Factorizations and Reductions of Order in Quadratic and other Non-recursive Higher Order Difference Equations

TL;DR

This paper develops conditions under which higher order difference equations, including quadratic ones, can be factored into lower order equations using form symmetries, extending previous group-based factorizations.

Contribution

It generalizes semiconjugate factorizations to non-recursive difference equations on arbitrary sets, including quadratic equations on algebraic fields.

Findings

Factorization conditions for higher order difference equations

Extension of semiconjugate factorizations beyond groups

New results for quadratic difference equations on algebraic fields

Abstract

A higher order difference equation may be generally defined in an arbitrary nonempty set S as: \[ f_{n}(x_{n},x_{n-1},...,x_{n-k})=g_{n}(x_{n},x_{n-1},...,x_{n-k}) \] where $f_{n},g_{n} :S^{k+1}\rightarrow S$ are given functions for $n=1,2,... $ and $k$ is a positive integer. We present conditions that imply the above equation can be factored into an equivalent pair of lower order difference equations using possible form symmetries (order-reducing changes of variables). These results extend and generalize semiconjugate factorizations of recursive difference equations on groups. We apply some of this theory to obtain new factorization results for the important class of quadratic difference equations on algebraic fields. We also discuss the nontrivial issue of the existence of solutions for quadratic equations.