A Frobenius-type theory for discrete systems

TL;DR

This paper introduces a Frobenius-like framework for analyzing singularities in discrete dynamical systems, extending classical methods to lattice-based models using Rota algebras.

Contribution

It develops a novel approach based on Rota algebra theory to study singularities in discrete systems, paralleling classical Frobenius theory for differential equations.

Findings

Applied the theory to Bessel, Hermite, and Airy equations.

Established a lattice-preserving Leibniz rule approach.

Extended classical singularity analysis to discrete systems.

Abstract

We develop an approach analogous to classical Frobenius theory for the analysis of singularities of ODEs in the case of discrete dynamical systems. Our methodology is based on the Roman-Rota theory of finite operators and relies crucially on the idea of preserving the Leibniz rule on a lattice of points by means of the notion of Rota algebras. The relevant cases of the Bessel, Hermite and Airy equations are discussed.