Some Properties of Overpartitions into Nonmultiples of Two Integers

TL;DR

This paper investigates overpartitions that are regular with respect to two coprime integers, establishing a combinatorial identity and exploring their congruence properties using generating functions and modular forms.

Contribution

It introduces a new seven-way combinatorial identity for overpartitions that are simultaneously { ext{ extellipsis}}-regular and { ext{ extellipsis}}-regular, expanding understanding of their structure and properties.

Findings

Proved a seven-way combinatorial identity for these overpartitions.

Established congruence properties using generating functions.

Applied modular forms and Radu's Algorithm to analyze partition congruences.

Abstract

We consider properties of overpartitions that are simultaneously {\ell}-regular and {\mu}-regular, where {\ell} and {\mu} are positive relatively prime integers. We prove a seven-way combinatorial identity related to these overpartitions. We also prove several congruence properties satisfied by this class of partitions (and a further related class) using both generating functions and modular forms with Radu's Algorithm.