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Infinite families of congruences modulo 5 and 9 for overpartitions

Gu C., Hirschhorn M. D., Sellers J. A., Xia E. X. W.

Bulletin of the Polish Academy of Sciences. Mathematics

2018

Vol. 66, no. 1

31--44

Let p͞(n) denote the number of overpartitions of n. Recently, a number of congruences modulo 5 and powers of 3 for p͞(n) were established by a number of authors. In particular, Treneer proved that the generating function for p͞(5n) modulo 5 is ∑∞n=0 p͞(5n)qn ≡ (q;q)6∞/(q2; q2)3∞ (mod 5). In this paper, employing elementary methods, we establish the generating function of p͞(5n) which yields the congruence due to Treneer. Furthermore, we prove some new congruences modulo 5 and 9 for p͞(n) by utilizing the fact that the generating functions for p͞(5n) modulo 5 and for p͞(3n) modulo 9 are eigenforms for half-integral weight Hecke operators.