On the Difference Equation x n = a n x n - k / ( b n + c n x n - 1 ⋯ x n - k )

Abstract

The behavior of well-defined solutions of the difference equation $x_n=a_n{x}_{n-k}/(b_n+c_n{x}_{n-1}\cdots x_{n-k}),\hspace{0.17em}n\in {\mathbb{N}}_0$, where $k\in \mathbb{N}$ is fixed, the sequences $a_n$, $b_n$ and $c_n$ are real, $(b_n,c_n)\ne (\mathrm{0,0})$, $n\in {\mathbb{N}}_0$, and the initial values $x_{-k},\dots ,x_{-1}$ are real numbers, is described.