Cranks and dissections in Ramanujan's lost notebook

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• 作者：Berndt ; Bruce C. ; Huat Chan ; Heng ; Heng Chan ; Song ; Liaw ; Wen-Chin
• 关键词：primary 11P82 ; secondary 11P83 ; Cranks ; Dissections ; Ramanujan's lost notebook ; Partition function Formula Not Shown ; Congruences for Formula Not Shown
• 刊名：Journal of Combinatorial Theory, Series A
• 出版年：2005
• 出版时间：January, 2005
• 年：2005
• 卷：109
• 期：1
• 页码：91-120
• 全文大小：347 K

文摘

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 Formula Not Shown of cranks. All are found, sometimes in abbreviated form, in the lost notebook, and imply dissections of Formula Not Shown . Consequences of our work are interesting new q-series identities and congruences in the spirit of Atkin and Swinnerton-Dyer.