2014/06/15 by Chen, William Y. C., Sun, Lisa H., Wang, Rong-Hua +1
#05A17 #11P83 #Combinatorics (math.CO) #FOS: Mathematics #Number Theory (math.NT)
paper · doi:10.48550/arxiv.1406.3801
Let p(n) denote the number of overpartitions of n. Hirschhorn and Sellers showed that p(4n+3)≡ 0 \pmod8 for n≥ 0. They also conjectured that p(40n+35)≡ 0 \pmod40 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 p(5n)≡ (-1)np(4⋅ 5n) \pmod5 for n ≥ 0 and p(n)≡ (-1)np(4n)\pmod8 for n ≥ 0 by using the relation of the generating function of p(5n) modulo 5 found by Treneer and the 2-adic expansion of the generating function of p(n) due to Mahlburg. As a consequence, we deduce that p(4k(40n+35))≡ 0 \pmod40 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 p(4k ⋅5ℓ2n)≡ 0 \pmod5, where k≥ 0 and ℓ is a prime such that ℓ≡3 \pmod5 and ((-n)/(ℓ))=-1. Moreover, we show that p(52n)≡ p(54n) \pmod5 for n ≥ 0. So we are led to the congruences p(4k52i+3(5n±1))≡ 0 \pmod5 for n, k, i≥ 0. In this way, we obtain various Ramanujan-type congruences for p(n) modulo 5 such as p(45(3n+1))≡ 0 \pmod5 and p(125(5n± 1))≡ 0 \pmod5 for n≥ 0.