Algebra of linear recurrence relations in arbitrary characteristic

TL;DR

This paper develops an algebraic framework for linear recurrence relations over arbitrary characteristics, utilizing representation theory and divided derivatives, inspired by umbral calculus, to unify and generalize classical results.

Contribution

It introduces an algebraic approach based on endomorphism representations and divided derivatives, extending the theory of linear recurrence relations to arbitrary characteristic fields.

Findings

Unified algebraic framework for recurrence relations

Use of divided derivatives for natural proofs

Generalization to arbitrary characteristic fields

Abstract

The goal of this paper is to present an algebraic approach to the basic results of the theory of linear recurrence relations. This approach is based on the ideas from the theory of representations of one endomorphisms (a special case of which is well known as the theory of the Jordan normal form of matrices). The notion of the divided derivatives, an analogue of the divided powers, turned out to be crucial for proving the results in a natural way and in their natural generality. The final form of our methods was influenced by the the umbral calculus of G.-C. Rota.