Some new congruences on biregular overpartitions

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.