Unification, refinements and companions of generalisations of Schur's theorem

TL;DR

This paper presents a unifying theorem for overpartitions with difference conditions, consolidating various generalisations of Schur's theorem and providing new companions and refinements using advanced combinatorial methods.

Contribution

It introduces a comprehensive theorem that unifies multiple generalisations of Schur's theorem for overpartitions, extending previous results with new companions and refinements.

Findings

Unified theorem for overpartitions generalising Schur's theorem

New companions and refinements of Andrews' overpartition theorems

Application of weighted words and $q$-difference equations techniques

Abstract

We prove a general theorem on overpartitions with difference conditions that unifies generalisations of Schur's theorem due to Alladi-Gordon, Andrews, Corteel-Lovejoy and the author. This theorem also allows one to give companions and refinements of the generalisations of Andrews' theorems to overpartitions. The proof relies on the method of weighted words of Alladi and Gordon and $q$-difference equation techniques introduced recently by the author.