On the solutions of a higher order recursive sequence

Abstract

In this paper, we introduce an explicit formula and discuss the global behavior of solutions of the recursive sequence
$$
x_{n+1}=\frac{a x_{n-2 k-1}}{b+c \prod_{l=0}^k x_{n-2 l-1}}, \quad n=0,1, \ldots,
$$
where $a, b, c$ are positive real numbers, the initial conditions $x_{-2 k-1}, x_{-2 k}, \ldots, x_{-1}, x_0$ are real numbers and $k$ is a nonnegative integer. We show that every admissible solution with $\prod_{l=0}^k x_{-2(l+1)+i}=\frac{a-b}{c}, i=1,2$ is periodic with prime period $2 k+2$. Otherwise, the solution converges to zero if $a<b$ or converges to a period-(2k+2) solution if $a>b$. We finally study some special cases and give illustrative examples.