Proof of a Conjecture of Hirschhorn and Sellers on Overpartitions
William Y.C. Chen and Ernest X.W. Xia
Abstract: Let (n) denote the number of overpartitions of n. It was conjectured by Hirschhorn and Sellers that (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 (40n + 35) modulo 5. Using the (p, k)-parametrization of theta functions given by Alaca, Alaca and Williams, we prove the congruence (40n + 35) ≡ 0 (mod 5). Combining this congruence and the congruence (4n + 3) ≡ 0 (mod 8) for n ≥ 0obtained by Hirschhorn and Sellers, and Fortin, Jacob and Mathieu, we confirm the conjecture of Hirschhorn and Sellers.
AMS Classification: 11P83, 05A17
Keywords: overpartition, congruence, theta function, dissection formula, (p, k)-parametrization