Arithmetic properties of $2^\alpha-$Regular overpartition pairs

Abstract

Recently, several mathematicians have investigated various partition functions with the goal of discovering Ramanujan-type congruences. One such function is $\overline{B}_{2^\alpha}(n)$, which represents the number of $2^\alpha-$regular overpartition pairs of $n$. In this context, we establish Ramanujan-type congruences modulo powers of $2$ for this function. For instance, we prove that \begin{equation*} \overline{B}_{2^{\alpha}}(2^{\alpha+\beta+1}(n+1)) \equiv 0\pmod{2^{3\beta+5}} \end{equation*} for all $n, \beta\geq 0,\, \alpha \in \mathbb{N}$.