Ramanujan-type formulae for \(1/\pi\): a second wind? (English) Zbl 1159.11053

Zudilin, Wadim

Yui, Noriko (ed.) et al., Modular forms and string duality. Proceedings of a workshop, Banff, Canada, June 3–8, 2006. Providence, RI: American Mathematical Society (AMS); Toronto: The Fields Institute for Research in Mathematical Sciences (ISBN 978-0-8218-4484-7/hbk). Fields Institute Communications 54, 179-188 (2008).

This paper contains a brief survey of the elliptic, modular, hypergeometric, and telescoping methods used to prove formulas for powers of \(1/\pi\) inspired by Ramanujan’s \(17\) original examples, one of which is \[ \sum_{n \geq 0} 2^{-6n}\binom{2n}{n}^3(4n+1)(-1)^n = \frac{2}{\pi}. \] Some directions for future research are indicated.
For the entire collection see [Zbl 1147.11005].

Reviewer: Jeremy Lovejoy (Paris)

MSC:

11Y60	Evaluation of number-theoretic constants
11F11	Holomorphic modular forms of integral weight
33C20	Generalized hypergeometric series, \({}_pF_q\)
33F10	Symbolic computation of special functions (Gosper and Zeilberger algorithms, etc.)

Keywords:

Ramanujan-type formulas for 1/pi; Apéry; elliptic functions; modular forms; hypergeometric series; creative telescoping