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

Guillera, Jesús; Zudilin, Wadim

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

Keywords:

Ramanujan-type identities for \(1/\pi\); hypergeometric series; arithmetic hypergeometric series; algebraic transformations; asymptotics