A generalisation of a second partition theorem of Andrews to overpartitions

TL;DR

This paper extends Andrews' second partition theorem of Rogers-Ramanujan type to overpartitions, employing a novel method involving $q$-difference and recurrence equations, broadening its applicability in combinatorics and algebra.

Contribution

The paper introduces a new generalisation of Andrews' second partition theorem to overpartitions using an innovative analytical technique.

Findings

Generalisation of Andrews' second theorem to overpartitions

Development of a method involving $q$-difference and recurrence equations

Potential applications in combinatorics and algebra

Abstract

In 1968 and 1969, Andrews proved two partition theorems of the Rogers-Ramanujan type which generalise Schur's celebrated partition identity (1926). Andrews' two generalisations of Schur's theorem went on to become two of the most influential results in the theory of partitions, finding applications in combinatorics, representation theory and quantum algebra. In a recent paper, the author generalised the first of these theorems to overpartitions, using a new technique which consists in going back and forth between $q$-difference equations on generating functions and recurrence equations on their coefficients. Here, using a similar method, we generalise the second theorem of Andrews to overpartitions.