Some new congruences for generalized overcubic partition function
Adam Paksok, Nipen Saikia
TL;DR
The paper extends the arithmetic of the generalized overcubic partition function $\overline a_c(n)$ by proving a universal congruence modulo powers of two: $\overline a_{2^\lambda m+t}(n)\equiv \overline a_t(n)\pmod{2^{\lambda+1}}$ for $\lambda\ge1$, $m\ge0$, $t\ge1$. The authors employ $q$-series techniques, Ramanujan theta-functions, and generating-function decompositions to establish the result and then extract new congruences modulo $8$ and $16$ for specific index families, such as $\overline a_{2m+1}(n)$, $\overline a_{2m+2}(n)$, and $\overline a_{8m+3}(n)$, including explicit arithmetic-progressions. These contributions broaden Ramanujan-type congruences for generalized overcubic partitions and provide tools for further congruence restrictions in partition theory.
Abstract
Amdeberhan et al. (2024) introduced the notion of a generalized overcubic partition function $\overline a_c (n)$ and proved an infinite family of congruences modulo a prime $p\ge 3$ and some Ramanujan type congruences. In this paper, we show that $\overline a_{2^λm+t}(n) \equiv \overline a_t (n) \pmod {2^{λ+1}}$, where $λ\geq1, m\geq0,$ and $t\geq1$ are integers. We also prove some new congruences modulo $8$ and $16$ for $\overline a_{2m+1}(n), \overline a_{2m+2}(n), \overline a_{8m+3}(n)$, where $m$ is any non-negative integer.