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

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- by Kathrin Bringmann and Holly Swisher
- Proc. Amer. Math. Soc.
**135**(2007), 2317-2326 - DOI: https://doi.org/10.1090/S0002-9939-07-08735-7
- Published electronically: March 21, 2007
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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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## Bibliographic 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