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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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On a conjecture of Koike on identities between Thompson series and Rogers-Ramanujan functions
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by Kathrin Bringmann and Holly Swisher PDF
Proc. Amer. Math. Soc. 135 (2007), 2317-2326 Request permission

Erratum: Proc. Amer. Math. Soc. 136 (2008), 1501-1501.

Abstract:

One of the many amazing things Ramanujan did in his lifetime was to list $40$ identities involving what are now called the Rogers-Ramanujan functions $G(q)$ and $H(q)$ on one side, and products of functions of the form $Q_m = \prod _{n=1}^\infty (1-q^{mn})$ on the other side. The identities are rather complicated and seem too difficult to guess. Recently however, Koike devised a strategy for finding (but not proving) these types of identities by connecting them to Thompson series. He was able to conjecture many new Rogers-Ramanujan type identities between $G(q)$ and $H(q)$, and Thompson series. Here we prove these identities.
References
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Additional Information
  • Kathrin Bringmann
  • Affiliation: Department of Mathematics, University of Wisconsin, Madison, Wisconsin 53706
  • MR Author ID: 774752
  • Email: bringman@math.wisc.edu
  • Holly Swisher
  • Affiliation: Department of Mathematics, The Ohio State University, Columbus, Ohio 43210
  • MR Author ID: 678225
  • Email: swisher@math.ohio-state.edu
  • Received by editor(s): January 31, 2006
  • Received by editor(s) in revised form: March 27, 2006
  • Published electronically: March 21, 2007
  • Communicated by: Ken Ono
  • © Copyright 2007 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 135 (2007), 2317-2326
  • MSC (2000): Primary 11F22, 33D15, 11F03
  • DOI: https://doi.org/10.1090/S0002-9939-07-08735-7
  • MathSciNet review: 2302552