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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

A theta function identity and its implications


Author: Zhi-Guo Liu
Journal: Trans. Amer. Math. Soc. 357 (2005), 825-835
MSC (2000): Primary 11F11, 11F12, 11F27, 33E05
Published electronically: September 2, 2004
MathSciNet review: 2095632
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Abstract: In this paper we prove a general theta function identity with four parameters by employing the complex variable theory of elliptic functions. This identity plays a central role for the cubic theta function identities. We use this identity to re-derive some important identities of Hirschhorn, Garvan and Borwein about cubic theta functions. We also prove some other cubic theta function identities. A new representation for $\prod_{n=1}^\infty(1-q^n)^{10}$is given. The proofs are self-contained and elementary.


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Additional Information

Zhi-Guo Liu
Affiliation: Department of Mathematics, East China Normal University, Shanghai 200062, People’s Republic of China
Email: zgliu@math.ecnu.edu.cn, liuzg18@hotmail.com

DOI: http://dx.doi.org/10.1090/S0002-9947-04-03572-X
PII: S 0002-9947(04)03572-X
Keywords: Elliptic functions, theta function identities, infinite products
Received by editor(s): September 18, 2003
Received by editor(s) in revised form: October 24, 2003
Published electronically: September 2, 2004
Additional Notes: The author was supported in part by Shanghai Priority Academic Discipline and the National Science Foundation of China
Article copyright: © Copyright 2004 American Mathematical Society