On a difference equation whose solutions are related to Fibonacci numbers

Abstract

In this paper, we consider the following fifth-order non-linear
difference equation
$$
z_{n}=\frac{z_{n-2}^{s}z_{n-3}z_{n-4}}{z_{n-1}\left( az_{n-5}^{s}+bz_{n-3}z_{n-4}\right) }, \ s,n\in \mathbb{N},
$$
where the initial values $z_{-j}$, $j=\overline{0,4}$ and the parameters $a$, $b$ are non-zero real numbers. In addition, the solutions of a more general difference equation defined by one to one functions are obtained. We will show, that both the solutions of the above mentioned equation and the solutions of the general difference equation are related to a generalized Fibonacci sequences.