Proof of a conjecture of Hirschhorn and Sellers on overpartitions

• Autorzy: William Y. C. Chen , Ernest X. W. Xia
• Języki publikacji: EN

Abstrakty

EN

Let p̅(n) denote the number of overpartitions of n. It was conjectured by Hirschhorn and Sellers that p̅(40n+35) ≡ 0 (mod 40) for n ≥ 0. Employing 2-dissection formulas of theta functions due to Ramanujan, and Hirschhorn and Sellers, we obtain a generating function for p̅(40n+35) modulo 5. Using the (p, k)-parametrization of theta functions given by Alaca, Alaca and Williams, we prove the congruence p̅(40n+35) ≡ 0 (mod 5) for n ≥ 0. Combining this congruence and the congruence p̅(4n+3) ≡ 0 (mod 8) for n ≥ 0 obtained by Hirschhorn and Sellers, and Fortin, Jacob and Mathieu, we confirm the conjecture of Hirschhorn and Sellers.

Kategorie tematyczne

• 11P83: Partitions; congruences and congruential restrictions
• 05A17: Partitions of integers

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Acta Arithmetica
• Rocznik: 2014
• Tom: 163
• Numer: 1
• Strony: 59-69

Daty

wydano

2014

Twórcy

autor

William Y. C. Chen

• Center for Applied Mathematics, Tianjin University, Tianjin 300072, P.R. China

autor

Ernest X. W. Xia

• Department of Mathematics, Jiangsu University, Zhenjiang, Jiangsu 212013, P.R. China

Identyfikatory

DOI

10.4064/aa163-1-5