## Abstract

Let $\stackrel{\u0304}{p}\left(n\right)$ denote the number of overpartitions of $n$. We show that $\stackrel{\u0304}{p}\left(3n\right)\equiv \stackrel{\u0304}{p}\left(9\cdot 3n\right)\phantom{\rule{0.3em}{0ex}}\left(mod\phantom{\rule{0.3em}{0ex}}3\right)$ for $n\ge 0$ and $\stackrel{\u0304}{p}\left(3n\right)\equiv {\left(-1\right)}^{n}\stackrel{\u0304}{p}\left(16\cdot 3n\right)\phantom{\rule{0.3em}{0ex}}\left(mod\phantom{\rule{0.3em}{0ex}}3\right)$ for $n\ge 0$ by using a relation of the generating function of $\stackrel{\u0304}{p}\left(3n\right)$ modulo $3$ and elementary dissection manipulations. Furthermore, by studying a 4-dissection formula of the generating function of $\stackrel{\u0304}{p}\left(3n\right)$ modulo 3 and iterating the above two congruence relations, we derive that $\stackrel{\u0304}{p}\left(1{6}^{\alpha}{9}^{\beta}\cdot \left(72n+51\right)\right)\equiv 0\phantom{\rule{0.3em}{0ex}}\left(mod\phantom{\rule{0.3em}{0ex}}3\right)$ for $\alpha ,\beta ,n\ge 0$. Moreover, applying the fact that $\varphi {\left(q\right)}^{5}$ is a Hecke eigenform in ${M}_{5\u22152}\left({\tilde{\mathrm{\Gamma}}}_{0}\left(4\right)\right)$, we obtain an infinite family of congruences $\stackrel{\u0304}{p}\left(1{6}^{\alpha}{9}^{\beta}\cdot 3{\ell}^{2}n\right)\equiv 0\phantom{\rule{0.3em}{0ex}}\left(mod\phantom{\rule{0.3em}{0ex}}3\right)$, where $\alpha ,\beta \ge 0$ and $\ell $ is a prime such that $\ell \equiv 1\phantom{\rule{0.3em}{0ex}}\left(mod\phantom{\rule{0.3em}{0ex}}3\right)$ and $\left(n\u2215\ell \right)=-1$. In this way, we find various Ramanujan-type congruences for $\stackrel{\u0304}{p}\left(n\right)$ modulo $3$ such as $\stackrel{\u0304}{p}\left(1029n+441\right)\equiv 0\phantom{\rule{0.3em}{0ex}}\left(mod\phantom{\rule{0.3em}{0ex}}3\right)$, $\stackrel{\u0304}{p}\left(1029n+735\right)\equiv 0\phantom{\rule{0.3em}{0ex}}\left(mod\phantom{\rule{0.3em}{0ex}}3\right)$ and $\stackrel{\u0304}{p}\left(1029n+882\right)\equiv 0\phantom{\rule{0.3em}{0ex}}\left(mod\phantom{\rule{0.3em}{0ex}}3\right)$ for $n\ge 0$.

## Citation

Li Zhang. "Ramanujan-type congruences for overpartitions modulo $3$." Rocky Mountain J. Math. 50 (6) 2257 - 2264, December 2020. https://doi.org/10.1216/rmj.2020.50.2257

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