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

Abstract

Abstract This paper deals with the solution, stability character and asymptotic behavior of the rational difference equation x n + 1 = α x n − 1 + β γ x n x n − 1 , n ∈ N 0 , $$\begin{array}{} \displaystyle x_{n+1}=\frac{\alpha x_{n-1}+\beta}{ \gamma x_{n}x_{n-1}},\qquad n \in \mathbb{N}_{0}, \end{array}$$ where ℕ0 = ℕ ∪ {0}, α, β, γ ∈ 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 x n + 1 = α x n − 1 + β γ y n x n − 1 , y n + 1 = α y n − 1 + β γ x n y n − 1 , n ∈ N 0 . $$\begin{array}{} \displaystyle x_{n+1} = \frac{\alpha x_{n-1} + \beta}{\gamma y_n x_{n-1}}, \qquad y_{n+1} = \frac{\alpha y_{n-1} +\beta}{\gamma x_n y_{n-1}} ,\qquad n\in \mathbb{N}_0. \end{array}$$