Linear recurrences indexed by $\mathbb{Z}$

TL;DR

This paper studies linear recurrences indexed by integers, providing a unique reduced form, a solution matrix that parametrizes solutions, and combinatorial insights into their structure.

Contribution

It introduces a method to find a unique reduced recurrence and constructs a solution matrix that characterizes the solution space.

Findings

Existence of a unique reduced recurrence with the same solutions

Construction of a solution matrix parametrizing solutions

Combinatorial characterization of bases and solution space dimension

Abstract

This note considers linear recurrences (also called linear difference equations) in unknowns indexed by the integers. We characterize a unique \emph{reduced} linear recurrence with the same solutions as a given linear recurrence, and construct a \emph{solution matrix} which parametrizes the space of solutions. Several properties of solution matrices are shown, including a combinatorial characterization of bases and dimension of the space of solutions.