In 1944 Dyson defined the rank of a partition as the largest part minus the number of parts, and conjectured that the residue of the rank mod $5$ divides the partitions of $5n+4$ into five equal classes. This gave a combinatorial explanation of Ramanujan's famous partition congruence mod $5$. He made an analogous conjecture for the rank mod $7$ and the partitions of $7n+5$. In 1954 Atkin and Swinnerton-Dyer proved Dyson's rank conjectures by constructing several Lambert-series identities basically using the theory of elliptic functions. In 2016 the author gave another proof using the theory of weak harmonic Maass forms. In this paper we describe a new and more elementary approach using Hecke-Rogers series.

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Created by F.G. Garvan (fgarvan@ufl.edu) on Tuesday, December 15, 2020.
Last update made Tue May 16 09:54:53 CDT 2023.

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