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

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: One of the many amazing things Ramanujan did in his lifetime was to list identities involving what are now called the Rogers-Ramanujan 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 Rogers-Ramanujan type identities between and , and Thompson series. Here we prove these identities.

**[BBP87]**Bruce C. Berndt, Anthony J. Biagioli, and James M. Purtilo,*Ramanujan’s modular equations of “large” prime degree*, J. Indian Math. Soc. (N.S.)**51**(1987), 75–110 (1988). MR**988310****[Bia89]**Anthony J. F. Biagioli,*A proof of some identities of Ramanujan using modular forms*, Glasgow Math. J.**31**(1989), no. 3, 271–295. MR**1021804**, 10.1017/S0017089500007850**[Bir75]**B. J. Birch,*A look back at Ramanujan’s notebooks*, Math. Proc. Cambridge Philos. Soc.**78**(1975), 73–79. MR**0379372****[CN79]**J. H. Conway and S. P. Norton,*Monstrous moonshine*, Bull. London Math. Soc.**11**(1979), no. 3, 308–339. MR**554399**, 10.1112/blms/11.3.308**[CY96]**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. 255–326. MR**1400423****[Dar21]**H. B. C. Darling,*Proof of certain identities and congruences enunciated by S. Ramanujan*, Proc. London Math. Soc. (2)**19**(1921), 350-372.**[Kim04]**Chang Heon Kim,*Borcherds products associated with certain Thompson series*, Compos. Math.**140**(2004), no. 3, 541–551. MR**2041767**, 10.1112/S0010437X03000770**[Koi04]**Masao Koike,*Thompson series and Ramanujan’s identities*, Galois theory and modular forms, Dev. Math., vol. 11, Kluwer Acad. Publ., Boston, MA, 2004, pp. 367–373. MR**2059774**, 10.1007/978-1-4613-0249-0_20**[Mar96]**Yves Martin,*Multiplicative 𝜂-quotients*, Trans. Amer. Math. Soc.**348**(1996), no. 12, 4825–4856. MR**1376550**, 10.1090/S0002-9947-96-01743-6**[Ono04]**Ken Ono,*The web of modularity: arithmetic of the coefficients of modular forms and 𝑞-series*, CBMS Regional Conference Series in Mathematics, vol. 102, Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 2004. MR**2020489****[Rog21]**L. J. Rogers,*On a type of modular relations*, Proc. London Math. Soc (2)**19**(1921), 387-397.**[Wat33]**G. N. Watson,*Proof of certain identities in combinatory analysis*, J. Indian Math. Soc.**20**(1933), 57-69.

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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.ohio-state.edu

DOI:
https://doi.org/10.1090/S0002-9939-07-08735-7

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