New congruences for overpartitions modulo 3 and 9

Abstract

Let $\overline{p}(n)$ denote the number of overpartitions of $n$. In this paper, we establish an infinite family of congruences $\overline{p}(9^{\alpha }{16}^{\beta }\cdot 3(8\ell n+8i+1))\equiv 0(\mathrm{mod}3)$, where $\alpha ,\beta ,n\ge 0$, $\ell$ is an arbitrary odd prime and is a nonnegative integer such that $\left(\frac{8i+1}{\ell }\right)=-1$. In this way, we find various congruences for $\overline{p}(n)$ modulo $3$ such as $\overline{p}(120n+51,99)\equiv 0(\mathrm{mod}3)$ and $\overline{p}(168n+51,99,123)\equiv 0(\mathrm{mod}3)$. Furthermore, by studying the generating function of $\overline{p}(3n)$ modulo 9 and applying the fact that $\varphi {(q)}^5$ is a Hecke eigenform, we obtain another infinite family of congruences $\overline{p}(3{\ell }^{2}n)\equiv 0(\mathrm{mod}9)$, where $\ell$ is a prime with $\ell \equiv 7(\mathrm{mod}9)$, and $n$ is a nonnegative integer such that $\left(\frac{n}{\ell }\right)=-1$. This leads to lots of congruences for $\overline{p}(n)$ modulo $9$ such as $\overline{p}(1029n+441,735,882)\equiv 0(\mathrm{mod}9)$.