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

TL;DR

This paper investigates the arithmetic properties of a specific partition function related to overpartition pairs, establishing Ramanujan-type congruences modulo powers of 2, thus contributing to the understanding of partition functions' divisibility properties.

Contribution

The paper proves new Ramanujan-type congruences modulo powers of 2 for the function counting 2^α-regular overpartition pairs, extending previous results in partition theory.

Findings

Established congruences modulo 2^{3β+5} for the overpartition pair function

Demonstrated divisibility properties for the function at specific arguments

Extended the theory of partition functions with Ramanujan-type congruences

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 $n, \beta\geq 0,\, \alpha \in \mathbb{N}$.