Explicit formulas for the solutions of a system of difference equations of fourth order in the complex domain

Abstract

The aim of this paper is to solve in closed-form the following fourth order system of difference equations
\begin{equation*}
x_{n+1}=\frac{y_{n}x_{n-2}y_{n-3}}{ay_{n-3}x_{n-2}+by_{n}y_{n-3}+cy_{n}x_{n-2}},\,y_{n+1}=\frac{y_{n-2}x_{n-1}}{\alpha
y_{n-2}+\beta x_{n-1}},\,n=0,1,\cdots,
\end{equation*}
where the initial values $x_{-2}$, $x_{-1}$, $x_{0}$, $y_{-3}$, $y_{-2}$, $y_{-1}$, $y_{0}$ are non-zero complex numbers and the parameters $a$, $b$, $c$, $\alpha$, $\beta$ are arbitrary real numbers. At the end, we conclude the results for the more general system of difference equations
\begin{equation*}
x_{n+1}=f^{-1}\left(\frac{g(y_{n}) f(x_{n-2}) g(y_{n-3})}{a
g(y_{n-3}) f(x_{n-2})+b g(y_{n})
g(y_{n-3})+cg(y_{n})f(x_{n-2})}\right), \quad n=0,1,\cdots,
\end{equation*}
\begin{equation*}y_{n+1}=g^{-1}\left(\frac{g(y_{n-2}) f(x_{n-1})}{\alpha g(y_{n-2})+\beta
f(x_{n-1})}\right), \quad n=0,1,\cdots,
\end{equation*}
where $f,g: I\rightarrow J$ are $(1-1)$ continuous functions on $I\subseteq \mathbb{C}^*$, the initial values $x_{-2}$, $x_{-1}$, $x_{0}$, $y_{-3}$, $y_{-2}$, $y_{-1}$, $y_{0}$ are non-zero complex numbers in $I$ and the parameters $a$, $b$, $c$, $\alpha$, $\beta$
are arbitrary real numbers.