Some new infinite families of congruences modulo 3 for overpartitions into odd parts

Ernest X. W. Xia¹

¹ Department of Mathematics Jiangsu University Zhenjiang, Jiangsu 212013, P.R. China

Colloquium Mathematicum, Tome 142 (2016) no. 2, pp. 255-266.

Résumé

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 $, $$ \bar{p}_o(4^\alpha(24n+17)) \equiv \bar{p}_o(4^\alpha(24n+23)) \equiv 0 \ ({\rm mod}\ 3). $$

Détail

DOI : 10.4064/cm142-2-6

Keywords: bar denote number overpartitions which only odd parts congruences modulo powers function bar have derived hirschhorn sellers lovejoy osburn paper employing dissections certain quotients theta functions due ramanujan prove infinite families ramanujan type congruences bar modulo example prove alpha geq bar alpha equiv bar alpha equiv mod