Liouville-Green (WKB) asymptotics for almost-diagonal linear second-order matrix difference equations

A Liouville--Green (or WKB) asymptotic approximation theory is developed for a class of almost--diagonal (''asymptotically diagonal'') linear second-order matrix difference equations. Rigorous and explicitly computable bounds for the error terms are obtained, the asymptotics being made with respect to both, the index and some parameter affecting the equation. The case of the associated inhomogeneous equations is also considered in detail. Some examples and a number of applications are presented for the purpose of illustration.