Asymptotic approximations for second-order linear difference equations in Banach algebras, II

Abstract

A Liouville-Green (WKB) asymptotic approximation theory is developed for some classes of linear second-order difference equations in Banach algebras. The special case of linear matrix difference equations (or, equivalently, of second-order systems) is emphasized. Rigorous and explicitly computable bounds for the error terms are obtained, and this when both, the sequence index and some parameter that may enter the coefficients, go to infinity. A simple application is made to orthogonal matrix polynomials in the Nevai class.

Asymptotic Integration of Differential and Difference Equations

“…Also see [140] for a generalization of such results to second-order linear difference equations in Banach algebras.…”

Canonical forms and discrete Liouville–Green asymptotics for second-order linear difference equations

“…4. It is possible to proceed similarly in case of matrix difference equations or, equivalently, for systems, but we will not do it in this paper; see, e.g., [4,11,12]. Finally, we summarize the high points of the paper in the short concluding Sect.…”