma}[{\cite[Lemma 8.3]{halburdkt:09}}] \label{order.lemma}
Let $T:[0,\infty)\to[0,\infty)$ be a non-decreasing continuous function
and let $s\in(0,\infty)$. If the hyper order of $T$ is strictly less than one, i.e.,
 $$
 \limsup_{r\to\infty}\frac{\log\log T (r)}{\log r}=\rho_2<1,
 $$
and $\delta\in(0,1-\rho_2)$, then
 $$
 T(r+s)=T(r)+o\Big(\frac{T(r)}{r^{\delta}}\Big),
 $$
where $r$ runs to infinity outside of a set of finite logarithmic measure.
\end{lemma}


The difference version of a result due to A. Mohon'ko and V. Mohon'ko
\cite{Mohonko-Mohonko}, is found by Halburd and Korhonen \cite{Halburd-Korhonen-2006},
and Laine and Yang \cite[ Theorem 2.4]{Laine-Yang}, independently.
Let us state this result as follows.

\begin{lemma}[{\cite[Corollary 3.4]{Halburd-Korhonen-2006}}]\label{P.lemma}
Let $w(z)$ be a nonconstant finite order meromorphic solution of
 $ P(z,w)=0$,
where $P(z,w)$ is a difference polynomial in $w(z)$.
If $P(z,a)\not\equiv0$ for a small meromorphic function related to $w$, then
 $$
 m\Big(\frac{1}{w-a}\Big)=S(r,w)
 $$
for all $r$ outside of a possible exceptional set with finite logarithmic measure.
\end{lemma}

The difference analogue of Clunie lemma, due to Laine and Yang, is the
generalization of the result in \cite[Theorem 3.1]{Halburd-Korhonen-2006},
which is stated as follows.

\begin{lemma}[{\cite[Theorem 2.3]{Laine-Yang}}] \label{clunie.lemma}
Let $w$ be a transcendental meromorphic solution of finite order $\rho$
of a difference equations of the form
 $$
 H(z,w)P(z,w)=Q(z,w)
 $$
where $H(z,f)$, $P(z,w)$, $Q(z,f)$ are difference polynomials such that total
degree $n$ in $w(z)$ and its shifts,
and the total degree of $Q(z,w)$ is $\leq n$. Then, for each $\varepsilon>0$,
 $$
 m(r,P(z,w))=O(r^{\rho-1+\varepsilon})+S(r,w),
 $$
possibly outside of an exceptional set of finite logarithmic measure.
\end{lemma}


\section{Difference Riccati equations}\label{Ri.sec}

About a century ago Malmquist \cite{malmquist:13} proved that if the
differential equation
 \begin{equation}\label{malmquist_start}
 w' = R(z,w(z)),
 \end{equation}
where $R(z,w(z))$ is rational in its both arguments, has a non-rational meromorphic
solution, then \eqref{malmquist_start} reduces into a Riccati equation
 $$
 w' = a(z) w^2 + b(z) w + c(z)
 $$
with rational coefficients. Similarly, if the difference equation
 \begin{equation}\label{malmquist_analogue_start}
 w(z+1) = R(z,w(z)),
 \end{equation}
where again $R(z,w(z))$ is rational in $z$ and $w(z)$, admits at least
one non-rational meromorphic solution of finite order
(or of hyper-order strictly less than one), then \eqref{malmquist_analogue_start}
must in fact be a difference Riccati equation
 $$
 w(z+1) = \frac{c(z)w(z) + d(z)}{a(z) w(z)+b(z)}
 $$
with rational coefficients. This is a known difference analogue of Malmquist's
theorem, which follows by a more general result due to
Yanagihara \cite{yanagihara:80}.

In what follows, the difference Riccati equations we mentioned have the
coefficients of small functions related to $w$. If $a(z)\equiv 0$, then
the difference Riccati equations are reduced to the linear difference equations.
Thus, let us consider the case $a(z)\not\equiv0$. In this section,
we discuss some basic properties of the solutions of the difference
Riccati equations, which will be used for investigating difference
Painlev\'e equations I and II in the next section.


\begin{lemma}\label{inverse.lemma}
If $w$ is the solution of
 \begin{equation}\label{diff-Ri.eq}
 w(z+1) = \frac{c(z) w(z)+d(z)}{a(z)w(z) + b(z)},
 \end{equation}
then $w$ also satisfies
 \begin{equation}\label{inv-diff-Ri.eq}
 w(z-1)=\frac{d(z-1) -b(z-1)w(z)}{a(z-1)w(z)-c(z-1)}.
 \end{equation}
\end{lemma}

\begin{proof}
If $w$ is the solution of \eqref{diff-Ri.eq}, then we have
 $$
 w(z) = \frac{c(z-1) w(z-1)+d(z-1)}{a(z-1)w(z-1) + b(z-1)}.
 $$
Thus, by a simple calculation it yields that
 $$
 w(z-1)=\frac{d(z-1) -b(z-1)w(z)}{a(z-1)w(z)-c(z-1)},
 $$
which is our assertion.
\end{proof}

Let us consider the rational solutions of the difference Riccati equations.
In this case, all the coefficients
of the difference Riccati equations are reduced to the constants.

\begin{lemma}\label{degree.lemma}
Let $w$ be a non-constant rational solution of the difference Riccati
equations \eqref{diff-Ri.eq},
which is written as $w(z)=P(z)/Q(z)$, where $P(z)$ and $Q(z)$ are polynomials,
then $\deg{P}\leq\deg{Q}$. Moreover, if $d\neq 0$, then $\deg{P}=\deg{Q}$.
\end{lemma}

\begin{proof}
Suppose that $w$ is a rational function, let us write $w$ as
 $$
 w(z)=\frac{P(z)}{Q(z)},
 $$
where $P(z)$ and $Q(z)$ are polynomials. If $w$ is a solution
of \eqref{diff-Ri.eq}, then it yields that by substituting it
to \eqref{diff-Ri.eq}
 $$
 \frac{P(z+1)}{Q(z+1)}=\frac{cP(z)+dQ(z)}{aP(z)+bQ(z)}.
 $$
From the equation above, we have the following equality
 \begin{equation}\label{PQdegree.eq}
 aP(z+1)P(z)+bP(z+1)Q(z)=cP(z)Q(z+1)+dQ(z)Q(z+1).
 \end{equation}
Let us denote $p=\deg{P}$ and $q=\deg{Q}$. If $p>q$, then the degree
of polynomials of both side of the equation \eqref{PQdegree.eq} are $2p$
and $p+q$ (or $2q$ if c=0), respectively. It implies that $p=q$,
which is a contradiction. Thus, we have $\deg{P}\leq\deg{Q}$.

