Asymptotic analysis of solutions of delay difference equations

The asymptotic behavior of solutions for the delay difference equation $x_{n+1}=x_{n}f\left(x_{n-k}\right),n>k,\text{for some}k\in N,$ is investigated, where $f$ has an asymptotic power series. These equations have been studied for some special cases. This paper analyzes other cases and presents asymptotic expansions of solutions for such higher-order difference equations. Several examples are provided.