# 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.

## Key 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.

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Source: https://tomesphere.com/paper/1906.04311