\documentclass[reqno]{amsart}
\usepackage{hyperref}
\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2020 (2020), No. 14, pp. 1--14.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu}
\thanks{\copyright 2020 Texas State University.}
\vspace{8mm}}
\begin{document}
\title[\hfilneg EJDE-2020/14\hfil $q$-difference Painlev\'e equations]
{Existence of rational solutions for $q$-difference Painlev\'e equations}
\author[H. Y. Xu, J. Tu \hfil EJDE-2020/14\hfilneg]
{Hong Yan Xu, Jin Tu}
\address{Hong Yan Xu \newline
School of Mathematics and Computer Science,
Shangrao Normal University,
Shangrao Jiangxi 334001, China}
\email{xhyhhh@126.com}
\address{Jin Tu \newline
Department of Mathematics,
Jiangxi Normal University,
Nanchan, Jiangxi 330022, China}
\email{tujin2008@sina.com}
\thanks{Submitted January 13, 2019. Published February 5, 2020.}
\subjclass[2010]{39A13, 30D35}
\keywords{Rational solution; $q$-difference Painlev\'e equation; fixed point}
\begin{abstract}
This article studies properties of meromorphic solutions
for several types of $q$-difference Painlev\'e equations.
We obtain conditions for the existence, and the form of rational solutions
for two classes of $q$-difference Painlev\'e equations.
Also for a solution $f$ we obtain results about the fixed points, the exponents
of convergence of poles of $f,\Delta_q f,(\Delta_qf)/f$.
Our results extend previous theorems given in the references.
\end{abstract}
\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{remark}[theorem]{Remark}
\newtheorem{example}[theorem]{Example}
\allowdisplaybreaks
\section{Introduction and statement of main results}
Painlev\'e equations have been an important research subject in the field
of the mathematics and physics, and they occur in many
physical situations: plasma physics, statistical mechanics, nonlinear waves, etc.
They appear as differential Painlev\'e equation, discrete Painlev\'e equation,
difference Painlev\'e, and so on; see \cite{chen14,fokas,grammat}.
Around 2006, with the development of Nevanlinna theory, Halburd-Korhonen \cite{hklondon}
and Chiang-Feng \cite{3} established independently important results about the complex
difference and difference operators. By utilizing these results,
Halburd-Korhonen \cite{5,hklondon,7} discussed the equation
\begin{equation}\label{e1}
f(z+1)+f(z-1)=R(z,f),
\end{equation}
where $R(z,f)$ is rational in $f$ and meromorphic in $z$. They pointed out that this
equation can be transformed into difference Painlev\'e I equations
\begin{gather}\label{e2}
f(z+1)+f(z-1)=\frac{az+b}{f(z)}+c,\\
\label{e3}
f(z+1)+f(z)+f(z-1)=\frac{az+b}{f(z)}+c,
\end{gather}
and into difference Painlev\'e II equations
\begin{equation}\label{e4}
f(z+1)+f(z-1)=\frac{(az+b)f(z)+c}{1-f(z)^2}.
\end{equation}
In 2010, Ronkainen \cite{ron} further investigated the meromorphic solutions of
the equation
\begin{equation} \label{P3}
f(z+1)f(z-1)=R(z,f)
\end{equation}
where $R(z,f)$ is a rational and irreducible in $f$ and meromorphic in $z$.
He proved that either $f$ satisfies the difference Riccati equation
$$
f(z+1)=\frac{A(z) f(z)+B(z)}{f(z)+C(z)},
$$
or equation \eqref{P3} can be transformed to one of the following equations
\begin{gather*}
f(z+1)f(z-1)=\frac{\eta(z)f(z)^2-\lambda(z)f(z)+\mu(z)}{(f(z)-1)(f(z)-\upsilon(z))},\\
f(z+1)f(z-1)=\frac{\eta(z)f(z)^2-\lambda(z)f(z)}{f(z)-1},\\
f(z+1)f(z-1)=\frac{\eta(z)(f(z)-\lambda(z))}{(f(z)-1)},\\
f(z+1)f(z-1)=h(z)f(z)^m,
\end{gather*}
where $\eta(z),\lambda(z),\upsilon(z)$ satisfy certain conditions.
