Arithmetic Properties of Generalized Cubic and Overcubic Partitions

TL;DR

This work studies arithmetic properties of generalized cubic partitions with $c$ colors, $a_c(n)$, and generalized overcubic partitions, $\bar{a}_c(n)$, presenting generating functions $\frac{1}{f_1 f_2^{c-1}}$ and $\frac{f_4^{c-1}}{f_1^2 f_2^{2c-3}}$ respectively. It combines elementary $q$-series with modular forms, employing eta-quotients, Radu's algorithm, Sturm bounds, and Hecke operators to derive isolated congruences for $a_c(n)$ and rich modular-behavior descriptions for $\bar{a}_c(n)$ modulo $4$, $8$, and $12$, as well as infinite families modulo powers of $2$ and $3$, plus lacunarity and density results. The main contributions include explicit isolated congruences for $a_c(n)$, a complete modulo-$4$ and modulo-$8$ characterization of $\bar{a}_c(n)$, and extensive infinite families of congruences, supported by both elementary and modular-form techniques. These results extend Ramanujan-type phenomena to generalized partitions and illuminate lacunarity and density properties through the interplay of $q$-series identities and modularity.

Abstract

We prove several congruences satisfied by the generalized cubic and generalized overcubic partition functions, recently introduced by Amdeberhan, Sellers, and Singh. We also prove infinite families of congruences modulo powers of $2$ and modulo $12$ satisfied by the generalized overcubic partitions, as well as some density results that they satisfy. We use both elementary $q$-series techniques as well as the theory of modular forms to prove our results.

Theorems & Definitions (44)

• Theorem 1.1
• Remark 1.2
• Theorem 1.3
• Theorem 1.4
• Remark 1.5
• Theorem 1.6
• Remark 1.7
• Theorem 1.8
• Theorem 1.9
• Theorem 1.10