In particular, if $d\neq 0$, let us denote
 $$
 P_*(z,w)=\overline{w}(aw+b)-(cw+d).
 $$
Since $w=0$ is not the solution of $P_*(z,w)=0$, it follows by Lemma \ref{P.lemma}
 that
 \begin{equation}\label{T1.eq}
 N\big(r,\frac{1}{w}\big)=q\log r+O(1)=T(r,w)+O(1).
 \end{equation}
Now let us rewrite difference Riccati equations \eqref{diff-Ri.eq} as follows
 $$
 a\overline{w}w=cw-b\overline{w}+d.
 $$
Since $a$, $b$, $c$ and $d$ are small functions of $w$, it yields by
$H(z,w)=aw$ and $Q(z,w)=cw-b\overline{w}+d$ that
 \begin{equation}\label{T2.eq}
 N(r,w)=p\log r+O(1)=T(r,w)+O(1).
 \end{equation}
according to Lemma \ref{clunie.lemma}. Therefore, we conclude from \eqref{T1.eq}
and \eqref{T2.eq} that
$\deg{P}=\deg{Q}$,
which is our assertion.
\end{proof}

Now we will find a linear transformation, by which every difference
 Riccati equation \eqref{diff-Ri.eq} can be reduced into a simple form
 \begin{equation}\label{simple-Ri.eq}
 \overline{w}=\frac{w+D}{w+1},
 \end{equation}
where $D$ related to $a,b,c,d$ is a small function of $w$.
Thus, we can only consider the solutions of difference Riccati equations
\eqref{simple-Ri.eq}.


\begin{lemma}\label{transfor.lemma}
Any difference Riccati equation \eqref{diff-Ri.eq} can be transformed by a 
linear mapping to the form \eqref{simple-Ri.eq}.
\end{lemma}

\begin{proof}
Since $w$ satisfies \eqref{diff-Ri.eq}, by the linear transformation $w=Af+B$
it yields
 $$
 \overline{Af}+\overline{B}=\frac{cAf+cB+d}{aAf+aB+b},
 $$
where $A,B$ are the small functions of $f$. By Nevanlinna First Main theory,
it follows that $A$ and $B$ are also the small functions of $w$.
Thus, $f$ satisfies
 $$
 \overline{Af}=\frac{(c-a\overline{B})
\big(f+\frac{cB+d-aB\overline{B}-b\overline{B}}{cA-aA\overline{B}}\big)}
{a\left(f+\frac{aB+b}{aA}\right)}
 $$
Let us set
 \begin{gather*}
 a\overline{A}=c-a\overline{B} \\
 aA=b+aB
 \end{gather*}
then, we have our assertion if $A$ and $B$ satisfy
\begin{gather*}
 A= \frac{1}{2}\big(\frac{\underline{c}}{\underline{a}}+\frac{b}{a}\big) \\
 B= \frac{1}{2}\big(\frac{\underline{c}}{\underline{a}}-\frac{b}{a}\big)
 \end{gather*}
Moveover, we can express $D$ by $a,b,c$ and $d$.
\end{proof}

\begin{remark} \rm
It follows that there are two linearly independent meromorphic solutions 
of difference Riccati equations by Lemma \ref{transfor.lemma} and
discussion in \cite{Ishizaki} and \cite{Praagman}. We can construct other 
meromorphic solutions of difference Riccati equations by these two linearly
independent solutions. Moveover, we could use it for investigating the 
solutions of difference Painleve equation II. Let us discuss it 
in the next section.
\end{remark}


\section{Difference Painlev\'e equations}\label{P.sec}

Halburd and Korhonen  proved that if the equation
 \begin{equation}\label{I-II.eq}
 \overline{w}+\underline{w}=R(z,w),
 \end{equation}
where $R(z,w)$ is rational in $w$ and meromorphic in $z$, has an 
admissible meromorphic solution of finite order, then either
 $w$ satisfies a difference Riccati equation
 $$
 \overline{w}=\frac{\overline{p}w+q}{w+p},
 $$
where $p,q$ are small functions related to $w$, or equations \eqref{I-II.eq}
 can be transformed by a linear change in $w$ to one of the following equations:
\begin{gather}{2}
\overline{w}+w+\underline{w}=\frac{\pi_1z+\pi_2}{w}+k_1 \label{one.eq} \\
\overline{w}-w+\underline{w}=\frac{\pi_1z+\pi_2}{w}+(-1)^zk_1 \\
\overline{w}+\underline{w}=\frac{\pi_1z+\pi_3}{w}+\pi_2 \label{One.eq}\\
\overline{w}+\underline{w}=\frac{\pi_1z+k_1}{w}+\frac{\pi_2}{w^2}\label{ONE.eq}\\
\overline{w}+\underline{w}=\frac{(\pi_1z+k_1)w+\pi_2}{(-1)^{-z}-w^2}\\
\overline{w}+\underline{w}=\frac{(\pi_1z+k_1)w+\pi_2}{1-w^2}\label{two.eq}\\
\overline{w}w+w\underline{w}=p\\
\overline{w}+\underline{w}=pw+q
\end{gather}
where $\pi_k, k_k$ are finite order small functions related to $w$ of period $k$.

Historically, Equation \eqref{one.eq} is known as the difference
 Painlev\'e equation I,
The choice of name stems from the fact that a continuous limit, such as 
$w = -1/2+ \varepsilon^2u$, $k_1=-3$, $\pi_1z+\pi_2=-(3+2\varepsilon^4t)/4$,
$\varepsilon\to 0$, may be used to map \eqref{one.eq} to the
 Painlev\'e equation I $u''=6u^2 +t$. Equations \eqref{One.eq} and \eqref{ONE.eq} 
are still known as difference Painlev\'e equation I, while equation \eqref{two.eq} 
is often referred to as the difference Painlev\'e II.
In this section, we consider the solutions of difference Painleve equations I and II.

