Proof of a Conjecture of Hirschhorn and Sellers on Overpartitions

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.