A note on congruences for generalized cubic partitions modulo primes

TL;DR

This paper revisits congruences for a generalized cubic partition function, providing alternative proofs and establishing new infinite families of congruences for primes not congruent to 1 mod 8.

Contribution

It offers a classical q-series proof of known congruences and extends the results to infinite families for certain primes.

Findings

Alternative proof of existing congruences using q-series.

Infinite families of congruences for primes p ≠ 1 mod 8.

Extensions of congruence results beyond initial cases.

Abstract

Recently, Amdeberhan, Sellers, and Singh introduced the notion of a generalized cubic partition function $a_c(n)$ and proved two isolated congruences via modular forms, namely, $a_3(7n+4)\equiv 0\pmod{7}$ and $a_5(11n+10)\equiv 0\pmod{11}$. In this paper, we provide another proof of these congruences by using classical $q$-series manipulations. We also give infinite families of congruences for $a_c(n)$ for primes $p\not\equiv 1\pmod{8}$.