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

TL;DR

This paper analyzes solutions, stability, and asymptotic behavior of a specific rational second-order difference equation, linking solutions to generalized Padovan sequences and extending to a two-dimensional case.

Contribution

It introduces new solutions related to generalized Padovan numbers and extends the analysis to a coupled two-dimensional difference equation.

Findings

Solutions are expressed in terms of generalized Padovan sequences.

The stability and asymptotic behavior of solutions are characterized.

The two-dimensional case generalizes previous results.

Abstract

This paper deals with the solution, stability character and asymptotic behavior of the rational difference equation \begin{equation*} x_{n+1}=\frac{\alpha x_{n-1}+\beta}{ \gamma x_{n}x_{n-1}},\qquad n \in \mathbb{N}_{0}, \end{equation*} where $\mathbb{N}_{0}=\mathbb{N}\cup \left\{0\right\}$, $\alpha,\beta,\gamma\in\mathbb{R}^{+}$, 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{\alpha x_{n-1} + \beta}{\gamma y_n x_{n-1}}, \qquad y_{n+1} = \frac{\alpha y_{n-1} +\beta}{\gamma x_n y_{n-1}} ,\qquad n\in \mathbb{N}_0, \end{equation*} and this generalizes the results presented in \cite{yazlik}