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 ${\mathbb{N}}_0=\mathbb{N}\cup \left\{0\right\}$, $\alpha ,\beta ,\gamma \in {\mathbb{R}}^{+}$, 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{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}

17 pages, 2 figures

References in corpus (1)

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

Cited by in corpus (3)

• On the solutions of a second-order difference equations in terms of generalized Padovan sequences
• On the Solutions of Systems of Difference Equations via Tribonacci Numbers
• On The Dynamics Of Solutions Of A Rational Difference Equation Via Generalized Tribonacci Numbers