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On a conjecture of Koike on identities between Thompson series and Rogers-Ramanujan functions

Authors: Kathrin Bringmann and Holly Swisher
Journal: Proc. Amer. Math. Soc. 135 (2007), 2317-2326
MSC (2000): Primary 11F22, 33D15, 11F03
Published electronically: March 21, 2007
Erratum: Proc. Amer. Math. Soc. 136 (2008), 1501-1501.
MathSciNet review: 2302552
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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.

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Additional Information

Kathrin Bringmann
Affiliation: Department of Mathematics, University of Wisconsin, Madison, Wisconsin 53706

Holly Swisher
Affiliation: Department of Mathematics, The Ohio State University, Columbus, Ohio 43210

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
Article copyright: © Copyright 2007 American Mathematical Society

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