Some New Congruences Modulo Powers of 2 For $(j,k)$-Regular Overpartition

TL;DR

This paper extends known results by establishing new infinite families of congruences modulo powers of 2 for certain overpartition functions, specifically for _{4,8}(n), _{6,12}(n), and _{8,16}(n), generalizing previous work.

Contribution

It introduces new infinite families of congruences modulo powers of 2 for _{4,8}(n), _{6,12}(n), and _{8,16}(n), expanding the understanding of overpartition congruences.

Findings

Proved that _{4,8}(5^{2\u03b1+1}(16(5n+j)+14)) \, ext{is divisible by 64 for all } n, \u03b1 0, j=1,2,3,4.

Extended the family of known congruences for overpartition functions.

Demonstrated that these congruences hold for all non-negative integers n and  .

Abstract

Let $\overline{p}_{j,k}(n)$ denotes the number of $(j,k)$-regular overpartitions of a positive integer $n$ such that none of the parts is congruent to $j$ modulo $k$. Naika et. al. (2021) proved infinite families of congruences modulo powers of 2 for $\overline{p}_{3,6}(n)$, $\overline{p}_{5,10}(n)$ and $\overline{p}_{9,18}(n)$. In this paper, we obtain infinite families of congruences modulo power of 2 for $\overline{p}_{4,8}(n)$, $\overline{p}_{6,12}(n)$ and $\overline{p}_{8,16}(n)$. For example, we prove that, for all integers $n\geq 0$ and $\alpha\geq 0$, $$ \overline{p}_{4,8}\left( 5^{2\alpha+1}\left( 16(5n+j)+14\right) \right) q^n\equiv 0\pmod{64}; \qquad j=1,2,3,4.$$