Further results on arithmetic properties of biregular overpartitions

TL;DR

The paper studies arithmetic properties of $\overline{R_{\ell,\mu}}(n)$, the number of overpartitions avoiding parts divisible by $\ell$ or $\mu$ with $\gcd(\ell,\mu)=1$, for coprime pairs $(2,3),(4,3),(4,9),(8,27),(16,81)$. It develops elementary proofs using Ramanujan theta functions and dissection identities to establish congruences modulo $3$ and powers of $2$, and presents a generic method for proving modulo $8$ without a fixed $2$-dissection. It provides explicit congruences such as $\overline{R_{2,3}}(9n+6)\equiv 0\pmod{6}$ and analogous results for the other pairs, offering alternative proofs to results of Nadji and AMS through purely theta-function techniques. The discussion suggests directions for extending to higher moduli and for proving additional residue-class congruences using only theta-function techniques. The methods provide self-contained proofs and broaden the understanding of biregular overpartition congruences with potential for broader applications.

Abstract

Recently there has been quite a bit of study carried out related to arithmetic properties of overpartitions into non-multiples of two co-prime integers. The paper [19] by Nadji et al. looked into congruences modulo $3$ and powers of $2$ for certain specific pairs of co-prime integers, while the paper [1] by Alanazi et al. investigated some congruences related to some similar and some different pairs of co-prime integers. In this paper we propose some elegant and elementary proofs of a subset of the congruences given in [1] by using only theta function and dissection identities. We also propose a generic method for proving congruences modulo $8$ which doesn't necessarily use any specific $2$-dissection.

Theorems & Definitions (24)

• Lemma 2.1
• Lemma 2.2
• Lemma 2.3
• Lemma 2.4
• Theorem 3.1
• proof : Proof of Equation \ref{['cong-1']}
• Remark 3.2
• Theorem 4.1
• proof
• Theorem 4.2