Some new congruences on biregular overpartitions

TL;DR

This paper establishes new infinite families of congruences for overpartition counts with specific divisibility restrictions, expanding previous results using advanced modular form techniques and identities.

Contribution

It introduces novel infinite families of congruences for overpartition functions involving pairs like (2,9), (5,2), (5,4), and (8,3), utilizing Hecke eigenforms and dissection formulas.

Findings

New congruences modulo 3 and powers of 2 for specific overpartition pairs.

Extension of previous results to broader parameter families such as (5,2^t) and (3,2^t).

Application of Hecke eigenform theory and Newman identities to overpartition congruences.

Abstract

Recently, Nadji, Ahmia and Ram\'{i}rez \cite{Nadji2025} investigate the arithmetic properties of ${\bar B}_{\ell_1,\ell_2}(n)$, the number of overpartitions where no part is divisible by $\ell_1$ or $\ell_2$ with $\gcd(\ell_1,\ell_2)$$=1$ and $\ell_1$,$\ell_2$$>1$. Specifically, they established congruences modulo $3$ and powers of $2$ for the pairs of $(\ell_1,\ell_2)$$\in$$\{(4,3),(4,9),(8,3),(8,9)\}$, using the concept of generating functions, dissection formulas and Smoot's implementation of Radu's Ramanujan-Kolberg algorithm. After that, Alanazi, Munagi and Saikia \cite{Alanazi2024} studied and found some congruences for the pairs of $(\ell_1,\ell_2)\in\{(2,3),(4,3),(2,5),(3,5),(4,9),(8,27),\\(16,81)\} $ using the theory of modular forms and Radu's algorithm. Recently Paudel, Sellers and Wang \cite{Paudel2025} extended several of their results and established infinitely many families of new congruences. In this paper, we find infinitely many families of congruences modulo $3$ and powers of $2$ for the pairs $(\ell_1,\ell_2)$ $ \in$ $\{(2,9),(5,2),(5,4),(8,3)\}$ and in general for $ (5,2^t)$ $\forall t\geq3$ and for $ (3,2^t)$,$ (4,3^t)$ $\forall t\geq2$, using the theory of Hecke eigenform, an identity due to Newman and the concept of dissection formulas and generating functions.