Proof of a conjecture of Hirschhorn and Sellers on overpartitions

William Y. C. Chen ¹   ; Ernest X. W. Xia ²

¹ Center for Applied Mathematics Tianjin University Tianjin 300072, P.R. China
² Department of Mathematics Jiangsu University Zhenjiang, Jiangsu 212013, P.R. China

Acta Arithmetica, Tome 163 (2014) no. 1, pp. 59-69

Résumé

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.

Détail

DOI : 10.4064/aa163-1-5

Keywords: bar denote number overpartitions conjectured hirschhorn sellers bar equiv mod geq employing dissection formulas theta functions due ramanujan hirschhorn sellers obtain generating function bar modulo nbsp using parametrization theta functions given alaca alaca williams prove congruence bar equiv mod geq combining congruence congruence bar equiv mod geq obtained hirschhorn sellers fortin jacob mathieu confirm conjecture hirschhorn sellers