A theta function identity and its implications
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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.References
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Additional Information
- Zhi-Guo Liu
- Affiliation: Department of Mathematics, East China Normal University, Shanghai 200062, People’s Republic of China
- MR Author ID: 364722
- Email: zgliu@math.ecnu.edu.cn, liuzg18@hotmail.com
- 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
- © Copyright 2004 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 357 (2005), 825-835
- MSC (2000): Primary 11F11, 11F12, 11F27, 33E05
- DOI: https://doi.org/10.1090/S0002-9947-04-03572-X
- MathSciNet review: 2095632