Cranks and dissections in Ramanujanʹs lost notebook

In his lost notebook, Ramanujan offers several results related to the crank, the existence of which was first conjectured by F. J. Dyson and later established by G.E. Andrews and F.G. Garvan. Using an obscure identity found on p. 59 of the lost notebook, we provide uniform proofs of several congruences in the ring of formal power series for the generating function F ( q ) of cranks. All are found, sometimes in abbreviated form, in the lost notebook, and imply dissections of F ( q ) . Consequences of our work are interesting new q-series identities and congruences in the spirit of Atkin and Swinnerton-Dyer.