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Ramanujan-type formulae for \(1/\pi\): the art of translation. (English) Zbl 1371.11162

Berndt, Bruce C. (ed.) et al., The legacy of Srinivasa Ramanujan. Proceedings of the international conference in celebration of the 125th anniversary of Ramanujan’s birth, University of Delhi, Delhi, India, December 17–22, 2012. Mysore: Ramanujan Mathematical Society (ISBN 978-93-80416-13-7/hbk). Ramanujan Mathematical Society Lecture Notes Series 20, 181-195 (2013).
Summary: We outline an elementary method for proving numerical hypergeometric identities, in particular, Ramanujan-type identities for \(1/\pi\). The principal idea is using algebraic transformations of arithmetic hypergeometric series to translate non-singular points into singular ones, where the required constants can be computed using asymptotic analysis.
For the entire collection see [Zbl 1300.11002].

MSC:

11Y60 Evaluation of number-theoretic constants
33C20 Generalized hypergeometric series, \({}_pF_q\)
11F11 Holomorphic modular forms of integral weight
65B10 Numerical summation of series
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