The Ramanujan–Dyson identities and George Beck’s congruence conjectures

Abstract

Dyson’s famous conjectures (proved by Atkin and Swinnerton-Dyer) gave a combinatorial interpretation of Ramanujan’s congruences for the partition function. The proofs of these results center on one of the universal mock theta functions that generate partitions according to Dyson’s rank. George Beck has generalized the study of partition function congruences related to rank by considering the total number of parts in the partitions of [Formula: see text]. The related generating functions are no longer part of the world of mock theta functions. However, George Beck has conjectured that certain linear combinations of the related enumeration functions do satisfy congruences modulo 5 and 7. The conjectures are proved here.