Partitions with Non-Repeating Odd Parts and Q-Hypergeometric Identities

Abstract

We obtain a series expansion for the product generating function of partitions in which the odd parts do not repeat. This is done by studying the 2-modular Ferrers graphs of such partitions via Durfee squares. This provides a unified approach to several fundamental identities in the theory of partitions and q-series such as those of Sylvester, Lebesgue, Gauss, and Rogers-Fine, and provides links with Gollnitz’s deep theorem.