Infinite families of congruences modulo 5 and 9 for overpartitions
Volume 66 / 2018
Bulletin Polish Acad. Sci. Math. 66 (2018), 31-44
MSC: 11P83, 05A17.
DOI: 10.4064/ba8129-9-2017
Published online: 23 October 2017
Abstract
Let $\bar{p}(n)$ denote the number of overpartitions of $n$. Recently, a number of congruences modulo 5 and powers of 3 for $\bar{p}(n)$ were established by a number of authors. In particular, Treneer proved that the generating function for $\bar{p}(5n)$ modulo 5 is $\sum_{n=0}^\infty \bar{p}(5n)q^n \equiv {(q;q)_\infty^6 }/{(q^2;q^2)_\infty^3} \pmod 5.$ In this paper, employing elementary methods, we establish the generating function of $\bar{p}(5n)$ which yields the congruence due to Treneer. Furthermore, we prove some new congruences modulo 5 and 9 for $\bar{p}(n)$ by utilizing the fact that the generating functions for $\bar{p}(5n)$ modulo 5 and for $\bar{p}(3n)$ modulo 9 are eigenforms for half-integral weight Hecke operators.
