Arithmetic properties of $2^{\hat{I}}\pm -$Regular overpartition pairs

Abstract

Recently, several mathematicians have investigated various partition functions with the goal of discovering Ramanujan-type congruences. One such function is ${\overline{B}}_{2^{\hat{I}}\pm }(n)$, which represents the number of $2^{\hat{I}}\pm -$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^α}(2^{α+β+1}(n+1)) \equiv 0\pmod{2^{3β+5}} \end{equation*} for all $n,{\hat{I}}^2\ge 0,\hat{I}\pm \in \mathbb{N}$.

14 pages