Exact generating functions for the number of partitions into distinct parts

Abstract

Let [Formula: see text] denote the number of partitions of a non-negative integer into distinct (or, odd) parts. We find exact generating functions for [Formula: see text], [Formula: see text] and [Formula: see text]. We deduce some congruences modulo 5 and 25. We employ Ramanujan’s theta function identities and some identities for the Rogers–Ramanujan continued fraction.