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December 2021 New congruences for overpartitions modulo $3$ and $9$
Li Zhang
Rocky Mountain J. Math. 51(6): 2269-2273 (December 2021). DOI: 10.1216/rmj.2021.51.2269

## Abstract

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

## Citation

Li Zhang. "New congruences for overpartitions modulo $3$ and $9$." Rocky Mountain J. Math. 51 (6) 2269 - 2273, December 2021. https://doi.org/10.1216/rmj.2021.51.2269

## Information

Received: 12 January 2021; Revised: 23 March 2021; Accepted: 12 April 2021; Published: December 2021
First available in Project Euclid: 22 March 2022

Digital Object Identifier: 10.1216/rmj.2021.51.2269

Subjects:
Primary: 05A17
Secondary: 11P83

Keywords: congruence , generating function , overpartition  