Arithmetic properties for generalized cubic partitions and overpartitions modulo a prime

TL;DR

This work investigates arithmetic properties of generalized cubic partitions and their overpartition analogs under prime moduli. It introduces $a_c(n)$ with generating function $F_c(q)=\sum_{n\ge0} a_c(n) q^n=\frac{1}{f_1 f_2^{\,c-1}}$ and derives an infinite family of Ramanujan-type congruences $a_{p-1}(pn+r) \equiv 0 \pmod p$ for odd primes $p$ whenever $8r+1$ is a quadratic nonresidue mod $p$, with an extension to $a_{kp-1}$. It also establishes isolated congruences $a_3(7n+4) \equiv 0 \pmod 7$ and $a_5(11n+10) \equiv 0 \pmod{11}$ via modular forms, and extends the theory to generalized cubic overpartitions, proving analogous congruences for $\overline{a}_{kp-1}(pn+r)$. The paper presents parallel elementary and modular-forms approaches, leveraging functional equations, eta-quotients, Sturm bounds, and Hecke operators, thereby broadening Ramanujan-type congruence phenomena to richer partition families and their overpartition variants.

Abstract

A cubic partition is an integer partition wherein the even parts can appear in two colors. In this paper, we introduce the notion of generalized cubic partitions and prove a number of new congruences akin to the classical Ramanujan-type. We emphasize two methods of proofs, one elementary (relying significantly on functional equations) and the other based on modular forms. We close by proving analogous results for generalized overcubic partitions.

Theorems & Definitions (17)

• Theorem 1.1
• Theorem 1.2
• Corollary 1.3
• Theorem 1.4
• Lemma 2.1
• proof
• proof : Proof of Theorem \ref{['a_p-1_mod_p']}
• proof : Proof of Corollary \ref{['a_kp-1_mod_p']}
• Definition 3.1
• Definition 3.2