A Note on the Recursive Sequence $x_{n + 1} = p_kx_n + p_{k − 1}x_{n − 1} +...+ p_1x_{n − k + 1}$

Abstract

We present some comments on the behavior of solutions of the difference equation $x_{n + 1} = p_kx_n + p_{k − 1}x_{n − 1} +...+ p_1x_{n − k + 1}$, $n = −1, 0, 1,…,$ where $p_i ≥ 0, i = 1,..., k, k ∈ N$, and $x_{−k},..., x_{−1} ∈ R$.