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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)


On some cubic modular identities

Author: Li-Chien Shen
Journal: Proc. Amer. Math. Soc. 119 (1993), 203-208
MSC: Primary 11F03
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.

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PII: S 0002-9939(1993)1152291-6
Keywords: Theta functions, quadratic transformation, modular identity
Article copyright: © Copyright 1993 American Mathematical Society

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