\begin{theorem}\label{Ri-I.theorem}
Let $w$ be a non-constant solution of the difference Riccati equation 
\eqref{diff-Ri.eq} with rational coefficients. Then $w$ is not the solution 
of the difference Painlev\'e equation I
 $$
 \overline{w}+\underline{w}=\frac{\pi_1z+\pi_3}{w}+\pi_2,
 $$
where $\pi_i (i=1,2,3)$ are the small functions related to $w$ with period of 
$i$ respectively, if $\pi_2\not\equiv0$.
\end{theorem}

\begin{proof}
Suppose that $w$ is the solution of
 $$
 \overline{w} = \frac{cw+d}{aw+b},
 $$
where $a, b, c$ and $d$ are the small functions of $w$. 
By Lemma \ref{inverse.lemma}, it follows that $w$ satisfies
 $$
 \underline{w}=\frac{\underline{d} -\underline{b}w}{\underline{a}w-\underline{c}}.
 $$
According to discussion above, it yields that $w$ satisfies that
 \begin{equation}\label{sum.eq}
 \begin{split}
 \overline{w}+\underline{w}
&=\frac{cw+d}{aw+b}+\frac{\underline{d} 
 -\underline{b}w}{\underline{a}w-\underline{c}}\\
&=\frac{(c\underline{a}-a\underline{b})w^2+(\underline{a}d
 -c\underline{c}+a\underline{d}-\underline{b}b)w+b\underline{d}-d\underline{c}}
 {a\underline{a}w^2+(b\underline{a}-a\underline{c})w-b\underline{c}}.
 \end{split}
 \end{equation}
Now, if $w$ is the solution of the difference Painlev\'e equation I \eqref{One.eq}, 
then we have
 $$
 \frac{(c\underline{a}-a\underline{b})w^2+(\underline{a}d-c\underline{c}+a\underline{d}-\underline{b}b)w+b\underline{d}-d\underline{c}}
 {a\underline{a}w^2+(b\underline{a}-a\underline{c})w-b\underline{c}}
 =\frac{\pi_1z+\pi_3}{w}+\pi_2.
 $$
By a simple calculation, it yields
 \begin{align*}
 &(c\underline{a}-a\underline{b})w^3+(\underline{a}d-c\underline{c}
+a\underline{d}-\underline{b}b)w^2+(b\underline{d}-d\underline{c})w\\
&=\pi_2a\underline{a}w^3+\pi_2(b\underline{a}-a\underline{c})w^2+\pi a\underline{a}w^2+\pi(b\underline{a}
 -a\underline{c})w-b\underline{c}k_1w-b\underline{c}\pi,
 \end{align*}
where $\pi=\pi_1z+\pi_3$. Therefore, by comparing the coefficients related 
to $w$ in both side of equations, we have
 $$
 b\underline{c}(\pi_1z+\pi_3)\equiv0.
 $$
It is well-known that $\pi_1z+\pi_3\not\equiv0$, then $b\equiv0$ or $c\equiv0$.
 Let us suppose that $b\equiv0$ at first, thus it follows that
 \begin{gather*}
 \pi_2=\frac{c}{a}\\
 \underline{a}d+a\underline{d}=\pi a\underline{a}\\
 -d\underline{c}=-a\underline{c}\pi
 \end{gather*}
by comparing the coefficients of equations related to $w$. 
Since $\pi_2\not\equiv0$, we have $c\not\equiv0$. Then it shows that
$\pi=d/a$. By submitting it to the Riccati difference equation, it yields that
 $$
 \overline{w}=\frac{\pi_1z+\pi_3}{w}+\pi_2
 $$
which implies that $\underline{w}\equiv0$. It is impossible.
Similarly, we have it is impossible if $c\equiv0$. It is a contradiction
with $w$ is a solution of difference Painlev\'e equation I \eqref{One.eq}.
\end{proof}

By the same way as the proof of Theorem \ref{Ri-I.theorem},
 we have the following result.

\begin{theorem}\label{Ri-I-A.theorem}
Let $w$ be a non-constant solution of the difference Riccati equation 
\eqref{diff-Ri.eq} with rational coefficients. Then $w$ is not the solution 
of the difference Painlev\'e equation I
 \begin{equation*}
 \overline{w}+w+\underline{w}=\frac{\pi_1z+\pi_2}{w}+k_1,
 \end{equation*}
where $\pi_i$ and $k_j (i=1,2; j=1)$ are the small functions related to 
$w$ with period of $i$ and $j$, respectively.
\end{theorem}

Let us discuss the case when the difference Painlev\'e equation I has 
a rational solution. It is well-known that the only small
functions related to the rational functions are constants. 
If $w$ is a rational solution of
 \begin{equation}\label{diff-pain-I-A.eq}
 \overline{w}+w+\underline{w}=\frac{\pi_1z+\pi_2}{w}+k_1,
 \end{equation}
where $\pi_i$ and $k_j (i=1,2; j=1)$ are the small functions related to 
$w$ with period of $i$ and $j$ respectively, then $\pi_1$, $\pi_2$ and 
$k_1$ are reduced to constants.
If $\pi_1\equiv 0$ and $\pi_2\equiv 0$, then \eqref{diff-pain-I-A.eq} is 
reduced to a linear difference equation, which is trivial.
Thus, we only need to consider the case that either $\pi_1\not\equiv 0$ 
or $\pi_2\not\equiv 0$.

\begin{theorem}\label{rational.theorem}
 Let $w$ be a non-constant solution of the difference Painlev\'e equation I 
defined as in \eqref{diff-pain-I-A.eq}, then $w$ is transcendental.
\end{theorem}


\begin{proof}
Suppose that $w$ is a rational solution of \eqref{diff-pain-I-A.eq}. 
Let us write $w$ as
 $$
 w(z)=\frac{P(z)}{Q(z)}=\frac{a_mz^m+a_{m-1}z^{m-1}+\cdots
+a_1z+a_0}{b_nz^n+b_{n-1}z^{n-1}+\cdots+b_1z+b_0}.
 $$
If we substitute $w$ into the difference Painlev\'e equation I 
\eqref{diff-pain-I-A.eq}, then we have
 $$
 \frac{P(z+1)}{Q(z+1)}+\frac{P(z)}{Q(z)}+\frac{P(z-1)}{Q(z-1)}=\frac{(\pi_1z+\pi_2)Q(z)}{P(z)}+k_1.
 $$
According to the equation above, it follows that
 \begin{equation}\label{PQ.eq}
 \begin{split}
 &P(z+1)P(z)Q(z)Q(z-1)+P(z)Q(z+1)Q(z-1)P(z)\\
 &+P(z-1)P(z)Q(z)Q(z+1)\\
 &=(\pi_1z+\pi_2)Q(z)Q(z+1)Q(z)Q(z-1) \\
&\quad +k_1Q(z+1)Q(z)Q(z-1)P(z).
 \end{split}
 \end{equation}
There are three terms related to $P(z)$ and $Q(z)$ on the left-hand side of
\eqref{PQ.eq} with the same degree.
Moreover, their coefficients of maximum degree are not different, so it 
is easy to see that the degree of the polynomial on the left-hand side of \eqref{PQ.eq} is $2m+2n$.

Suppose that $\pi_1\neq0$. If $n>m-1$, then the degree of the right-hand 
side of \eqref{PQ.eq} is $4n+1$. It implies that
$2m=2n+1$, which is impossible. If $n<m-1$, then the degree of the right-hand 
side of \eqref{PQ.eq} is $3n+m$. It implies that
$m=n$, which is impossible. If $m=n+1$, then the degree of the right-hand side
 of \eqref{PQ.eq} is not larger than $3n+m$ or $4n+1$. It implies that
$2m-1\leq2n$, which is impossible. Therefore, we have $\pi_1=0$.

