New Congruences on Biregular Overpartitions

Abstract

Recently, Nadji, Ahmia and Ram\'{i}rez \cite{Nadji2025} investigated 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 $(\ell_1, \ell_2)$ $\in$ $\{(4,3),(4,9),(8,3),(8,9)\}$, using the concept of generating functions, dissection formulas and Smoot's implementation of Radu's Ramanujan-Kolberg algorithm. Further, Alanazi, Munagi and Saikia \cite{Alanazi2024} established some congruences for the pairs $(\ell_1,\ell_2)$ $\in$ $\{(2,3),(4,3),(2,5),(3,5),(4,9),(8,27),(16,81)\}$ 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 $(\ell_1,\ell_2)$ $\in$ $ \{(5,2^t), (4,3^t)\}$ $\forall t\geq1$ with $t\in\mathbb{N}$ and for $(3,2^t)$ $\forall t\geq2 $ with $t\in\mathbb{N}$, using the theory of Hecke eigenforms, an identity due to Newman \cite{Newman1959}, the concept of dissection formulas.