The asymptotic stability of \(x_{n+1}-ax_ n+bx_{n-k}=0\) (Q1343960)

For linear difference equations of the form \[ x_{n+1}= ax_n+ bx_{n-k}, \qquad n\in \mathbb{N}\cup \{0\} \] necessary and sufficient conditions for the asymptotic stability are presented in terms of the real coefficients \(a\), \(b\) and the delay \(k\in \mathbb{N}\cup \{0\}\). The proof of the main result is based on a careful analysis of the dependence of the roots of the characteristic polynomial on the parameters \(a\), \(b\) and \(k\).