Extending Recent Congruence Results on $(\ell,\mu)$-Regular Overpartitions

TL;DR

This paper extends recent results on the arithmetic properties of overpartition functions avoiding multiples of certain integers, establishing new congruences through elementary methods.

Contribution

It significantly broadens the scope of known congruences for overpartition functions using elementary $q$-series techniques.

Findings

Established infinitely many new congruences for overpartition functions.

Extended previous congruence results to broader classes of $(\, ext{l}, ext{μ})$-regular overpartitions.

Provided elementary proofs relying on classical $q$-series manipulations.

Abstract

Recently, Alanazi, Munagi, and Saikia employed the theory of modular forms to investigate the arithmetic properties of the function $\overline{R_{\ell,\mu}}(n)$, which enumerates the overpartitions of $n$ where no part is divisible by either $\ell$ or $\mu$, for various integer pairs $(\ell, \mu)$. In this paper, we substantially extend several of their results and establish infinitely many families of new congruences. Our proofs are entirely elementary, relying solely on classical $q$-series manipulations and dissection formulas.