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.