Ramanujan-type Congruences for Overpartitions Modulo 5
William Y.C. Chen, Lisa H. Sun, Rong-Hua Wang and Li Zhang
Abstract: Let (n) denote the number of overpartitions of n. Hirschhorn and Sellers showed that (4n + 3) ≡ 0 (mod 8) for n ≥ 0. They also conjectured that (40n + 35) ≡ 0 (mod 40) for n ≥ 0. Chen and Xia proved this conjecture by using the (p, k)-parametrization of theta functions given by Alaca, Alaca and Williams. In this paper, we show that (5n) ≡ (-1)^{n}(4 ⋅ 5n) (mod 5) for n ≥ 0 and (n) ≡ (-1)n^{p}(4n) (mod 8) for n ≥ 0 by using the relation of the generating function of (5n) modulo 5 found by Treneer and the 2-adic expansion of the generating function of (n) due to Mahlburg. As a consequence, we deduce that (4^{k}(40n + 35)) ≡ 0 (mod 40) for n, k ≥ 0. Furthermore, applying the Hecke operator on φ(q)^{3} and the fact that φ(q)^{3} is a Hecke eigenform, we obtain an infinite family of congrences (4^{k} ⋅ 5l^{2}n) ≡ 0 (mod 5), where k ≥ 0 and l is a prime such that l ≡ 3 (mod 5) and (-n/l) = -1. Moreover, we show that (5^{2}n) ≡ (5^{4}n) (mod 5) for n ≥ 0. So we are led to the congruences (4^{k}5^{2i+3}(5n ± 1)) ≡ 0 (mod 5) for n, k, i ≥ 0. In this way, we obtain various Ramanujan-type congruences for (n) modulo 5 such as (45(3n + 1)) ≡ 0 (mod 5) and (125(5n ± 1)) ≡ 0 (mod 5) for n ≥ 0. AMS Classification: 05A17, 11P83 Keywords: overpartition, Ramanujan-type congruence, modular form, Hecke operator, Hecke eigenform Download: pdf |