Linked partition ideals and a family of quadruple summations

TL;DR

This paper extends classical partition relations using a new quadruple summation identity linked to overpartitions, involving advanced hypergeometric series techniques.

Contribution

It introduces a uniform extension of Andrews' relations for specific partitions via a novel quadruple summation identity and hypergeometric series methods.

Findings

Established a new Rogers--Ramanujan type identity.

Connected overpartition generating functions with quadruple summations.

Extended classical partition relations in a uniform framework.

Abstract

Recently, $4$-regular partitions into distinct parts are connected with a family of overpartitions. In this paper, we provide a uniform extension of two relations due to Andrews for the two types of partitions. Such an extension is made possible with recourse to a new trivariate Rogers--Ramanujan type identity, which concerns a family of quadruple summations appearing as generating functions for the aforementioned overpartitions. More interestingly, the derivation of this Rogers--Ramanujan type identity is relevant to a certain well-poised basic hypergeometric series.