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Conference paper

Abstract

We explain the use and set grounds about applicability of algebraic transformations of arithmetic hypergeometric series for proving Ramanujan’s formulae for \(1/\pi\) and their generalisations.

Keywords

\(\pi\) Ramanujan Arithmetic hypergeometric series Algebraic transformation Modular function 

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References

  1. 1.
    G. Almkvist, D. van Straten and W. Zudilin, Generalizations of Clausen’s formula and algebraic transformations of Calabi–Yau differential equations, Proc. Edinburgh Math. Soc. 54 (2011), 273–295.MATHCrossRefGoogle Scholar
  2. 2.
    W. N. Bailey, Generalized hypergeometric series, Cambridge Tracts in Math. 32 (Cambridge Univ. Press, Cambridge 1935); 2nd reprinted ed. (Stechert-Hafner, New York–London 1964).Google Scholar
  3. 3.
    G. Bauer, Von den Coefficienten der Reihen von Kugelfunctionen einer Variablen, J. Reine Angew. Math. 56 (1859), 101–121.MATHCrossRefGoogle Scholar
  4. 4.
    B. C. Berndt, Ramanujan’s Notebooks, Part V (Springer, New York 1998).MATHCrossRefGoogle Scholar
  5. 5.
    J. M. Borwein and P. B. Borwein, Pi and the AGM (Wiley, New York 1987).MATHGoogle Scholar
  6. 6.
    H. H. Chan and S. Cooper, Rational analogues of Ramanujan’s series for \(1/\pi\), Math. Proc. Cambridge Philos. Soc. 153 (2012), 361–383. DOI 10.1017/S0305004112000254.Google Scholar
  7. 7.
    H. H. Chan and W. Zudilin, New representations for Apéry-like sequences, Mathematika 56 (2010), 107–117.MathSciNetMATHCrossRefGoogle Scholar
  8. 8.
    D. V. Chudnovsky and G. V. Chudnovsky, Approximations and complex multiplication according to Ramanujan, in Ramanujan revisited (Urbana-Champaign, IL 1987) (Academic Press, Boston, MA 1988), pp. 375–472.Google Scholar
  9. 9.
    S. B. Ekhad and D. Zeilberger, A WZ proof of Ramanujan’s formula for \(\pi\), in Geometry, Analysis, and Mechanics, J. M. Rassias (ed.) (World Scientific, Singapore 1994), pp. 107–108.Google Scholar
  10. 10.
    J. Guillera, Some binomial series obtained by the WZ-method, Adv. in Appl. Math. 29 (2002), 599–603.MathSciNetMATHCrossRefGoogle Scholar
  11. 11.
    J. Guillera, Generators of some Ramanujan formulas, Ramanujan J. 11 (2006), 41–48.MathSciNetMATHCrossRefGoogle Scholar
  12. 12.
    J. Guillera, On WZ-pairs which prove Ramanujan series, Ramanujan J. 22 (2010), 249–259.MathSciNetMATHCrossRefGoogle Scholar
  13. 13.
    J. Guillera, A new Ramanujan-like series for \(1{/\pi }^{2}\), Ramanujan J. 26 (2011), 369–374.MathSciNetMATHCrossRefGoogle Scholar
  14. 14.
    M. Petkovšek, H. S. Wilf and D. Zeilberger, A = B (A. K. Peters, Wellesley, MA 1997).Google Scholar
  15. 15.
    S. Ramanujan, Modular equations and approximations to \(\pi\), Quart. J. Math. (Oxford) 45 (1914), 350–372.Google Scholar
  16. 16.
    M. D. Rogers, New 5 F 4 hypergeometric transformations, three-variable Mahler measures, and formulas for \(1/\pi\), Ramanujan J. 18 (2009), 327–340.MathSciNetMATHCrossRefGoogle Scholar
  17. 17.
    L. J. Slater, Generalized hypergeometric functions (Cambridge Univ. Press, Cambridge 1966).MATHGoogle Scholar
  18. 18.
    J. Wan and W. Zudilin, Generating functions of Legendre polynomials: A tribute to Fred Brafman, J. Approximation Theory 164 (2012), 488–503.MathSciNetMATHCrossRefGoogle Scholar
  19. 19.
    W. Zudilin, Quadratic transformations and Guillera’s formulas for \(1{/\pi }^{2}\), Math. Notes 81 (2007), 297–301.MathSciNetMATHCrossRefGoogle Scholar
  20. 20.
    W. Zudilin, More Ramanujan-type formulae for \(1{/\pi }^{2}\), Russian Math. Surveys 62 (2007), 634–636.MathSciNetMATHCrossRefGoogle Scholar
  21. 21.
    W. Zudilin, Ramanujan-type formulae for \(1/\pi\): A second wind?, in Modular forms and string duality (Banff, June 3–8, 2006), N. Yui et al. (eds.), Fields Inst. Commun. Ser. 54 (Amer. Math. Soc., Providence, RI 2008), 179–188.Google Scholar
  22. 22.
    W. Zudilin, Ramanujan-type supercongruences, J. Number Theory 129 (2009), 1848–1857.MathSciNetMATHCrossRefGoogle Scholar
  23. 23.
    W. Zudilin, Arithmetic hypergeometric series, Russian Math. Surveys 66 (2011), 369–420.MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  1. 1.School of Mathematical and Physical SciencesThe University of NewcastleCallaghanAustralia

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