Semiconjugate Factorizations of Higher Order Linear Difference Equations in Rings

TL;DR

This paper introduces a nonlinear method for factorizing higher order linear difference equations in rings, using eigensequences derived from solutions, which generalizes classical operator factorization and applies to equations with variable coefficients.

Contribution

It presents a novel semiconjugate factorization approach for linear difference equations in rings, extending classical methods to variable coefficients and nonlinear contexts.

Findings

Decomposition of difference equations into lower-order equations using eigensequences.

Application of the method to equations with periodic coefficients.

Formulas for solutions of functional recurrences related to special functions.

Abstract

We study linear difference equations with variable coefficients in a ring using a new nonlinear method. In a ring with identity, if the homogeneous part of the linear equation has a solution in the unit group of the ring (i.e., a unitary solution) then we show that the equation decomposes into two linear equations of lower orders. This decomposition, known as a semiconjugate factorization in the nonlinear theory, generalizes the classical operator factorization in the linear context. Sequences of ratios of consecutive terms of a unitary solution are used to obtain the semiconjugate factorization. Such sequences, known as eigensequences are well-suited to variable coefficients; for instance, they provide a natural context for the expression of the classical Poincar\'{e}-Perron Theorem. We discuss some applications to linear difference equations with periodic coefficients and also derive formulas for the general solutions of linear functional recurrences satisfied by the classical special functions such as the modified Bessel and Chebyshev.