The asymptotic stability of x<inf>n+1</inf>-a²x<inf>n-1</inf>+bx<inf>n-k</inf>=0

Abstract

We give the necessary and sufficient conditions for the asymptotic stability of the linear delay difference equation xn+1-a2xn-1+bxn-k= 0, n=0, 1, ..., where a and b are arbitrary real numbers and k is a positive integer greater than 1. The obtained conditions are given in terms of parameters a and b of difference equations. The method of proof is based on arithematic of complex numbers as well as properties of analytic functions.