Some New Congruences For Overpartition Function With $\ell$-Regular Non-Overlined Parts

Abstract

Alanzi et al. (2022) investigated overpartition of a positive integer $n$ with $\ell$-regular non-overlined parts denoted by ${\overline{R}}_{\ell }^{\ast }(n)$, and proved some results for the case $\ell =3$. As extension to the results of Alanzi et al., Sellers (2024) proved some new congruences for ${\overline{R}}_3^{\ast }(n)$. In this paper, we prove some new infinite families and particular congruences for ${\overline{R}}_{\ell }^{\ast }(n)$ for $\ell =4,5k,6$, and 8, where $k$ is any positive integer. We also offer some congruences connecting ${\overline{R}}_{\ell }^{\ast }(n)$ with some other partition functions.

14 pages