Ramanujan-type Congruences for Overpartitions Modulo 5

Abstract

Let $\overline{p}(n)$ denote the number of overpartitions of $n$. Hirschhorn and Sellers showed that $\overline{p}(4n+3)\equiv 0(\mathrm{mod}8)$ for $n\ge 0$. They also conjectured that $\overline{p}(40n+35)\equiv 0(\mathrm{mod}40)$ for $n\ge 0$. Chen and Xia proved this conjecture by using the $(p,k)$-parametrization of theta functions given by Alaca, Alaca and Williams. In this paper, we show that $\overline{p}(5n)\equiv (-1{)}^n\overline{p}(4\cdot 5n)(\mathrm{mod}5)$ for $n\ge 0$ and $\overline{p}(n)\equiv (-1{)}^n\overline{p}(4n)(\mathrm{mod}8)$ for $n\ge 0$ by using the relation of the generating function of $\overline{p}(5n)$ modulo $5$ found by Treneer and the $2$-adic expansion of the generating function of $\overline{p}(n)$ due to Mahlburg. As a consequence, we deduce that $\overline{p}(4^k(40n+35))\equiv 0(\mathrm{mod}40)$ for $n,k\ge 0$. Furthermore, applying the Hecke operator on $\varphi (q{)}^3$ and the fact that $\varphi (q{)}^3$ is a Hecke eigenform, we obtain an infinite family of congrences $\overline{p}(4^k\cdot 5{\ell }^{2}n)\equiv 0(\mathrm{mod}5)$, where $k\ge 0$ and $\ell$ is a prime such that $\ell \equiv 3(\mathrm{mod}5)$ and $\left(\frac{-n}{\ell }\right)=-1$. Moreover, we show that $\overline{p}(5^{2}n)\equiv \overline{p}(5^{4}n)(\mathrm{mod}5)$ for $n\ge 0$. So we are led to the congruences $\overline{p}(4^k{5}^{2i+3}(5n\pm 1))\equiv 0(\mathrm{mod}5)$ for $n,k,i\ge 0$. In this way, we obtain various Ramanujan-type congruences for $\overline{p}(n)$ modulo $5$ such as $\overline{p}(45(3n+1))\equiv 0(\mathrm{mod}5)$ and $\overline{p}(125(5n\pm 1))\equiv 0(\mathrm{mod}5)$ for $n\ge 0$.

11 pages