Some new infinite families of congruences modulo 3 for overpartitions into odd parts
Volume 142 / 2016
Colloquium Mathematicum 142 (2016), 255-266
MSC: Primary 11P83; Secondary 05A17.
DOI: 10.4064/cm142-2-6
Abstract
Let $ \bar{p}_o(n)$ denote the number of overpartitions of $n$ in which only odd parts are used. Some congruences modulo 3 and powers of 2 for the function $ \bar{p}_o(n)$ have been derived by Hirschhorn and Sellers, and Lovejoy and Osburn. In this paper, employing 2-dissections of certain quotients of theta functions due to Ramanujan, we prove some new infinite families of Ramanujan-type congruences for $ \bar{p}_o(n)$ modulo 3. For example, we prove that for $n, \alpha\geq 0 $, $$ \bar{p}_o(4^\alpha(24n+17)) \equiv \bar{p}_o(4^\alpha(24n+23)) \equiv 0 \ ({\rm mod}\ 3). $$
