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On some cubic modular identities


Author: Li-Chien Shen
Journal: Proc. Amer. Math. Soc. 119 (1993), 203-208
MSC: Primary 11F03
DOI: https://doi.org/10.1090/S0002-9939-1993-1152291-6
MathSciNet review: 1152291
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Abstract: Let $ 4K$ and $ 2iK'$ be the periods of $ \operatorname{sn} z$. By evaluating the functional equations $ {\operatorname{sn} ^2}z + {\operatorname{cn} ^2}z = 1$ and $ {k^2}{\operatorname{sn} ^2}z + {\operatorname{dn} ^2}z = 1$ at $ z = iK'/3$, we deduce a set of cubic modular identities from which the familiar modular equation of degree $ 3$ follows directly as a corollary.


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

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1993-1152291-6
Keywords: Theta functions, quadratic transformation, modular identity
Article copyright: © Copyright 1993 American Mathematical Society

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