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

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 $n\ge 0$ and $\alpha \ge 0$, $${\overline{p}}_{4,8}\left(5^{2\alpha +1}\left(16(5n+j)+14\right)\right)q^n\equiv 0(\mathrm{mod}64);j=1,2,3,4.$$