Proof of a conjecture of Hirschhorn and Sellers on overpartitions
Volume 163 / 2014
Acta Arithmetica 163 (2014), 59-69
MSC: Primary 11P83; Secondary 05A17.
DOI: 10.4064/aa163-1-5
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 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 prove the congruence $\bar{p}(40n+35)\equiv 0\ ({\rm mod\ } 5)$ for $n\geq 0$. Combining this congruence and the congruence $\bar{p}(4n+3)\equiv 0\ ({\rm mod\ } 8)$ for $n\geq 0$ obtained by Hirschhorn and Sellers, and Fortin, Jacob and Mathieu, we confirm the conjecture of Hirschhorn and Sellers.
