On a system of difference equations of third order solved in closed form

Abstract

In this note we show the the system of difference equations $$x_{n+1}=\frac{a{y}_{n-2}{x}_{n-1}{y}_n+b{x}_{n-1}{y}_{n-2}+c{y}_{n-2}+d}{y_{n-2}{x}_{n-1}{y}_n},$$ $$y_{n+1}=\frac{a{x}_{n-2}{y}_{n-1}{x}_n+b{y}_{n-1}{x}_{n-2}+c{x}_{n-2}+d}{x_{n-2}{y}_{n-1}{x}_n},$$ where $n\in {\mathbb{N}}_0$, the initial values $x_{-2}$, $x_{-1}$, $x_0$, $y_{-2}$, $y_{-1}$ and $y_0$ are arbitrary nonzero real numbers and the parameters $a$, $b$, $c$ and $d$ are arbitrary real numbers with $d\ne 0$, can be solved in a closed form. We will see that when $a=b=c=d=1$ the solutions are expressed using the famous Teteranacci numbers. In particular, the results obtained here extend those in our work \cite{arxiv}.

Manuscript submitted to a journal for publication on 20.08.2019