## 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

- 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** - 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**, DOI 10.1017/S0017089500007850 - B. J. Birch,
*A look back at Ramanujan’s notebooks*, Math. Proc. Cambridge Philos. Soc.**78**(1975), 73–79. MR**379372**, DOI 10.1017/S0305004100051501 - J. H. Conway and S. P. Norton,
*Monstrous moonshine*, Bull. London Math. Soc.**11**(1979), no. 3, 308–339. MR**554399**, DOI 10.1112/blms/11.3.308 - 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** - H. B. C. Darling,
*Proof of certain identities and congruences enunciated by S. Ramanujan*, Proc. London Math. Soc. (2)**19**(1921), 350–372. - Chang Heon Kim,
*Borcherds products associated with certain Thompson series*, Compos. Math.**140**(2004), no. 3, 541–551. - 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**, DOI 10.1007/978-1-4613-0249-0_{2}0 - Yves Martin,
*Multiplicative $\eta$-quotients*, Trans. Amer. Math. Soc.**348**(1996), no. 12, 4825–4856. MR**1376550**, DOI 10.1090/S0002-9947-96-01743-6 - 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** - L. J. Rogers,
*On a type of modular relations*, Proc. London Math. Soc (2)**19**(1921), 387–397. - G. N. Watson,
*Proof of certain identities in combinatory analysis*, J. Indian Math. Soc.**20**(1933), 57–69.

## 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