If $\pi_1=0$ and $\pi_2\neq0$, then the equation we considered is
 \begin{equation}\label{PainI.eq}
 \overline{w}+w+\underline{w}=\frac{\pi_2}{w}+k_1,
 \end{equation}
where $k_1$ and $\pi_2$ are reduced to constants if $w$ is a rational 
solution of \eqref{PainI.eq}. Now let us denote
 $$
 P_1(z,w)=w(\overline{w}+w+\underline{w})-\pi_2-k_1w.
 $$
Since $w=0$ is not the solution of $P_1(z,w)$, it follows from 
Lemma \ref{P.lemma} that
 \begin{equation}\label{Npole.eq}
 N\big(r,\frac{1}{w}\big)=m\log r+O(1)=T(r,w)+O(1).
 \end{equation}
Since $\pi_2$ and $k_1$ are the small function of $w$, it yields by 
Lemma \ref{clunie.lemma} that
 \begin{equation}\label{Nzero.eq}
 N(r,w)=n\log r+O(1)=T(r,w)+O(1).
 \end{equation}
Therefore, we have $m=n$ from \eqref{Npole.eq} and \eqref{Nzero.eq}.

Let $A=\{z_i\in\mathbb{C}; i\in\mathbb{N}\}$ be the set of all zeros of $w(z)=0$,
and its cardinality of the set $A\cap\{z\in\mathbb{C}: |z|\leq r\}$ is denoted by
$n_A\big(r,\frac{1}{w}\big)$ according to its multiplicity, its counting functions
is denoted by $N_A\big(r,\frac{1}{w}\big)$. Suppose that $z_0\in A$,
then from \eqref{PainI.eq} $z_0$ is a pole of $w(z+1)$ or $w(z-1)$. 
If $z_0$ is the pole of $w(z+1)$ or $w(z-1)$, then $z_0+2$ or $z_0-2$ 
is also the pole of $w(z)$ by \eqref{PainI.eq}.

Let $B$ be the subset of $A$ of all points such that
 \[
 w(z_0-1)=\infty \quad\text{and} \quad w(z_0+1)=\infty
 \]
hold. We denote by $N_B\big(r,\frac{1}{w}\big)$ the corresponding counting
function, and
 $$
 N_{A\setminus B}\big(r,\frac{1}{w}\big)
=N_A\big(r,\frac{1}{w}\big)-N_B\big(r,\frac{1}{w}\big)
 $$
the counting function for those points $z_0\in A\setminus B$. 
Therefore, for each points in $B\in\{z\in\mathbb{C}: |z|\leq r\}$ there
are at least two poles in the disc $\{z\in\mathbb{C}: |z|\leq r+1\}$,
which implies that
 \begin{equation}\label{NBI.eq}
 N_B\big(r,\frac{1}{w}\big)\leq \frac{1}{2}N_B\big(r+1,w\big).
 \end{equation}
Similarly, one point in $A\setminus B\cap\{z\in\mathbb{C}; |z|\leq r\}$
 there exists exactly one pole in the disc $\{z\in\mathbb{C}: |z|\leq r+1\}$.
Therefore, we have
 \begin{equation}\label{NABI.eq}
 N_{A\setminus B}\big(r,\frac{1}{w}\big)\leq N_{A\setminus B}(r+1,w).
 \end{equation}
We proceed to prove that the points in the set $A$ are ``almost all" 
in the set $A\setminus B$. Suppose that
 $$
 N_{A\setminus B}(r+1,w)=\alpha T(r,w)+S(r,w),
 $$
where $0\leq\alpha\leq 1$. From \eqref{Npole.eq} to \eqref{NABI.eq}, we have
 \begin{equation}
 \begin{split}
 T(r,w)
 &=N\big(r,\frac{1}{w}\big)+S(r,w)=N_A\big(r,\frac{1}{w}\big)+S(r,w)\\
 &=N_B\big(r,\frac{1}{w}\big)+N_{A\setminus B}\big(r,\frac{1}{w}\big)+S(r,w)\\
 &\leq \frac{1}{2}N_B(r+1,w)+N_{A\setminus B}(r+1,w)\\
 &= \frac{1}{2}N_A(r+1,w)+\frac{1}{2}N_{A\setminus B}(r+1,w)\\
 &\leq (\frac{1}{2}+\frac{\alpha}{2})T(r+1,w)+S(r,w).
 \end{split}
 \end{equation}
It yields that $\alpha=1$ by Lemma \ref{order.lemma}, since $w$ is 
a meromorphic function of finite order. Therefore, we have
 $$
 N_B(r+1,w)=S(r,w).
 $$
It reveals that for ``almost all" $z_0\in A$, then $w(z_0+1)=\infty$ and 
$w(z_0-1)=\infty$ can not hold at the same time. If $w$ is a rational function,
then there is no point $z_0\in A$ such that $w(z_0+1)=\infty$ and 
$w(z_0-1)=\infty$ hold at the same time.

Suppose that $z_0$ is the zero of $w$, then $z_0$ is the pole of $w(z+1)$ or 
$w(z-1)$. Let us suppose $w(z_0+1)=\infty$. Then
$w(z_0+2)=\infty$ and $w(z_0+1)+w(z_0+2)=k_1$. Since $w$ can be written as the 
quotient of $P(z)$ and $Q(z)$, then all the zeros of
$P(z)$ are the zeros of $Q(z+1)$ and $Q(z+2)$. Keeping in mind that 
$\deg{P}=\deg{Q}$, it shows that
 $$
 P(z)=\alpha Q(z+1)=\beta Q(z+2),
 $$
where $\alpha$ and $\beta$ are constants. According to 
\cite[Lemma 2.4]{Wen-Heittokangas-Laine}, it follows that $P(z)$ and $Q(z)$
 are constants, which is impossible.
Thus, $w$ is not a rational solution of \eqref{PainI.eq}. Now we have our assertion.
\end{proof}


Chen and Shen investigate the rational solutions of the difference Painlev\'e 
equation I \eqref{One.eq} in \cite{Chen-Shon}. They show that if $\pi_1=0$ and
$w=P(z)/Q(z)$ is the solution of \eqref{One.eq}, where $P(z)$ and $Q(z)$ are 
polynomials, then $\deg{P}\leq\deg{Q}$. By using the same way as Theorem
\ref{rational.theorem}, we conclude that there is no rational solution 
of \eqref{One.eq}, which improves the result of Chen and Shon.


