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

Abstract

We show that the difference equation $x_{n+1}=x_n{x}_{n-k}/x_{n-k+1}(a+b{x}_n{x}_{n-k}),n\in {\mathbb{N}}_0$, where $k\in \mathbb{N}$, the parameters $a$, $b$ and initial values $x_{-i}$, $i=\overline{0,k}$ are real numbers, can be solved in closed form considerably extending the results in the literature. By using obtained formulae, we investigate asymptotic behavior of well-defined solutions of the equation.