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 $$ x_{n+1}=\dfrac{ay_{n-2}x_{n-1}y_n+bx_{n-1}y_{n-2}+cy_{n-2}+d}{y_{n-2}x_{n-1}y_n},$$ $$y_{n+1}=\dfrac{ax_{n-2}y_{n-1}x_n+by_{n-1}x_{n-2}+cx_{n-2}+d}{x_{n-2}y_{n-1}x_n},$$ 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}.