New analogues of Clausenʼs identities arising from the theory of modular forms

Around 1828, T. Clausen discovered that the square of certain hypergeometric F 1 2 function can be expressed as a hypergeometric F 2 3 function. Special cases of Clausenʼs identities were later used by S. Ramanujan in his derivation of 17 series for 1 / π . Since then, there were several attempts to...

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Bibliographic Details
Published in:Advances in mathematics (New York. 1965) 2011-10, Vol.228 (2), p.1294-1314
Main Authors: Chan, Heng Huat, Tanigawa, Yoshio, Yang, Yifan, Zudilin, Wadim
Format: Article
Language:English
Subjects:
Clausenʼs identities
Modular forms of one variable
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Summary:Around 1828, T. Clausen discovered that the square of certain hypergeometric F 1 2 function can be expressed as a hypergeometric F 2 3 function. Special cases of Clausenʼs identities were later used by S. Ramanujan in his derivation of 17 series for 1 / π . Since then, there were several attempts to find new analogues of Clausenʼs identities with the hope to derive new classes of series for 1 / π . Unfortunately, none were successful. In this article, we will present three new analogues of Clausenʼs identities. Their discovery is motivated by the study of relations between modular forms of weight 2 and modular functions associated with modular groups of genus 0.
ISSN:0001-8708
1090-2082
DOI:10.1016/j.aim.2011.06.011