Generally speaking, the above four equation can be called as the difference
Painlev\'e III equations.
In the past two decades, many mathematicians paid consideration attention to the
value distribution of solutions for complex difference equations, and obtained
lots of important results on the properties of solutions for difference
Painlev\'e I-III equations (see \cite{zheng19,chen14,5,hklondon,7,lf18,wen16,zhongliu}).
In 2010, Chen-Shon \cite{cs10} considered the difference Painlev\'e I equation \eqref{e2}
and obtained the following theorem.
\begin{theorem}[{see \cite[Theorem 4]{cs10}}] \label{thm1.1}
Let $a,b,c$ be constants, where $a,b$ are not both equal to zero. Then
\begin{itemize}
\item[(i)] if $a\neq 0$, then \eqref{e2} has no rational solution;
\item[(ii)] if $a=0$, and $b\neq0$, then \eqref{e2} has a nonzero constant solution
$w(z)=A$, where $A$ satisfies $2A^2-cA-b=0$.
\end{itemize}
The other rational solution is $w(z)=\frac{P(z)}{Q(z)}+A$, where $P(z)$ and $Q(z)$
are relatively prime polynomials and satisfy $\deg P<\deg Q$.
\end{theorem}
In 2014, Zhang-Yang \cite{zhangy2014} studied the difference Painlev\'e III
equations with the constant coefficients, and obtained the following result.
\begin{theorem}[\cite{zhangy2014}]
If $f$ is a transcendental finite-order meromorphic solution of
$$
f(z+1)f(z-1)(f(z)-1)=\eta w(z)\text{ or } f(z+1)f(z-1)(f(z)-1)=f(z)^2-\lambda w(z),
$$
where $\eta(\neq0),\lambda(\neq0,1)$ are constants, then
\begin{itemize}
\item[(i)] $\lambda(f)=\sigma(f)$;
\item[(ii)] $f$ has at most one non-zero Borel exceptional value for $\sigma(f)>0$.
\end{itemize}
\end{theorem}
The Logarithmic Derivative Lemma on $q$-difference operators
was established by Barnett, Halburd, Korhonen and
Morgan \cite{1} in 2007.
Then the interest in studying the properties on the existence and value distribution
of solutions has increased considerably for some $q$-difference equation
which are formed by replacing the $q$-difference $f(qz)$,
$q \in \mathbb{C} \setminus\{0, 1\}$ with $f(z + c)$ of meromorphic function
in some expression concerning complex difference equations;
see \cite{du18jia,gund,kw16,laineyang,liu2013,qiyang,ru,xlz2017,xu19,
xu20,xu16,zhangkorhonen,zhch2, zhch}.
In 2015, Qi-Yang \cite{qiyang} considered the equations
\begin{align}\label{e5}
f(qz)+f\big(\frac{z}{q}\big)=\frac{az+b}{f(z)}+c,
\end{align}
which can be seen as $q$-difference analogues of \eqref{e2},
and obtained the following result.
\begin{theorem}[{\cite[Theorem 1.1]{qiyang}}]
Let $f(z)$ be a transcendental meromorphic solution with zero order of equation \eqref{e5},
and let $a, b, c$ be constants such that $a, b$ cannot vanish
simultaneously. Then
\begin{itemize}
\item[(i)] $f(z)$ has infinitely many poles.
\item[(ii)] If $a \neq 0$ and any $d\in \mathbb{C}$, then $f(z)-d$ has infinitely many zeros.
\item[(iii)] If $a = 0$ and $f(z)$ takes a finite value $A$ finitely often, then $A$
is a solution of ~$2z^2-cz-b=0$.
