New Congruences on Biregular Overpartitions

Abstract

Recently, Nadji, Ahmia and Ramírez \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 $({\ell }_1,{\ell }_2)$ $\in$ $\{(5,2^t),(4,3^t)\}$ $\forall t\ge 1$ with $t\in \mathbb{N}$ and for $(3,2^t)$ $\forall t\ge 2$ with $t\in \mathbb{N}$, using the theory of Hecke eigenforms, an identity due to Newman \cite{Newman1959}, the concept of dissection formulas.

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