Triple product identity, Quintuple product identity and Ramanujan’s differential equations for the classical Eisenstein series

Chan, Heng Huat

doi:10.1090/S0002-9939-07-08723-0

			Journals Home Search Author Info Subscribe Tech Support My Subscriptions Browser Compatibility Info Help
ISSN 1088-6826(online) ISSN 0002-9939(print)

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
Published electronically: March 2, 2007
MathSciNet review: 2299470
Full-text PDF Free Access

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. George E. Andrews, Richard Askey, and Ranjan Roy, Special functions, Encyclopedia of Mathematics and its Applications, vol. 71, Cambridge University Press, Cambridge, 1999. MR 1688958 (2000g:33001)
2. Bruce C. Berndt, Ramanujan’s notebooks. Part III, Springer-Verlag, New York, 1991. MR 1117903 (92j:01069)
3. Bruce C. Berndt, Song Heng Chan, Zhi-Guo Liu, and Hamza Yesilyurt, A new identity for (𝑞;𝑞)¹⁰_{∞} with an application to Ramanujan’s partition congruence modulo 11, Q. J. Math. 55 (2004), no. 1, 13–30. MR 2043004 (2004k:11166), http://dx.doi.org/10.1093/qjmath/55.1.13
4. Bruce C. Berndt and Ae Ja Yee, A page on Eisenstein series in Ramanujan’s Lost Notebook, Glasg. Math. J. 45 (2003), no. 1, 123–129. MR 1972702 (2004a:11032), http://dx.doi.org/10.1017/S0017089502001076
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), http://dx.doi.org/10.1090/S0002-9939-1972-0289316-2
6. Shaun Cooper, The quintuple product identity, Int. J. Number Theory 2 (2006), no. 1, 115–161. MR 2217798 (2007f:33021), http://dx.doi.org/10.1142/S1793042106000401
7. James G. Huard, Zhiming M. Ou, Blair K. Spearman, and Kenneth S. Williams, Elementary evaluation of certain convolution sums involving divisor functions, Number theory for the millennium, II (Urbana, IL, 2000) A K Peters, Natick, MA, 2002, pp. 229–274. MR 1956253 (2003j:11008)
8. Serge Lang, Introduction to modular forms, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 222, Springer-Verlag, Berlin, 1995. With appendixes by D. Zagier and Walter Feit; Corrected reprint of the 1976 original. MR 1363488 (96g:11037)
9. Zhi-Guo Liu, A three-term theta function identity and its applications, Adv. Math. 195 (2005), no. 1, 1–23. MR 2145792 (2006c:11050), http://dx.doi.org/10.1016/j.aim.2004.07.006
10. S. Ramanujan, On certain arithmetical functions, Trans. Cambridge Philos. Soc. 22 (1916), 159-184.
11. Srinivasa Ramanujan, The lost notebook and other unpublished papers, Springer-Verlag, Berlin, 1988. With an introduction by George E. Andrews. MR 947735 (89j:01078)
12. R. A. Rankin, Elementary proofs of relations between Eisenstein series, Proc. Roy. Soc. Edinburgh Sect. A 76 (1976/77), no. 2, 107–117. MR 0441870 (56 #261)
13. Nils-Peter Skoruppa, A quick combinatorial proof of Eisenstein series identities, J. Number Theory 43 (1993), no. 1, 68–73. MR 1200809 (94f:11029), http://dx.doi.org/10.1006/jnth.1993.1007
14. K. Venkatachaliengar, Development of elliptic functions according to Ramanujan, Technical Report, 2. Madurai Kamaraj University, Department of Mathematics, Madurai, 1988.