A short Proof of a conjecture by Hirschhorn and Sellers on Overpartitions

Abstract

Let $\overline{p}(n)$ be the number of overpartitions of $n$, we establish and give a short elementary proof of the following congruence \[\overline{p}({{4}^{\alpha }}(40n+35))\equiv 0 \, (\bmod \, 40),\] where $\alpha ,n $ are nonnegative integers. By letting $\alpha =0$ we proved a conjecture of Hirschhorn and Sellers. Some new congruences for $\overline{p}(n)$ modulo 3 and 5 have also been found, including the following two infinite families of Ramanujan-type congruences: for any integers $n\ge 0$ and $\alpha \ge 1$, \[\overline{p}({{5}^{2\alpha +1}}(5n+1))\equiv \overline{p}({{5}^{2\alpha +1}}(5n+4))\equiv 0 \, (\bmod \, 5).\]