Overpartitions with parts separated by parity

TL;DR

This paper extends the concept of partitions separated by parity to overpartitions, deriving generating functions and proving identities related to modular forms, hypergeometric series, and mock modular forms.

Contribution

It introduces two generalizations of Andrews' partitions separated by parity to overpartitions and establishes new $q$-series identities for these families.

Findings

Derived generating functions for 16 overpartition families

Proved identities connecting overpartitions to modular and mock modular forms

Established relations to $q$-hypergeometric series

Abstract

In this paper, we generalize Andrews' partitions separated by parity to overpartitions in two ways. We investigate the generating functions for 16 overpartition families whose parts are separated by parity, and we prove various $q$-series identities for these functions. These identities include relations to modular forms, $q$-hypergeometric series, and mock modular forms.