Representation of solutions of delayed linear discrete systems with permutable or nonpermutable matrices and second-order differences

Nonhomogeneous delayed linear higher-order discrete systems with second-order differences and multiple delays with linear parts defined by permutable or nonpermutable matrices are considered. Representations of their solutions are obtained by using Z\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {Z}}$$\end{document}-transform. Our results extend previous results. Finally, examples are given to illustrate our theoretical results.