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