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Proof of a conjecture of Hirschhorn and Sellers on overpartitions

Volume 163 / 2014

William Y. C. Chen, Ernest X. W. Xia Acta Arithmetica 163 (2014), 59-69 MSC: Primary 11P83; Secondary 05A17. DOI: 10.4064/aa163-1-5

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 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 prove the congruence $\bar{p}(40n+35)\equiv 0\ ({\rm mod\ } 5)$ for $n\geq 0$. Combining this congruence and the congruence $\bar{p}(4n+3)\equiv 0\ ({\rm mod\ } 8)$ for $n\geq 0$ obtained by Hirschhorn and Sellers, and Fortin, Jacob and Mathieu, we confirm the conjecture of Hirschhorn and Sellers.

Authors

  • William Y. C. ChenCenter for Applied Mathematics
    Tianjin University
    Tianjin 300072, P.R. China
    e-mail
  • Ernest X. W. XiaDepartment of Mathematics
    Jiangsu University
    Zhenjiang, Jiangsu 212013, P.R. China
    e-mail

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