D-module approach to Liouville's Theorem for difference operators

TL;DR

This paper extends Liouville's theorem to difference operators by using residue maps derived from Weyl algebra structures, revealing new boundedness conditions for complex functions.

Contribution

It introduces a general framework for Liouville's theorem analogues for difference operators using residue maps and algebraic structures.

Findings

Established Liouville's theorem analogues for various difference operators.

Connected boundedness conditions to algebraic residue maps.

Unified approach using Weyl and q-Weyl algebra structures.

Abstract

We establish analogues of Liouville's theorem in the complex function theory, with the differential operator replaced by various difference operators. This is done generally by the extraction of (formal) Taylor coefficients using a residue map which measures the obstruction having local "anti-derivative". The residue map is based on a Weyl algebra or $q$-Weyl algebra structure satisfied by each corresponding operator. This explains the different senses of "boundedness" required by the respective analogues of Liouville's theorem in this article.