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Transactions of the American Mathematical Society

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On the additive formulae of the theta functions and a collection of Lambert series pertaining to the modular equations of degree $ 5$


Author: Li-Chien Shen
Journal: Trans. Amer. Math. Soc. 345 (1994), 323-345
MSC: Primary 33D10; Secondary 11F27
MathSciNet review: 1250827
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Abstract: We examine the connection between the additive formulae of the theta functions, the Fourier series expansion of the 12 elliptic functions, and the logarithmic derivatives of the theta functions. As an application, we study the Lambert series related to the modular equations of degree 5 and many interesting identities of Ramanujan are found in this process.


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DOI: http://dx.doi.org/10.1090/S0002-9947-1994-1250827-3
Keywords: Theta functions, imaginary transformation, Lambert series, Jacobian elliptic functions
Article copyright: © Copyright 1994 American Mathematical Society