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Rational analogues of Ramanujan’s series for \(1/\pi\). (English) Zbl 1268.11165

Summary: A general theorem is stated that unifies 93 rational Ramanujan-type series for \(1/\pi \), 40 of which are believed to be new. Moreover, each series is shown to have a companion identity, thereby giving another 93 series, the majority of which are new.

MSC:

11Y60 Evaluation of number-theoretic constants
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[1] DOI: 10.1017/S000497270003834X · Zbl 1078.33020
[2] DOI: 10.1112/plms/pdp007 · Zbl 1248.11031
[3] Chudnovsky, Approximations and complex multiplication according to Ramanujan pp 375– (1987)
[4] DOI: 10.1112/S0024611505015364 · Zbl 1089.33012
[5] Chan, Math. Res. Lett. 16 pp 405– (2009) · Zbl 1193.11038
[6] DOI: 10.1016/j.aim.2011.06.011 · Zbl 1234.33009
[7] DOI: 10.4064/aa126-4-2 · Zbl 1123.11042
[8] DOI: 10.1017/S0024610701002241 · Zbl 1110.11300
[9] DOI: 10.1007/BF02762264 · Zbl 0922.11040
[10] DOI: 10.1016/j.aim.2003.07.012 · Zbl 1122.11087
[11] Beukers, Progr. Math. (1983)
[12] DOI: 10.2307/2325206 · Zbl 0672.10017
[13] Borwein, Organic Mathematics pp 89– (1995)
[14] DOI: 10.1016/0377-0427(93)90302-R · Zbl 0818.65010
[15] Zudilin, Ramanujan-type formulae for 1/{\(\pi\)}: a second wind? (2008) · Zbl 1159.11053
[16] Borwein, Pi and the AGM – A Study in Analytic Number Theory and Computational Complexity (1987)
[17] Zagier, Integral solutions of Apéry-like recurrence equations pp 349– (2009)
[18] Borwein, More Ramanujan-type series for 1/{\(\pi\)} pp 359– (1988) · Zbl 0652.10019
[19] DOI: 10.1112/jlms/s1-4.1.39 · JFM 55.0273.01
[20] DOI: 10.1093/qmath/hag038 · Zbl 1060.11063
[21] DOI: 10.1006/jnth.2000.2615 · Zbl 1005.33009
[22] DOI: 10.1016/S0377-0427(99)00033-3 · Zbl 0953.11005
[23] DOI: 10.1007/s11139-007-9040-x · Zbl 1226.11113
[24] Ramanujan, Notebooks (2 volumes) (1957)
[25] Ramanujan, Quart. J. Math. 45 pp 350– (1914)
[26] Hirschhorn, Contemp. Math. (2000)
[27] Berndt, Illinois J. Math. 45 pp 75– (2001)
[28] DOI: 10.1017/S1446788700019728
[29] Berndt, Trans. Amer. Math. Soc. 347 pp 4163– (1995)
[30] Berndt, Number theory in the spirit of Ramanujan (2006) · Zbl 1117.11001
[31] Farkas, Theta constants, Riemann surfaces and the modular group. An introduction with applications to uniformization theorems, partition identities and combinatorial number theory (2001) · Zbl 0982.30001
[32] DOI: 10.1007/978-1-4612-1624-7
[33] Elaydi, An Introduction to Difference Equations (2005) · Zbl 1071.39001
[34] DOI: 10.1007/978-1-4612-0965-2
[35] DOI: 10.1090/S0273-0979-05-01047-5 · Zbl 1109.11026
[36] DOI: 10.1515/crll.1859.56.101 · ERAM 056.1478cj
[37] DOI: 10.1093/qmath/6.1.193 · Zbl 0065.27602
[38] DOI: 10.4169/193009709X458555 · Zbl 1229.11162
[39] Andrews, Special Functions (1999)
[40] Cooper, J. Ramanujan Math. Soc. 27 pp 75– (2012)
[41] DOI: 10.1017/S0013091509000959 · Zbl 1223.33007
[42] DOI: 10.4064/aa141-1-2 · Zbl 1226.11133
[43] DOI: 10.1073/pnas.86.21.8178 · Zbl 0683.65008
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