On the rational difference equation $x_{n+1}=\frac{x_n\cdot (\overline{a}x_{n-k}+ax_{n-k+1})}{bx_{n-k+1}+cx_{n-k}}$

Abstract

In this paper, we will determine an explicit and a constructive type of solution for the difference equation
$$x_{n+1}=\frac{x_n\cdot (\overline{a}x_{n-k}+ax_{n-k+1})}{bx_{n-k+1}+cx_{n-k}},\quad n=0,1,\ldots,$$
where $\overline{a}\geq 0,a>0,b>0,c>0$ and $k\geq 1$ is an integer, with initial conditions $x_{-k},x_{-k+1},\ldots ,x_{-1},x_0$. We also will determine the global behavior of this solution. For the case when $\overline{a}=0$, the method presented here gives us the particular solution obtained by G\"um\"u\c{s} and Abo-Zeid that establishes an inductive type of proof.