Ramanujan-type congruences for overpartitions modulo 5

Let (p) over bar (n) denote the number of overpartitions of n. In this paper, we show that (p) over bar (5n) (-1)(n)(p) over bar (4 . 5n) (mod 5) for n >= 0 and (p) over bar (n) (-1)(n)(p) over bar (4n) (mod 8) for n >= 0 by using the relation of the generating function of (p) over bar (5n) modulo 5 found by Treneer and the 2-adic expansion of the generating function of (p) over bar (n) due to Mahlburg. As a consequence, we deduce that (p) over bar (4(k)(40n + 35)) 0 (mod 40) for n, k >= 0. When k = 0, it was conjectured by Hirschhorn and Sellers, and confirmed by Chen and Xia. Furthermore, applying the Hecke operator on phi(q)(3) and the fact that phi(q)(3) is a Hecke eigenform, we obtain an infinite family of congruences (p) over bar (4(k) . 5l(2)n) 0 (mod 5), where k >= 0 and.e is a prime such that l 3 (mod 5) and (-n/l) = -1. Moreover, we show that (p) over bar (5(2)n) (p) over bar (5(4)n) (mod 5) for n >= 0. So we are led to the congruences (p) over bar (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 (p) over bar (n) modulo 5 such as (p) over bar (45(3n + 1)) 0 (mod 5) and (p) over bar (125(5n +/- 1)) 0 (mod 5) for n >= 0. (C) 2014 Elsevier Inc. All rights reserved.