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Ranks and Cranks, Part II

In: Ramanujan's Lost Notebook

Author

Listed:
  • George E. Andrews

    (The Pennsylvania State University, Department of Mathematics)

  • Bruce C. Berndt

    (University of Illinois at Urbana–Champaign, Department of Mathematics)

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 .

Suggested Citation

  • George E. Andrews & Bruce C. Berndt, 2012. "Ranks and Cranks, Part II," Springer Books, in: Ramanujan's Lost Notebook, edition 127, chapter 3, pages 45-70, Springer.
    • Handle: RePEc:spr:sprchp:978-1-4614-3810-6_3
    DOI: 10.1007/978-1-4614-3810-6_3
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