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Infinite families of congruences modulo 5 and 9 for overpartitions

Let p̄(n) denote the number of overpartitions of n. Recently, a number of congruences modulo 5 and powers of 3 for p̄(n) were established by a number of authors. In particular, Treneer proved that the generating function for p̄(5n) modulo 5 is ∑∞ n=0 p̄(5n)q n ≡ (q; q)∞/(q; q)∞ (mod 5). In this paper, employing elementary methods, we establish the generating function of p̄(5n) which yields the congruence due to Treneer. Furthermore, we prove some new congruences modulo 5 and 9 for p̄(n) by utilizing the fact that the generating functions for p̄(5n) modulo 5 and for p̄(3n) modulo 9 are eigenforms for half-integral weight Hecke operators.