Solutions of a System of Two Higher-Order Difference Equations in Terms of Lucas Sequence
Solutions of a System of Two Higher-Order Difference Equations in Terms of Lucas Sequence
In this paper we give some theoretical explanations related to the representation for the general solution of the system of the higher-order rational difference equations $$ x_{n+1} = \frac{5 y_{n-k}-5}{y_{n-k}}, \qquad y_{n+1} = \frac{5 x_{n-k}-5}{x_{n-k}} ,\qquad n, k\in \mathbb{N}_0, $$ where $\mathbb{N}_{0}=\mathbb{N}\cup \left\{0\right\}$, and the initial conditions $x_{-k}$, $x_{-k+1},\ldots$, $x_{0}$, …