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

Author(s): Heng Huat Chan; Yao Lin Ong
Journal: Proc. Amer. Math. Soc. 127 (1999), 1735-1744.
MSC (1991): Primary 33E05, 11Y60
Posted: February 11, 1999
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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.

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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

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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

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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

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S. Ramanujan, Collected Papers, Chelsea, New York, 1962.

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Additional Information:

Heng Huat Chan
Affiliation: National University of Singapore, Department of Mathematics, Kent Ridge, Singapore 119260, Republic of Singapore
Email: chanhh@math.nus.sg

Yao Lin Ong
Affiliation: National Chung Cheng University, Department of Mathematics, Min-hsiung, Chiayi 621, Taiwan, Republic of China
Email: d8521002@willow.math.ccu.edu.tw

DOI: 10.1090/S0002-9939-99-04832-7
PII: S 0002-9939(99)04832-7
Keywords: Eisenstein series, modular equations, elliptic functions
Received by editor(s): September 11, 1997
Posted: February 11, 1999
Communicated by: David E. Rohrlich
Copyright of article: Copyright 1999, American Mathematical Society

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