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arXiv · 1307.4155

Proof of a Conjecture of Hirschhorn and Sellers on Overpartitions

William Y. C. Chen·Ernest X. W. Xia
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Abstract

Let $\bar{p}(n)$ denote the number of overpartitions of $n$. It was conjectured by Hirschhorn and Sellers that $\bar{p}(40n+35)\equiv 0\ ({\rm mod\} 40)$ for $n\geq 0$. Employing 2-dissection formulas of quotients of theta functions due to Ramanujan, and Hirschhorn and Sellers, we obtain a generating function for $\bar{p}(40n+35)$ modulo 5. Using the $(p, k)$-parametrization of theta functions given by Alaca, Alaca and Williams, we give a proof of the congruence $\bar{p}(40n+35)\equiv 0\ ({\rm mod\} 5)$. Combining this congruence and the congruence $\bar{p}(4n+3)\equiv 0\ ({\rm mod\} 8)$ obtained by Hirschhorn and Sellers, and Fortin, Jacob and Mathieu, we give a proof of the conjecture of Hirschhorn and Sellers.

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William Y. C. Chen, Ernest X. W. Xia. 2013-07-16. Proof of a Conjecture of Hirschhorn and Sellers on Overpartitions. https://arxiv.org/abs/1307.4155

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