On a conjecture of Koike on identities between Thompson series and RogersRamanujan functions
Authors:
Kathrin Bringmann and Holly Swisher
Journal:
Proc. Amer. Math. Soc. 135 (2007), 23172326
MSC (2000):
Primary 11F22, 33D15, 11F03
Published electronically:
March 21, 2007
Erratum:
Proc. Amer. Math. Soc. 136 (2008), 15011501.
MathSciNet review:
2302552
Fulltext PDF Free Access
Abstract 
References 
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Additional Information
Abstract: One of the many amazing things Ramanujan did in his lifetime was to list identities involving what are now called the RogersRamanujan functions and on one side, and products of functions of the form 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 RogersRamanujan type identities between and , and Thompson series. Here we prove these identities.
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 Imin Chen and Noriko Yui, Singular values of Thompson series, Groups, difference sets, and the Monster (Columbus, OH, 1993), Ohio State Univ. Math. Res. Inst. Publ., vol. 4, de Gruyter, Berlin, 1996, pp. 255326. MR MR1400423 (98a:11051)
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 H. B. C. Darling, Proof of certain identities and congruences enunciated by S. Ramanujan, Proc. London Math. Soc. (2) 19 (1921), 350372.
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 Chang Heon Kim, Borcherds products associated with certain Thompson series, Compos. Math. 140 (2004), no. 3, 541551. MR MR2041767 (2004k:11058)
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 Masao Koike, Thompson series and Ramanujan's identities, Galois theory and modular forms, Dev. Math., vol. 11, Kluwer Acad. Publ., Boston, MA, 2004, pp. 367373. MR MR2059774 (2005c:11051)
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 Yves Martin, Multiplicative quotients, Trans. Amer. Math. Soc. 348 (1996), no. 12, 48254856. MR MR1376550 (97d:11070)
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 Ken Ono, The web of modularity: arithmetic of the coefficients of modular forms and series, CBMS Regional Conference Series in Mathematics, vol. 102, Published for the Conference Board of the Mathematical Sciences, Washington, DC, 2004. MR MR2020489 (2005c:11053)
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 G. N. Watson, Proof of certain identities in combinatory analysis, J. Indian Math. Soc. 20 (1933), 5769.
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Additional Information
Kathrin Bringmann
Affiliation:
Department of Mathematics, University of Wisconsin, Madison, Wisconsin 53706
Email:
bringman@math.wisc.edu
Holly Swisher
Affiliation:
Department of Mathematics, The Ohio State University, Columbus, Ohio 43210
Email:
swisher@math.ohiostate.edu
DOI:
http://dx.doi.org/10.1090/S0002993907087357
PII:
S 00029939(07)087357
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
