An Overpartition Companion of Andrews and Keith's 2-colored $q$-series Identity

TL;DR

This paper generalizes Andrews and Keith's 2-colored $q$-series identity for overpartitions, extending their Schmidt type partition theorem and revealing broader underlying structures in partition theory.

Contribution

It introduces a new generalization of the Schmidt type theorem, providing a companion identity for overpartitions and suggesting a more comprehensive underlying framework.

Findings

Derived a generalized Schmidt type partition theorem

Established a companion identity for overpartitions

Indicated these identities are special cases of a broader result

Abstract

Andrews and Keith recently produced a general Schmidt type partition theorem using a novel interpretation of Stockhofe's bijection, which they used to find new $q$-series identities. This includes an identity for a trivariate 2-colored partition generating function. In this paper, their Schmidt type theorem is further generalized akin to how Franklin classically extended Glaisher's theorem. As a consequence, we obtain a companion to Andrews and Keith's 2-colored identity for overpartitions. These identities appear to be special cases of a much more general result.