Some New Congruences For Overpartition Function With $\ell$-Regular Non-Overlined Parts
Nipen Saikia, Adam Paksok
TL;DR
This work studies overpartitions with ${\ell}$-regular non-overlined parts, denoted ${\overline R_\ell^*}(n)$, extending previous results from ${\ell}=3$ to new infinite families of congruences for ${\ell}=4, 5k, 6, 8$ and establishing connections to classical partition functions. It leverages Ramanujan theta-function techniques and a central generating function ${\sum_{n\ge0} {\overline R_\ell^*(n)} q^n = \dfrac{f_2 f_\ell}{f_1^2}}$, employing modular-type congruence identities such as ${f_p \equiv f_1^p} \pmod{p}$ to derive congruences across arithmetic progressions and prime-power moduli. The results include explicit congruences like ${\overline R_4^*(4n+\xi) \equiv 0 \pmod{4}}$ for ${\xi=2,3}$, alongside broader families for ${\ell=5k,6,8}$ expressed via theta-functions and related $q$-series objects, and they also relate ${\overline R_\ell^*}(n)$ to ${\overline p(n)}$ and ${p(n)}$. Overall, the paper advances the arithmetic theory of restricted overpartitions and links to classical partition theory using $q$-series methods.
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.