In what follows, we consider the difference Painlev\'e equation II as follows,
 \begin{equation}\label{diff-pain-II.eq}
 \overline{w}+\underline{w}=\frac{(\pi_1z+k_1)w+\pi_2}{1-w^2},
 \end{equation}
where $\pi_i$ and $k_j (i=1,2; j=1)$ are the small functions related to $w$ 
with period of $i$ and $j$ respectively.
If $\pi_1=0$, $\pi_2=0$ and $k_1=0$, then the difference Painlev\'e equation II 
\eqref{diff-pain-II.eq} can be reduced into a linear difference equation. 
So we are interested in the case when not all of the coefficients 
of \eqref{diff-pain-II.eq} are zeros.

We proceed to consider the rational solutions of the difference Painlev\'e 
equation II as follows. Now let us consider the case $\pi_2=0$ at first.

\begin{theorem}\label{rational-II.theorem}
Let $w$ be a non-constant meromorphic solution of finite order of the difference 
Painlev\'e equation II \eqref{diff-pain-II.eq}, if
$\pi_1\neq 0$ and $\pi_2=0$, then $w$ is transcendental.
\end{theorem}

\begin{proof}
If $\pi_2=0$, then the difference Painlev\'e equation we considered as
 \begin{equation}\label{pi2=0.eq}
 \overline{w}+\underline{w}=\frac{(\pi_1z+k_1)w}{1-w^2},
 \end{equation}
where $\pi_1$ and $k_1$ are the small functions related to $w$ with period of 1.
The equation \eqref{pi2=0.eq} can be read
as follows by $w(z)=P(z)/Q(z)$, where $P(z)$ and $Q(z)$ are polynomials, 
if $w$ is a rational solution of \eqref{pi2=0.eq}
 $$
 (\overline{P}\underline{Q}+\underline{P}\overline{Q})(Q^2-P^2)=(\pi_1z+k_1)PQ\overline{Q}\underline{Q}.
 $$
We conclude that $\deg{Q}=\deg{P}$. In fact, if $\deg{P}>\deg{Q}$, then 
the degree of the polynomial of the left-hand side is $3\deg{P}+\deg{Q}$, 
and the degree of the polynomial of the
right-hand side is $\deg{P}+3\deg{Q}+1$, which implies that 
$2\deg{Q}+1=2\deg{P}$. It is impossible.
If $\deg{P}<\deg{Q}$, then the degree of the polynomial of the left-hand side
 is $3\deg{Q}+\deg{P}$, and the degree of the polynomial of the
right-hand side is also $\deg{P}+3\deg{Q}+1$, which is impossible. 
Therefore, we have $\deg{Q}=\deg{P}$.

Now, dividing by $w$ on the both side of \eqref{pi2=0.eq} implies that
 $$
 \frac{\overline{w}}{w}+\frac{\underline{w}}{w}=\frac{(\pi_1z+k_1)}{1-w^2}.
 $$
Since $w$ is a rational function, Lemma \ref{diff-deri.lemma} tells us that
 \begin{equation}\label{mw.eq}
 m\Big(r,\frac{(\pi_1z+k_1)w}{1-w^2}\Big)
=m\Big(\frac{\overline{w}}{w}+\frac{\underline{w}}{w}\Big)=O(1).
 \end{equation}
From \eqref{mw.eq}, it yields that by $w(z)=P(z)/Q(z)$
 $$
 m\Big(r,\frac{(\pi_1z+k_1)PQ}{Q^2-P^2}\Big)=O(1),
 $$
which implies that $\deg{P}+\deg{Q}+1\leq\max\{2\deg{P},2\deg{Q}\}$. 
It is a contradiction with $\deg{Q}=\deg{P}$. The proof is complte.
\end{proof}

In the following, we consider another case of the difference Painlev\'e equation 
II that $\pi_1=0$ and $\pi_2=0$. We state our result as follows.


\begin{theorem}\label{rational-II-B.theorem}
Let $w$ be a non-constant rational solution of the difference Painlev\'e 
equation II \eqref{diff-pain-II.eq}, if
$\pi_1= 0$ and $\pi_2=0$, then $w=\frac{1}{z+\alpha}$, where 
$\alpha\in\mathbb{C}$, and $k_1=2$. Moreover, $-w$ is also a rational solution
of the difference Painlev\'e equation II.
\end{theorem}

\begin{proof}
If $\pi_1=0$ and $\pi_2=0$, then the difference Painlev\'e equation II we considered is
 \begin{equation}\label{pi12=0.eq}
 \overline{w}+\underline{w}=\frac{k_1w}{1-w^2},
 \end{equation}
where $k_1$ is a small function related to $w$ with period of 1. 
If $w$ is a rational solution of \eqref{pi12=0.eq}, then
$k_1$ is reduced to a constant.

Let $A=\{z_i\in\mathbb{C}; i\in\mathbb{N}\}$ be the set of all zeros of
 \begin{alignat*}{1}
 w(z)-1=0 \quad\text{and} \quad w(z)+1=0,
 \end{alignat*}
and its cardinality of the set $A\cap\{z\in\mathbb{C}: |z|\leq r\}$ is denoted by
 $n_A(r,\frac{1}{w\pm1})$ according to its multiplicity, its counting functions
is denoted by $N_A(r,\frac{1}{w\pm1})$. Suppose that $z_0\in A$, then 
from \eqref{pi12=0.eq} $z_0$ is the pole of $w(z+1)$ or $w(z-1)$. 
If $z_0$ is the pole
of $w(z+1)$ or $w(z-1)$, then $z_0+2\in A$ or $z_0-2\in A$.

