On the solutions of a second-order difference equations in terms of generalized Padovan sequences

Abstract

This paper deals with the solution, stability character and asymptotic behavior of the rational difference equation \begin{equation*} x_{n+1}=\frac{αx_{n-1}+β}{ γx_{n}x_{n-1}},\qquad n \in \mathbb{N}_{0}, \end{equation*} where $\mathbb{N}_{0}=\mathbb{N}\cup \left\{0\right\}$, $α,β,γ\in\mathbb{R}^{+}$, and the initial conditions $x_{-1}$ and $x_{0}$ are non zero real numbers such that their solutions are associated to generalized Padovan numbers. Also, we investigate the two-dimensional case of the this equation given by \begin{equation*} x_{n+1} = \frac{αx_{n-1} + β}{γy_n x_{n-1}}, \qquad y_{n+1} = \frac{αy_{n-1} +β}{γx_n y_{n-1}} ,\qquad n\in \mathbb{N}_0, \end{equation*} and this generalizes the results presented in \cite{yazlik}