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

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.