Let $B$ be the subset of $A$ of all points such that
 \begin{alignat*}{1}
 w(z_0-1)=\infty \quad\text{and} \quad w(z_0+1)=\infty
 \end{alignat*}
hold. We denote by $N_B\Big(r,\frac{1}{w\pm1}\Big)$ the corresponding
counting function, and
 $$
 N_{A\setminus B}\Big(r,\frac{1}{w\pm1}\Big)
=N_A\Big(r,\frac{1}{w\pm1}\Big)-N_B\Big(r,\frac{1}{w\pm1}\Big)
 $$
counting function for those points $z_0\in A\setminus B$. 
Therefore, for each points in $B\in\{z\in\mathbb{C}: |z|\leq r\}$ there
is exactly one pole in the disc $\{z\in\mathbb{C}: |z|\leq r+1\}$, which implies that
 \begin{equation}\label{NB.eq}
 N_B(r-1,w)\leq N_B\Big(r,\frac{1}{w\pm1}\Big)\leq N_B(r+1,w).
 \end{equation}
Similarly, two points in $A\setminus B\cap\{z\in\mathbb{C}; |z|\leq r\}$
there exists one pole in the disc $\{z\in\mathbb{C}: |z|\leq r+1\}$. Therefore, we have
 \begin{equation}\label{NAB.eq}
 N_{A\setminus B}\Big(r,\frac{1}{w\pm1}\Big)\leq 2N_{A\setminus B}(r+1,w).
 \end{equation}
Since $w$ is the finite order meromorphic solution of \eqref{pi12=0.eq}, let
 $$
 H(z,w)=(\overline{w}+\underline{w})(1-w^2)-k_1w.
 $$
It is easy to check that $w=\pm1$ is not the solutions of $H(z,w)=0$. 
It follows from Lemma \ref{P.lemma} that
 \begin{equation}\label{N.eq}
 N_A\Big(r,\frac{1}{w\pm1}\Big)=N\Big(r,\frac{1}{w\pm1}\Big)=2T(r,w)+S(r,w)
 \end{equation}
In the following, we proceed to prove that the points in the set $A$ are ``almost all" in the set $A\setminus B$. Suppose that
 $$
 N_{A\setminus B}(r+1,w)=\alpha T(r,w)+S(r,w),
 $$
where $0\leq\alpha\leq 1$. From \eqref{NB.eq} to \eqref{N.eq}, we have
 \begin{equation}
 \begin{split}
 2T(r,w)
&=N\Big(r,\frac{1}{w\pm1}\Big)+S(r,w)=N_A\Big(r,\frac{1}{w\pm1}\Big)+S(r,w)\\
 &=N_B\Big(r,\frac{1}{w\pm1}\Big)+N_{A\setminus B}
 \Big(r,\frac{1}{w\pm1}\Big)+S(r,w)\\
 &\leq N_B(r+1,w)+2N_{A\setminus B}(r+1,w)\\
 &= N_A(r+1,w)+N_{A\setminus B}(r+1,w)\\
 &\leq (1+\alpha)T(r+1,w)+S(r,w).
 \end{split}
 \end{equation}
It yields that $\alpha=1$ by Lemma \ref{order.lemma}, since $w$ is a
 meromorphic function of finite order. Therefore, we have
 $$
 N_{B}(r+1,w)=S(r,w).
 $$
It reveals that for ``almost all" $z_0\in A$, then $w(z_0+1)=\infty$ 
and $w(z_0-1)=\infty$ can not hold at the same time. If $w$ is a rational function,
then there is no point $z_0\in A$ such that $w(z_0+1)=\infty$ and 
$w(z_0-1)=\infty$ hold at the same time. Now, set
 $$
 w(z)=P(z)/Q(z)
 $$
where $P(z)$ and $Q(z)$ are polynomials, we can rewrite \eqref{pi12=0.eq} as follows
 \begin{equation}\label{PQ-II.eq}
 \frac{P(z-1)Q(z+1)+P(z+1)Q(z-1)}{Q(z+1)Q(z-1)}=\frac{k_1P(z)Q(z)}{Q^2(z)-P^2(z)}.
 \end{equation}
Then the numerator and the denominator on the left-hand side are relatively prime. 
In fact, if it is not true, then
there is a point such that $Q(z_0+1)=0$ and $Q(z_0-1)=0$ hold, which is a 
contradiction with there is no point $z_0$ such that
$w(z_0+1)=\infty$ and $w(z_0-1)=\infty$ hold, if $w$ is a rational function.
 Moreover, the numerator and the denominator
on the right-hand side are also relatively prime. Indeed, if this is not true, 
there is a point such that $P(z_1)=0$ and $Q(z_1)=0$
hold, which is impossible. Therefore, we have
 \begin{equation}\label{Q1.eq}
 Q(z+1)Q(z-1)=Q^2(z)-P^2(z)
 \end{equation}
holds, and at the same time the  equality
 \begin{equation}\label{Q2.eq}
 P(z-1)Q(z+1)+P(z+1)Q(z-1)=k_1P(z)Q(z)
 \end{equation}
holds. Comparing the degree of both side of \eqref{Q1.eq}, we conclude that 
$\deg{Q}>\deg{P}$. Let $n=\deg{Q}$ and $m=\deg{P}$, it is clear that
$n>m$ and $n\geq 1$. Suppose that
\begin{gather*}
 P(z)=a_mz^m+a_{m-1}z^{m-1}+\ldots+a_1z+a_0, \\
 Q(z)=b_nz^n+b_{n-1}z^{n-1}+\ldots+b_1z+b_0,
\end{gather*}
where $a_i (0\leq i\leq n)$ and $b_j (0\leq j\leq m)$ are constants, and 
$a_n\neq 0$ and $b_m\neq 0$.
If we substitute $P(z)$ and $Q(z)$ into \eqref{Q1.eq}, and compare the 
coefficients of degree $2n-2$ of both side of
\eqref{Q1.eq}, then we have
 \begin{equation}\label{ab.eq}
 a_m^2=nb^2_n,
 \end{equation}
which implies that $m=n-1$. Now, let us compare the coefficients of degree 
$n$ of both side of \eqref{Q2.eq}, then it shows that
 $ k_1=2$.

Suppose that $n>2$, then we proceed to compare the coefficients of degree 
$2n-3$ of both side of \eqref{Q2.eq}, it yields that
 $$
 (n-1)(n-2)a_{n-1}b_n=0,
 $$
which is impossible. Thus, it shows that $n\leq 2$. Suppose that $n=2$, 
then we can write
 $$
 P(z)=a_1z+a_0 \quad\text{and} \quad Q(z)=b_2z^2+b_1z+b_0.
 $$
If we substitute $P(z)$ with degree 1 and $Q(z)$ with degree 2 into 
\eqref{Q1.eq} and \eqref{Q2.eq}, and compare the coefficients
of both side of two equations, then we have
 \begin{gather}
 a_1^2 =2b_2^2 \label{A.eq} \\
 a_0a_1 =b_1b_2\label{B.eq} \\
 a_1b_1 =a_0b_2 \label{C.eq} \\
 b_2^2+2b_0b_2 =b_1^2-a_0^2 .\label{D.eq}
 \end{gather}
From \eqref{B.eq} and \eqref{C.eq}, we conclude that $a_1^2=b_2^2$,
 which is a contradiction with \eqref{A.eq}. From the discussion above,
it is known that $n=1$. Therefore, it follows that by \eqref{ab.eq}
 $$
 w=\frac{1}{z+\alpha},
 $$
where $\alpha\in\mathbb{C}$. Moreover, $-w$ is also a solution of
difference Painlev\'e equations II \eqref{pi12=0.eq}. Therefore, we have 
our assertion.
\end{proof}

From Theorem \ref{rational-II-B.theorem}, we know that if the difference 
Painlev\'e equation II \eqref{diff-pain-II.eq} with
$\pi_1=0$ and $\pi_2=0$ solves a rational solution, then $k_1=2$, 
Thus, we have the following corollary.


