Ramanujan-type formulae for \(1/\pi: q\)-analogues. (English) Zbl 1436.11024

Guo, Victor J. W.; Zudilin, Wadim

Integral Transforms Spec. Funct. 29, No. 7, 505-513 (2018).

Summary: The hypergeometric formulae designed by Ramanujan more than a century ago for efficient approximation of \(\pi\), Archimedes’ constant, remain an attractive object of arithmetic study. In this note we discuss some \(q\)-analogues of Ramanujan-type evaluations and of related supercongruences.

MSC:

11B65	Binomial coefficients; factorials; \(q\)-identities
11Y60	Evaluation of number-theoretic constants
33C20	Generalized hypergeometric series, \({}_pF_q\)
33D15	Basic hypergeometric functions in one variable, \({}_r\phi_s\)

Keywords:

\(1/\pi\); Ramanujan; \(q\)-analogue; WZ pair; basic hypergeometric function; (super)congruence

References:

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