Factorization of Difference Equations by Semiconjugacy with Application to Non-autonomous Linear Equations

TL;DR

This paper explores how semiconjugacy can be used to factorize higher order difference equations into simpler systems, especially in non-autonomous linear equations with variable coefficients, broadening the scope of applicable equations.

Contribution

It introduces time-dependent form symmetries to identify semiconjugate properties in a wider class of non-autonomous difference equations, including general linear equations with variable coefficients.

Findings

Semiconjugacy enables transformation into lower-order systems.

Time-dependent symmetries expand applicability to more equations.

Includes non-autonomous, non-homogeneous linear equations.

Abstract

The existence of a semiconjugate relation permits the transformation of a higher order difference equation on a group into an equivalent triangular system of two difference equations of lower orders. Introducing time-dependent form symmetries in this paper enables us to identify the semiconjugate property in a larger set of non-autonomous difference equations than previously considered. We show that there is a substantial class of equations having this feature that includes the general (non-autonomous, non-homogeneous) linear equation with variable coefficients in an arbitrary algebraic field.