Ranks and Cranks, Part II

George E. Andrews () and Bruce C. Berndt ()
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George E. Andrews: The Pennsylvania State University, Department of Mathematics
Bruce C. Berndt: University of Illinois at Urbana–Champaign, Department of Mathematics

Chapter 3 in Ramanujan's Lost Notebook, 2012, pp 45-70 from Springer

Abstract: Abstract Chapter 3 continues the discussion of ranks and cranks from Chapter 2 . In his lost notebook, sometimes in an oblique fashion, Ramanujan provided the 2-, 3-, 5-, 7-, and 11-dissections for the generating function of cranks. These dissections can be expressed in either of two equivalent formulations – identities or congruences –, although the equivalence is not obvious. In his lost notebook, Ramanujan states one of the dissections as an identity, two as congruences, and two in anomalous ways, because only the quotients of theta functions appearing in the dissections are given. In this chapter, we consider the dissections as congruences and provide two different methods for obtaining proofs. One of them stems from an identity that is recorded cryptically by Ramanujan in his lost notebook and which is proved in Chapter 4 .

Keywords: Power Series; Theta Function; Minimal Polynomial; Primitive Root; Addition Formula (search for similar items in EconPapers)
Date: 2012
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Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-1-4614-3810-6_3

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DOI: 10.1007/978-1-4614-3810-6_3

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