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Ramanujan-type formulae for \(1/\pi: q\)-analogues. (English) Zbl 1436.11024
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, \({}_r\phi_s\)
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[1] q-analogues of two Ramanujan-type formulas for \(##?##\)J. Diff. Equ. ApplForthcoming
[2] Ramanujan, S., Modular equations and approximations to π, Quart J Math Oxford Ser (2), 45, 350-372, (1914) · JFM 45.1249.01
[3] Gasper, G; Rahman, M., Basic hypergeometric series, (2004), Cambridge University Press, Cambridge · Zbl 1129.33005
[4] Guillera, J., Generators of some Ramanujan formulas, Ramanujan J, 11, 41-48, (2006) · Zbl 1109.33029
[5] On q-analogues of some series for π and \(##?##\). Preprint [2018 February]; 11 pages. Available from:
[6] Series de Ramanujan: Generalizaciones y conjeturas [PhD thesis]. Zaragoza (Spain): Universidad de Zaragoza; 2007
[7] Rahman, M., Some quadratic and cubic summation formulas for basic hypergeometric series, Canad J Math, 45, 394-411, (1993) · Zbl 0774.33012
[8] Zudilin, W., Ramanujan-type supercongruences, J Number Theory, 129, 1848-1857, (2009) · Zbl 1231.11147
[9] A q-microscope for supercongruences. Preprint [2018 March]; 24 pages. Available from:
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