Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



On Eisenstein series and $\sum _{m,n=-\infty }^{\infty }q^{m^{2}+mn+2n^{2}}$

Authors: Heng Huat Chan and Yao Lin Ong
Journal: Proc. Amer. Math. Soc. 127 (1999), 1735-1744
MSC (1991): Primary 33E05, 11Y60
Published electronically: February 11, 1999
MathSciNet review: 1600120
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: In this paper, we derive some new identities satisfied by the series ${\sum _{m,n=-\infty }^{\infty }q^{m^{2}+mn+2n^{2}}}$ using Ramanujan's identities for $L(q)$, $M(q)$ and $N(q)$. Our work is motivated by an attempt to develop a theory of elliptic functions to the septic base.

References [Enhancements On Off] (What's this?)

  • 1. B.C. Berndt, Ramanujan Notebooks Part III, Springer-Verlag, New York, 1991. MR 92j:01069
  • 2. B.C. Berndt, Ramanujan Notebooks Part IV, Springer-Verlag, New York, 1994. MR 95e:11028
  • 3. B.C. Berndt, S. Bhargava, and F.G. Garvan, Ramanujan's theories of elliptic functions to alternative bases, Trans. Amer. Math. Soc. 347 (1995), 4163-4244. MR 97h:33034
  • 4. H.H. Chan, On Ramanujan's cubic transformation formula for $_{2}F_{1}(1/3,2/3;1;z)$, Math. Proc. Cambridge Philos. Soc. 124 (1998), 193-204. CMP 98:15
  • 5. S. Raghavan and S.S. Rangachari, On Ramanujan's elliptic integrals and modular identities, Number Theory and Related Topics, Oxford University Press, Bombay, 1989, pp. 119 - 149. MR 98b:11045
  • 6. S. Ramanujan, Collected Papers, Chelsea, New York, 1962.
  • 7. S. Ramanujan, On certain arithmetical functions, Trans. Cambridge Philos. Soc. 22 (1916), 159-184.
  • 8. J.B. Sohn, Private communications.
  • 9. K. Ventkatachaliengar, Development of elliptic functions according to Ramanujan, Madurai Kamaraj University, Madurai, 1988.

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (1991): 33E05, 11Y60

Retrieve articles in all journals with MSC (1991): 33E05, 11Y60

Additional Information

Heng Huat Chan
Affiliation: National University of Singapore, Department of Mathematics, Kent Ridge, Singapore 119260, Republic of Singapore

Yao Lin Ong
Affiliation: National Chung Cheng University, Department of Mathematics, Min-hsiung, Chiayi 621, Taiwan, Republic of China

Keywords: Eisenstein series, modular equations, elliptic functions
Received by editor(s): September 11, 1997
Published electronically: February 11, 1999
Communicated by: David E. Rohrlich
Article copyright: © Copyright 1999 American Mathematical Society

American Mathematical Society