Ramanujan-type congruences for overpartitions modulo 5

Let p¯(n) denote the number of overpartitions of n. In this paper, we show that p¯(5n)≡(−1)np¯(4⋅5n)(mod5) for n≥0 and p¯(n)≡(−1)np¯(4n)(mod8) 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(mod40) 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 ϕ(q)3 and the fact that ϕ(q)3 is a Hecke eigenform, we obtain an infinite family of congruences p¯(4k⋅5ℓ2n)≡0(mod5), where k≥0 and ℓ is a prime such that ℓ≡3(mod5) and (−nℓ)=−1. Moreover, we show that p¯(52n)≡p¯(54n)(mod5) for n≥0. So we are led to the congruences p¯(4k52i+3(5n±1))≡0(mod5) 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(mod5) and p¯(125(5n±1))≡0(mod5) for n≥0.