A theta function identity and its implications

Liu, Zhi-Guo

doi:10.1090/S0002-9947-04-03572-X

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
DOI: https://doi.org/10.1090/S0002-9947-04-03572-X
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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