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Ramanujan’s Forty Identities for the Rogers–Ramanujan Functions

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

The Rogers-Ramanujan identities are perhaps the most important identities in the theory of partitions. They were first proved by L.J. Rogers in 1894 and rediscovered by Ramanujan prior to his departure for England. Since that time, they have inspired a huge amount of research, including many analogues and generalizations. Published with the lost notebook is a manuscript providing 40 identities satisfied by these functions. In contrast to the Rogers-Ramanujan identities, the identities in this manuscript are identities between the two Rogers-Ramanujan functions at different powers of the argument. In other words, they are modular equations satisfied by the functions. The theory of modular forms can be invoked to provide proofs, but such proofs provide us with little insight, in particular, with no insight on how Ramanujan might have discovered them. Thus, for nearly a century, mathematicians have attempted to find “elementary” proofs of the identities. In this chapter, “elementary” proofs are given for each identity, with the proofs of the most difficult identities found only recently by Hamza Yesilyurt.

Suggested Citation

  • George E. Andrews & Bruce C. Berndt, 2012. "Ramanujan’s Forty Identities for the Rogers–Ramanujan Functions," Springer Books, in: Ramanujan's Lost Notebook, edition 127, chapter 8, pages 217-335, Springer.
    • Handle: RePEc:spr:sprchp:978-1-4614-3810-6_8
    DOI: 10.1007/978-1-4614-3810-6_8
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