We give the necessary and sufficient conditions for the asymptotic stability of the linear delay difference equation $x_{n+1}-a^{2}x_{n-1}+bx_{n-k}=0,quad n=0,1,cdots$, 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.

Keywords: the asymptotic stability, linear delay difference equation