Asymptotic Formulae for the Solutions of a Linear Delay Difference Equation−

Consider the linear delay difference equation xn+1 − xn = anxnk, where an′s are real numbers and k is a positive integer. In this paper we establish asymptotic formulae for the solutions of the above equation assuming Σ∞n=0 |an|p < ∞ for some p ≥ 1. The asymptotic formulae are given by an appropriate recursive sequence. For p ∈ {1, 2, 3, 4} we present explicit formulae. We also construct an example which shows that some comparison results in a recent paper of Yan and Qian are incorrect.