Cranks and dissections in Ramanujan's lost notebook

Bruce C. Berndt, Heng Huat Chan, Song Heng Chan, Wen Chin Liaw

Research output: Contribution to journalArticlepeer-review

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.

Original languageEnglish (US)
Pages (from-to)91-120
Number of pages30
JournalJournal of Combinatorial Theory. Series A
Volume109
Issue number1
DOIs
StatePublished - Jan 2005

Keywords

  • Congruences for p(n)
  • Cranks
  • Dissections
  • Partition function p(n)
  • Ramanujan's lost notebook

ASJC Scopus subject areas

  • Theoretical Computer Science
  • Discrete Mathematics and Combinatorics
  • Computational Theory and Mathematics

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