Triple product identity, Quintuple product identity and Ramanujan's differential equations for the classical Eisenstein series
Author:
Heng Huat Chan
Journal:
Proc. Amer. Math. Soc. 135 (2007), 1987-1992
MSC (2000):
Primary 14K25
DOI:
https://doi.org/10.1090/S0002-9939-07-08723-0
Published electronically:
March 2, 2007
MathSciNet review:
2299470
Full-text PDF
Abstract | References | Similar Articles | Additional Information
Abstract: In this article, we use the triple product identity and the quintuple product identity to derive Ramanujan's famous differential equations for the Eisenstein series.
- 1. G.E. Andrews, R. Askey and R. Roy, Special functions, Cambridge University Press, Cambridge, 1999. MR 1688958 (2000g:33001)
- 2. B.C. Berndt, Ramanujan's Notebooks, Part III, Springer-Verlag, New York, 1991. MR 1117903 (92j:01069)
- 3.
B. C. Berndt, S. H. Chan, Z.-G. Liu and H. Yesilyurt, A new identity for
with an application to Ramanujan's partition congruence modulo 11, Quart. J. Math., 55 (2004), 13-30. MR 2043004 (2004k:11166)
- 4. B. C. Berndt and A. J. Yee, A page on Eisenstein series in Ramanujan's lost notebook, Glasgow Math. J., 45 (2003), 123-129. MR 1972702 (2004a:11032)
- 5. L. Carlitz and M.V. Subbarao, A simple proof of the quintuple product identity, Proc. Amer. Math. Soc. 32 (1972), 42-44. MR 0289316 (44:6507)
- 6. S. Cooper, The quintuple product identity, Int. J. Number Theory 2 (2006), 115-161. MR 2217798
- 7. J.G. Huard, Z.M. Ou, B.K. Spearman and K.S. Williams, Elementary evaluation of certain convolution sums involving divisor functions, in Number Theory for the Millennium, vol. 2, M.A. Bennett, B.C. Berndt, N. Boston, H.G. Diamond, A.J. Hildebrand and W. Philipp, eds., A K Peters, Natick, MA, 2002, pp. 229-274. MR 1956253 (2003j:11008)
- 8. S. Lang, Introduction to modular forms, Springer-Verlag, New York, 1995. MR 1363488 (96g:11037)
- 9. Z.G. Liu, A three term theta function identity and its applications, Adv. Math. 195 (2005), 1-23. MR 2145792 (2006c:11050)
- 10. S. Ramanujan, On certain arithmetical functions, Trans. Cambridge Philos. Soc. 22 (1916), 159-184.
- 11. S. Ramanujan, The lost notebook and other unpublished papers, Narosa, New Delhi, 1988. MR 0947735 (89j:01078)
- 12. R.A. Rankin, Elementary proofs of relations between Eisenstein series, Proc. Roy. Soc. Edinburgh 76A (1976) 107-117. MR 0441870 (56:261)
- 13. N.-D. Skoruppa, A quick combinatorial proof of Eisenstein series identities, J. Number Theory 43 (1993), 68-73. MR 1200809 (94f:11029)
- 14. K. Venkatachaliengar, Development of elliptic functions according to Ramanujan, Technical Report, 2. Madurai Kamaraj University, Department of Mathematics, Madurai, 1988.
Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 14K25
Retrieve articles in all journals with MSC (2000): 14K25
Additional Information
Heng Huat Chan
Affiliation:
Department of Mathematics, National University of Singapore, 2 Science Drive 2, Singapore 117543
Email:
matchh@nus.edu.sg
DOI:
https://doi.org/10.1090/S0002-9939-07-08723-0
Received by editor(s):
June 27, 2005
Received by editor(s) in revised form:
December 27, 2005, and March 23, 2006
Published electronically:
March 2, 2007
Communicated by:
Wen-Ching Winnie Li
Article copyright:
© Copyright 2007
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.