Proof of a Conjecture of Hirschhorn and Sellers on Overpartitions

TL;DR

This paper proves a conjecture by Hirschhorn and Sellers that overpartition counts satisfy a specific divisibility property, using advanced theta function techniques and modular congruences.

Contribution

It introduces a novel proof of the Hirschhorn-Sellers conjecture by combining theta function dissection formulas and modular arithmetic techniques.

Findings

Proved that overpartition counts satisfy $ar{p}(40n+35) ot\equiv 0 mod 40$ for all n.

Established a new generating function for overpartitions modulo 5.

Confirmed the conjecture using combined congruences and theta function identities.

Abstract

Let $\bar{p}(n)$ denote the number of overpartitions of $n$. It was conjectured by Hirschhorn and Sellers that $\bar{p}(40n+35)\equiv 0\ ({\rm mod\} 40)$ for $n\geq 0$. Employing 2-dissection formulas of quotients of theta functions due to Ramanujan, and Hirschhorn and Sellers, we obtain a generating function for $\bar{p}(40n+35)$ modulo 5. Using the $(p, k)$-parametrization of theta functions given by Alaca, Alaca and Williams, we give a proof of the congruence $\bar{p}(40n+35)\equiv 0\ ({\rm mod\} 5)$. Combining this congruence and the congruence $\bar{p}(4n+3)\equiv 0\ ({\rm mod\} 8)$ obtained by Hirschhorn and Sellers, and Fortin, Jacob and Mathieu, we give a proof of the conjecture of Hirschhorn and Sellers.