\end{itemize}
\end{theorem}
In 2018, Liu-Zhang \cite{liu18} studied the difference equation
\begin{equation} \label{l18}
Y(\omega z)+Y(z)+Y(\frac{z}{\omega})=\frac{V(z)}{Y(z)}+c,
\end{equation}
which is a $q$-difference analogues of \eqref{e3}, and obtained the following result.
\begin{theorem}[{\cite[Thereom 1.2]{liu18}}]
Let $c\in \mathbb{C}\setminus\{0\}$, $|\omega|\neq1$, and $V(z)= \frac{X(z)}{B(z)}$
be an irreducible rational function,
where $X(z)$ and $B(z)$ are polynomials with $\deg X(z) = x$ and $\deg B(z) = b$.
\begin{itemize}
\item[(i)] Suppose that $x\geq b$ and $x-b$ is zero or an even number.
If \eqref{l18} has an irreducible rational solution
$Y(z) = \frac{I(z)}{J(z)}$, where $I(z)$ and $J(z)$ are polynomials with
$\deg I(z) = i$ and $\deg J(z) = j$, then
$i-j=\frac{x-b}{2}$.
\item[(ii)] Suppose that $x**0$, then $P(z)/Q(z)=az^s(1+o(1))$ as $|z|=r\to \infty$.
Thus, by virtue of \eqref{e8}, it follows that
$$
a^3z^{3s}(1+o(1))(az^s(1+o(1))-1)=\mu,\quad r\to \infty,
$$
this is impossible for $a\neq 0$.
If $s<0$, then as $r\to \infty$, it follows that $\frac{P(z)}{Q(z)}=o(1)$ and
$$
\frac{P(qz)}{Q(qz)}=o(1),\quad \frac{P(\frac{z}{q})}{Q(\frac{z}{q})}=o(1).
$$
Substituting these into \eqref{e8}, we get $o(1)=\mu$ as $r\to \infty$, this is
a contradiction for $\mu\neq 0$.
Thus, it yields that $s=0$ and $p=t$. From the assumptions of this theorem,
we know that the zeros of $Q(z)$ are not the zeros of $P(z)$ and $P(z)-Q(z)$.
Hence, in view of \eqref{e9}, it follows that all the zeros of $Q^2(z)$ are the zeros
of $P(qz)P(\frac{z}{q})$.
Since $\deg_z [Q(z)^2]=\deg_z[P(qz)P(\frac{z}{q})]=2p$,
then it yields from \eqref{e8} that
\begin{gather} \label{e9}
P(qz)P(\frac{z}{q})=a^2Q(z)^2,\\
\label{e10}
P(z)(P(z)-Q(z))=a(a-1)Q(qz)Q(\frac{z}{q}).
\end{gather}
Next, we confirm that the orders of all the zeros of $P(z)$ are even.
Let $z_0$ be a zero of $P(z)$ with the order $k$. If $z_0\neq 0$ and $k$ is an
odd integer. Then $P(z)$ has the term $(z-z_0)^k$, and $P(qz)P(\frac{z}{q})$
has the term
\begin{equation}
\label{e11}
(z-qz_0)^k\Big(z-\frac{z_0}{q}\Big)^k.
\end{equation}
It means that $qz_0$ and $\frac{z_0}{q}$ are both zeros of $P(qz)P(\frac{z}{q})$
with the order at least $k$.
In addition, since $P(z)$ and $Q(z)$ are relatively prime polynomials,
in view of \eqref{e10}, it follows that $Q(qz)Q(\frac{z}{q})$ has the term $(z-z_0)^k$.
Suppose that $Q(qz)$ and $Q(\frac{z}{q})$ have the terms $(z-z_0)^m$ and $(z-z_0)^l$
respectively, where $m,l \in N$ and $m+l=k$. Obviously, in view of \eqref{e11},
we have $m\neq 0$ and $l\neq 0$. Thus, $Q(z)$ has the term $(z-qz_0)^m(z-\frac{z_0}{q})^l$,
that is, $Q(z)^2$ has the term $(z-qz_0)^{2m}(z-\frac{z_0}{q})^{2l}$.
So, in view of \eqref{e10}, it follows that $P(qz)P(\frac{z}{q})$ has the term
$$
(z-qz_0)^{2m}(z-\frac{z_0}{q})^{2l}.
$$
In view of $m+l=k$ and $k$ is an odd integer, without loss of generality, assume that $mk$. Thus, $qz_0$ is a zero of $P(qz)P(\frac{z}{q})$ with the
order $2m**