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Level 14 and 15 analogues of Ramanujan's elliptic functions to alternative bases


Authors: Shaun Cooper and Dongxi Ye
Journal: Trans. Amer. Math. Soc. 368 (2016), 7883-7910
MSC (2010): Primary 11F11; Secondary 33C05
DOI: https://doi.org/10.1090/tran6658
Published electronically: November 16, 2015
MathSciNet review: 3546787
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Abstract: We briefly review Ramanujan's theories of elliptic functions to alternative bases, describe their analogues for levels 5 and 7, and develop new theories for levels 14 and 15. This gives rise to a rich interplay between theta functions, eta-products and Eisenstein series. Transformation formulas of degrees five and seven for hypergeometric functions are obtained, and the paper ends with some series for $ 1/\pi $ similar to ones found by Ramanujan.


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Additional Information

Shaun Cooper
Affiliation: Institute of Natural and Mathematical Sciences, Massey University-Albany, Private Bag 102904, North Shore Mail Centre, Auckland, New Zealand
Email: s.cooper@massey.ac.nz

Dongxi Ye
Affiliation: Department of Mathematics, University of Wisconsin, 480 Lincoln Drive, Madison, Wisconsin 53706
Email: lawrencefrommath@gmail.com

DOI: https://doi.org/10.1090/tran6658
Keywords: Dedekind eta function, Eisenstein series, hypergeometric function, modular form, pi, Ramanujan's theories of elliptic functions to alternative bases.
Received by editor(s): November 15, 2013
Received by editor(s) in revised form: January 10, 2015
Published electronically: November 16, 2015
Article copyright: © Copyright 2015 American Mathematical Society

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