\begin{corollary}\label{rational-II-B.corollary}
There is no rational solutions of the difference Painlev\'e equation II
 $$
 \overline{w}+\underline{w}=\frac{k_1w}{1-w^2},
 $$
if $k_1\neq 2$, where $k_1$ is a small function related to $w$ with period $1$.
\end{corollary}

The next example shows that there exists a rational solution of the difference
Painlev\'e equation II \eqref{diff-pain-II.eq}, if $\pi_1=0$ and $\pi_2=0$.

\begin{example} \rm
Let $w=1/z$. Then $w$ is a rational solution of the difference 
Painlev\'e equation II
 $$
 \overline{w}+\underline{w}=\frac{2w}{1-w^2}.
 $$
Also $-w$ is a rational solution of this equation.
\end{example}

In fact, we can obtain \eqref{Q1.eq}  the same way if $\pi_2\neq 0$ and $\pi_1=0$. 
Thus, for the difference Painlev\'e equation II becomes
 \begin{equation}\label{pi1=0.eq}
 \overline{w}+\underline{w}=\frac{k_1w+\pi_2}{1-w^2},
 \end{equation}
where $\pi_2$ and $k_1$ are non-zero small functions related to $w$, 
which are of period 2 and 1 respectively.
If $w$ is a rational solution of \eqref{pi1=0.eq}, then $k_1$ and $\pi_2$ 
are reduced to the non-zero constants.
If we write $w(z)=P(z)/Q(z)$, where $P(z)$ and $Q(z)$ are polynomials, 
then \eqref{Q1.eq} holds. It shows that
$\deg{P}<\deg{Q}$. Let
 $$
 H_1(z,w)= (\overline{w}+\underline{w})(1-w^2)-(k_1w+\pi_2).
 $$
It yields that $w=0$ is not a solution of $H_1(z,w)=0$ according to 
$\pi_2\neq 0$. Thus, it follows from Lemma \ref{P.lemma} that
 $$
 m\big(r,\frac{1}{w}\big)=S(r,w),
 $$
which implies that $\deg{p}\geq\deg{Q}$. It is a contradiction. 
Therefore, $w$ is not a rational solution of \eqref{pi1=0.eq}.

\begin{theorem}\label{rational-II-C.theorem}
Let $w$ be a non-constant meromorphic solution of finite order of the difference 
Painlev\'e equation II \eqref{diff-pain-II.eq}. If
$\pi_1= 0$ and $\pi_2\neq 0$, then $w$ is transcendental.
\end{theorem}

The next theorem is a partial result on the rational solutions of the 
difference Painlev\'e equation II with $\pi_1\neq 0$ and $\pi_2\neq 0$.

\begin{theorem}\label{rational-II-D.theorem}
Suppose that $w(z)=P(z)/Q(z)$ is a non-constant rational function. 
Let $P(z)$ and $Q(z)$ be the polynomials 
\begin{gather*}
 P(z)=a_mz^m+\ldots+a_1z+a_0, \\
 Q(z)=b_nz^n+\ldots+b_1z+b_0,
\end{gather*}
where $a_i (0\leq i\leq m)$ and $b_j (0\leq j\leq n)$ are constants. 
If $w$ is a rational solution of the difference Painlev\'e equation II 
\eqref{diff-pain-II.eq} such that $\pi_1\neq 0$ and $\pi_2\neq0$, 
then $n=m+1$ and $(\pi_2/\pi_1)^2=n\in\mathbb{N}$.
\end{theorem}

\begin{proof}
Suppose that $w$ is a rational solution of \eqref{diff-pain-II.eq}, 
then we have
 \begin{equation}\label{Q.eq}
 \begin{split}
 &Q(z+1)Q(z-1)((\pi_1z+k_1)P(z)Q(z)+\pi_2Q^2(z))\\
 &=(Q^2(z)-P^2(z))(P(z+1)Q(z-1)+P(z-1)Q(z+1)).
 \end{split}
 \end{equation}
Let us compare the degree of both sides of \eqref{Q.eq}, 
then we have $n=m+1$ and
 \begin{equation}\label{pi1pi2.eq}
 \pi_1a_{n-1}+\pi_2b_n=0.
 \end{equation}
By the similar way as \eqref{ab.eq} in Theorem \ref{rational-II-B.theorem}, 
it follows that
 $ a^2_{n-1}=nb^2_n$.
This and \eqref{pi1pi2.eq}, yield 
 $$
 \frac{\pi^2_2}{\pi^2_1}=n.
 $$
Therefore, we have our assertion.
\end{proof}

From  Theorem \ref{rational-II-D.theorem}, we know that if the difference 
Painlev\'e equation II \eqref{diff-pain-II.eq} with
$\pi_1\neq 0$ and $\pi_2\neq 0$ solves a rational solution, 
then $\pi_2^2/\pi_1^2$ is an integer, Thus, we have the following corollary.

\begin{corollary}\label{rational-II-D.corollary}
There is no rational solutions of the difference Painlev\'e equation II
 $$
 \overline{w}+\underline{w}=\frac{(\pi_1z+k_1)w+\pi_2}{1-w^2},
 $$
if $\pi_2^2/\pi_1^2$ is not an integer, where $\pi_i$ and 
$k_j (i=1,2; j=1)$ are the small functions related to $w$ 
with period of $i$ and $j$, respectively.
\end{corollary}

The next example shows that there exists a rational solution of the difference
Painlev\'e equation II \eqref{diff-pain-II.eq}, if $\pi_1\neq 0$ and $\pi_2\neq 0$.

\begin{example} \rm
Let $w=\frac{1}{z+\beta}$ and $\beta=\frac{k_1-2}{\pi_1}$, then $w$ is a 
rational solution of the difference Painlev\'e equation II
 $$
 \overline{w}+\underline{w}=\frac{(\pi_1z+k_1)w+\pi_2}{1-w^2},
 $$
if $\pi_2+\pi_1=0$.
\end{example}


Next we proceed to seek new solutions from a known solution of the 
difference Painlev\'e equation II \eqref{diff-pain-II.eq}.
To seek a new solution, let us reveal the relation between the solutions 
of the difference Riccati equation
and the solutions of the difference Painlev\'e equation II in the following theorem.

\begin{theorem}\label{Ri-II.theorem}
If $w$ is a solution of the difference Riccati equation
 $$
 \overline{w}=\frac{w+\left(1+\frac{\pi_2}{2}-\frac{\pi_1z+k_1}{2}\right)}{1+w},
 $$
where $\pi_i$ and $k_j$ $(i=1,2; j=1)$ are the small functions related to
 $w$ with period of $i$ and $j$, respectively,
then $w$ is the solution of the difference Painlev\'e equation II 
\eqref{diff-pain-II.eq}.
\end{theorem}

\begin{proof}
Since $w$ is a solution of difference Riccati equation, then by Lemma 
\ref{inverse.lemma}, $w$ satisfies
 $$
 \underline{w}=\frac{w-\left(1-\frac{\pi_2}{2}-\frac{\pi_1z+k_1}{2}\right)}{1-w}.
 $$
Therefore, 
 \begin{align*}
 \overline{w}+\underline{w}
&=\frac{w+(1+\frac{\pi_2}{2}-\frac{\pi_1z+k_1}{2})}{1+w}
 +\frac{w-(1-\frac{\pi_2}{2}-\frac{\pi_1z+k_1}{2})}{1-w}\\
 &=\frac{(\pi_1z+k_1)w+\pi_2}{1-w^2}.\\
 \end{align*}
\end{proof}

In what follows, we try to construct the meromorphic solutions of the 
difference Painlev\'e equation II .
If we find a meromorphic solution $w$ of the difference Riccati equations, 
then $w$ solves the difference Painlev\'e equation II according to 
Theorem \ref{Ri-II.theorem}. The main idea is from \cite{Ishizaki}.

