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Complex series for \(1/\pi\). (English) Zbl 1336.11079
A method is presented that allows to prove series of complex type for \(1/\pi\).
MSC:
11Y60 Evaluation of number-theoretic constants
11F11 Holomorphic modular forms of integral weight
11F03 Modular and automorphic functions
33C05 Classical hypergeometric functions, \({}_2F_1\)
33C20 Generalized hypergeometric series, \({}_pF_q\)
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[1] Apostol, T.M.: Modular Functions and Dirichlet Series in Number Theory, 2nd edn. Graduate Text in Math, vol. 41. Springer, New York (1990) · Zbl 0697.10023
[2] Bauer, G.: Von den Coeffizienten der Reihen von Kugelfunctionen einer Variablen. J. Reine Angew. Math. 56, 101–121 (1859) · ERAM 056.1478cj
[3] Berndt, B.C., Chan, H.H., Liaw, W.-C.: On Ramanujan’s quartic theory of elliptic functions. J. Number Theory 88(1), 129–156 (2001) · Zbl 1005.33009
[4] Borwein, J.M., Borwein, P.B.: Pi and the AGM; A Study in Analytic Number Theory and Computational Complexity. Wiley, New York (1987) · Zbl 0611.10001
[5] Chan, H.H., Zudilin, W.: New representations for Apéry-like sequences. Mathematika 56(1), 107–117 (2010) · Zbl 1275.11035
[6] Chan, H.H., Chan, S.H., Liu, Z.G.: Domb’s numbers and Ramanujan–Sato type series for 1/{\(\pi\)}. Adv. Math. 186, 396–410 (2004) · Zbl 1122.11087
[7] Chan, H.H., Wan, J., Zudilin, W.: Legendre polynomials and Ramanujan-type series for 1/{\(\pi\)}. Isr. J. Math. (to appear) · Zbl 1357.11123
[8] Chudnovsky, D.V., Chudnovsky, G.V.: Approximations and complex multiplication according to Ramanujan. In: Ramanujan Revisited, Urbana-Champaign, IL, 1987, pp. 375–472. Academic Press, Boston (1988) · Zbl 0647.10002
[9] Guillera, J., Zudilin, W.: ”Divergent” Ramanujan-type supercongruences. Proc. Am. Math. Soc. 140, 765–777 (2012) · Zbl 1276.11027
[10] Ramanujan, S.: Modular equations and approximations to {\(\pi\)}. Q. J. Math. 45, 350–372 (1914) · JFM 45.1249.01
[11] Sato, T.: Apéry numbers and Ramanujan’s series for 1/{\(\pi\)}. Abstract of a talk presented at the annual meeting of the Mathematical Society of Japan (28–31 March 2002)
[12] Sun, Z.-W.: List of conjectural series for powers of {\(\pi\)} and other constants. Preprint arXiv:1102.5649 [math.CA] (2011)
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