Extending Recent Congruence Results on $(\ell,μ)$-Regular Overpartitions

TL;DR

This paper studies the arithmetic of $(\ell,\mu)$-regular overpartitions via elementary $q$-series methods, extending several congruences previously obtained through modular forms. By carefully applying dissection formulas and modular reductions to generating functions, the authors derive infinite families of congruences for the cases $(\ell,\mu)=(2,3)$, $(4,3)$, $(4,9)$, including new modulo-$3$, modulo-$4$, modulo-$8$, modulo-$12$, modulo-$16$, modulo-$24$, and modulo-$32$ results. A central achievement is obtaining elementary proofs for these congruences and establishing new and broadened progressions such as $\overline{R_{2,3}}(3^{\beta}(3n+2)) \equiv 0 \pmod{3}$ and $\overline{R_{4,9}}(4(kn+r)) \equiv 0 \pmod{24}$ for suitable $k,r$. The work expands the landscape of congruences for overpartitions and provides accessible, self-contained proofs that complement prior modular-form approaches, with potential applicability to further $(\ell,\mu)$-regular families.

Abstract

Recently, Alanazi, Munagi, and Saikia employed the theory of modular forms to investigate the arithmetic properties of the function $\overline{R_{\ell,μ}}(n)$, which enumerates the overpartitions of $n$ where no part is divisible by either $\ell$ or $μ$, for various integer pairs $(\ell, μ)$. In this paper, we substantially extend several of their results and establish infinitely many families of new congruences. Our proofs are entirely elementary, relying solely on classical $q$-series manipulations and dissection formulas.

Theorems & Definitions (22)

• Theorem 1.1
• Corollary 1.2
• Theorem 1.3
• Theorem 1.4
• Conjecture 1.5
• Theorem 1.6
• Theorem 1.7
• Theorem 1.8
• Lemma 2.1
• proof