Ramanujan-type congruences for overpartitions modulo $3$

Abstract

Let $\bar{p}\left(n\right)$ denote the number of overpartitions of $n$. We show that $\bar{p}\left(3n\right)\equiv \bar{p}\left(9\cdot 3n\right)\left(mod3\right)$ for $n\ge 0$ and $\bar{p}\left(3n\right)\equiv {\left(-1\right)}^n\bar{p}\left(16\cdot 3n\right)\left(mod3\right)$ for $n\ge 0$ by using a relation of the generating function of $\bar{p}\left(3n\right)$ modulo $3$ and elementary dissection manipulations. Furthermore, by studying a 4-dissection formula of the generating function of $\bar{p}\left(3n\right)$ modulo 3 and iterating the above two congruence relations, we derive that $\bar{p}\left(16^{\alpha }{9}^{\beta }\cdot \left(72n+51\right)\right)\equiv 0\left(mod3\right)$ for $\alpha ,\beta ,n\ge 0$. Moreover, applying the fact that $\varphi {\left(q\right)}^5$ is a Hecke eigenform in $M_{5∕2}\left({\tilde{\mathrm{\Gamma }}}_0\left(4\right)\right)$, we obtain an infinite family of congruences $\bar{p}\left(16^{\alpha }{9}^{\beta }\cdot 3{\ell }^{2}n\right)\equiv 0\left(mod3\right)$, where $\alpha ,\beta \ge 0$ and $\ell$ is a prime such that $\ell \equiv 1\left(mod3\right)$ and $\left(n∕\ell \right)=-1$. In this way, we find various Ramanujan-type congruences for $\bar{p}\left(n\right)$ modulo $3$ such as $\bar{p}\left(1029n+441\right)\equiv 0\left(mod3\right)$, $\bar{p}\left(1029n+735\right)\equiv 0\left(mod3\right)$ and $\bar{p}\left(1029n+882\right)\equiv 0\left(mod3\right)$ for $n\ge 0$.