Cranks and dissections in Ramanujan's lost notebook

Abstract

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.

On the divergence of the Rogers-Ramanujan continued fraction on the unit circle

“…One area of research has been to try to evaluate R(q) for various q inside the unit circle, and indeed many explicit evaluations of R(e −π √ n ) and R(−e −π √ n ) have been given for n ∈ Q + , some of which were asserted by Ramanujan without proof (see, for example, [2], [3], [12] and [16]). …”

Cranks—really, the final problem

“…94-96] for proofs and a history of earlier proofs. Further references to other proofs of Ramanujan's five dissections for F a (q) and to proofs of certain other dissections of F a (q) can be found in [4]. The largest value of m for which a dissection of F a (q) has been given is m = 11.…”

“…Let N (m, n) denote the number of partitions of n with rank m, and let N (m, t, n) denote the number of partitions of n with rank congruent to m modulo t. Then Dyson conjectured that 4) and…”

“…The present authors [4] devised uniform proofs of Ramanujan's dissections in the language of congruences in two distinct ways. One of the methods arises from obscure statements found on page 59 of the lost notebook.…”

“…that leads to one set of the aforementioned uniform proofs of Ramanujan's dissections [4]. See [5, pp.…”