Linear recurrences indexed by $\mathbb{Z}$

Greg Muller

Published 2019-06-10Version 1

This note collects several general results about linear recurrences (also called linear difference equations) in unknowns indexed by the integers. We characterize a unique \emph{reduced} system equivalent to a given linear recurrence, and construct a \emph{solution matrix} which parametrizes the space of solutions to the original system. Several properties of solution matrices are shown, including a combinatorial characterization of bases and dimension of the space of solutions in terms of \emph{juggling patterns}.