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arXiv · 1407.5430

A short Proof of a conjecture by Hirschhorn and Sellers on Overpartitions

Liuquan Wang
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Abstract

Let $\overline{p}(n)$ be the number of overpartitions of $n$, we establish and give a short elementary proof of the following congruence \[\overline{p}({{4}^{α}}(40n+35))\equiv 0 \, (\bmod \, 40),\] where $α,n $ are nonnegative integers. By letting $α=0$ we proved a conjecture of Hirschhorn and Sellers. Some new congruences for $\overline{p}(n)$ modulo 3 and 5 have also been found, including the following two infinite families of Ramanujan-type congruences: for any integers $n\ge 0$ and $α\ge 1$, \[\overline{p}({{5}^{2α+1}}(5n+1))\equiv \overline{p}({{5}^{2α+1}}(5n+4))\equiv 0 \, (\bmod \, 5).\]

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Liuquan Wang. 2014-07-21. A short Proof of a conjecture by Hirschhorn and Sellers on Overpartitions. https://arxiv.org/abs/1407.5430

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