An overpartition analogue of the Andrews-G\"ollnitz-Gordon theorem

TL;DR

This paper extends the Andrews-G"ollnitz-Gordon theorem to overpartitions, providing new combinatorial identities and interpretations using Bailey's lemma and G"ollnitz-Gordon marking.

Contribution

It introduces a general overpartition analogue of the Andrews-G"ollnitz-Gordon theorem and related identities, broadening the combinatorial framework.

Findings

Established an overpartition analogue of Bressoud's identity.

Provided a combinatorial interpretation via G"ollnitz-Gordon marking.

Extended Lovejoy's overpartition theorem to the general case.

Abstract

In 1967, Andrews found a combinatorial generalization of the G\"ollnitz-Gordon theorem, which can be called the Andrews-G\"ollnitz-Gordon theorem. In 1980, Bressoud derived a multisum Rogers-Ramanujan-type identity, which can be considered as the generating function counterpart of the Andrews-G\"ollnitz-Gordon theorem. Lovejoy gave an overpartition analogue of the Andrews-G\"ollnitz-Gordon theorem for $i=k$. In this paper, we give an overpartition analogue of this theorem in the general case. By using Bailey's lemma and a change of base formula due to Bressoud, Ismail and Stanton, we obtain an overpartition analogue of Bressoud's identity. We then give a combinatorial interpretation of this identity by introducing the G\"ollnitz-Gordon marking of an overpartition, which yields an overpartition analogue of the Andrews-G\"ollnitz-Gordon theorem.