Suppose that $w$ is a meromorphic solution of the difference Ricciti equation
 \begin{equation}\label{W.eq}
 \underline{w}=\frac{w-\left(1-\frac{\pi_2}{2}-\frac{\pi_1z+k_1}{2}\right)}{1-w},
 \end{equation}
where $\pi_i$ and $k_j (i=1,2; j=1)$ are the small functions related to $w$ 
with period of $i$ and $j$, respectively. By denoting
 \begin{equation}\label{diff-w.eq}
 w=\frac{f(z+1)-f(z)}{f(z)},
 \end{equation}
we have the meromorphic function $f(z)$ satisfies the following equality
 $$
 \frac{f(z+2)-f(z+1)}{f(z+1)}
=\frac{\frac{f(z+1)-f(z)}{f(z)}+\pi}{1+\frac{f(z+1)-f(z)}{f(z)}},
 $$
where $\pi=1-\frac{\pi_2}{2}-\frac{\pi_1z+k_1}{2}$, which implies that 
$f(z)$ is a solution of the following second order linear homogeneous 
difference equation
 \begin{equation}\label{linear.eq}
 f(z+2)-2f(z+1)+(1-\pi)f(z)=0.
 \end{equation}
In what follows, let us consider the simplest case that $\pi$ is a constant. Thus,
 $$
 f(z)=(1+\sqrt{\pi})^zQ_1(z)+(1-\sqrt{\pi})^zQ_2(z)
 $$
is a solution of \eqref{linear.eq}, where $Q_1$ and $Q_2$ are functions 
with period 1. Therefore,
 $$
 w=\frac{\sqrt{\pi}(1+\sqrt{\pi})^zQ_1(z)-\sqrt{\pi}(1-\sqrt{\pi})^zQ_2(z) }
{(1+\sqrt{\pi})^zQ_1(z)+(1-\sqrt{\pi})^zQ_2(z) }
 $$
is a solutions of the difference Riccati equation \eqref{W.eq} by \eqref{diff-w.eq}.
 We conclude that $w$ is a solution of the difference
Painlev\'e equation II by Theorem \ref{Ri-II.theorem}.


Now we seek a new meromorphic solution by a known solution of the difference 
Painlev\'e equations II. For example, $w_1=1/z$
is also the solution of the difference Riccati equation
 \begin{equation}\label{w-Ri.eq}
 \overline{w}=\frac{w}{1+w}.
 \end{equation}
It follows from theorem \ref{Ri-II.theorem} that $w_1$ is a solution of 
the difference Painlev\'e equation II
 \begin{equation}\label{w-II}
 \overline{w}+\underline{w}=\frac{2}{1-w^2}.
 \end{equation}
If $f_1(z)$ is the solution of \eqref{diff-w.eq}, then let us set
 \begin{equation}\label{f.eq}
 f_1(z+1)=\frac{z+1}{z}f_1(z),
 \end{equation}
then according to \eqref{linear.eq}, we have $f_1(z)$ is a solution of 
the linear difference equation
 $$
 f(z+2)-2f(z+1)+f(z)=0.
 $$
To seek a new solution from $w_1=1/z$, we need the following lemma. 
We state it as complex analysis version in \cite{Ishizaki}.
It also works for the real line case, see \cite[p.48]{Kelley-Peterson} 
and \cite[pp.115-116]{Kohno}.

\begin{lemma}\label{solution.lemma}
Let $R(z)$ be a rational function. We write $R(z)$ in the form
 $$
 R(z)=\rho\frac{\Pi_{k=1}^n(z-\alpha_k)}{\Pi_{j=1}^m(z-\beta_j)},
 $$
where $\rho\neq 0$, $\alpha_k$, $k=1,\ldots, n$ and $j=1,\ldots,m$ 
are complex numbers. The first order linear homogeneous equations
 $$
 y(z+1)=R(z)y(z)
 $$
can be solved as
 $$
 y(z)=Q(z)\rho^z\frac{\Pi_{k=1}^n\Gamma(z-\alpha_k)}{\Pi_{j=1}^m\Gamma(z-\beta_j)}.
 $$
\end{lemma}

Since $f_1(z)$ is a solution of the first order difference equation \eqref{f.eq}, 
by Lemma \ref{solution.lemma} we have 
 $$
 f_1(z)=Q_1(z)\frac{\Gamma(z+1)}{\Gamma(z)}=zQ_1(z),
 $$
where $Q_1(z)$ is a function with period 1. 
It is known that $f_2(z)=Q_2(z)$ is another solution of \eqref{f.eq}, where
$Q_2(z)$ is a function with period 1. By \cite{Praagman}, we know that 
\eqref{f.eq} has two linearly independent meromorphic
solutions $Q_1(z)$ and $Q_2(z)$. Thus, every meromorphic solution of 
\eqref{f.eq} can be formed by
 $$
 f(z)=zQ_1(z)+Q_2(z).
 $$
Therefore, by using the transformation \eqref{diff-w.eq}, we have another 
new meromorphic solution of \eqref{w-Ri.eq}, which can be written as
 \begin{equation}\label{w-solution.eq}
 w(z)=\frac{Q_1(z)}{zQ_1(z)+Q_2(z)}.
 \end{equation}
Now we have found a new meormorphic solution \eqref{w-solution.eq} of 
difference Painlev\'e equation II \eqref{w-II} from a known solution
$w_1=1/z$. In particular,
 $$
 w=\frac{1}{z+\alpha}
 $$
solves the difference Painlev\'e equation II \eqref{w-II}, where 
$\alpha$ is any complex constant.

\subsection*{Acknowledgements}
This project was supported by the National Natural Science Foundation for
the Youth of China (No. 11501402), and by the Shanxi Scholarship council of
China (No. 2015-043).


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\end{document}