Holonomic Alchemy and Series for \(1/\pi \)

  • Shaun CooperEmail author
  • James G. Wan
  • Wadim Zudilin
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 221)


We adopt the “translation” as well as other techniques to express several identities conjectured by Z.-W. Sun by means of known formulas for \(1/\pi \) involving Domb and other Apéry-like sequences.


Apéry-like sequence Domb numbers Eisenstein series Holonomic function Modular form Modular parameterization Ramanujan’s series for \(1/\pi \) Sun’s conjectures Translation technique Zeilberger’s algorithm. 

2010 Mathematics Subject Classification

Primary 11Y60 33C20 Secondary 11B65 11F11 11Y55 65B10 



We thank the referee for helpful comments and suggestions.


  1. 1.
    G.E. Andrews, R. Askey, R. Roy, Special Functions (Cambridge University Press, Cambridge, 1999)CrossRefzbMATHGoogle Scholar
  2. 2.
    J.M. Borwein, P.B. Borwein, Pi and the AGM: A Study in Analytic Number Theory and Computational Complexity (Wiley, New York, 1987)zbMATHGoogle Scholar
  3. 3.
    H.H. Chan, S.H. Chan, Z.-G. Liu, Domb’s numbers and Ramanujan-Sato type series for \(1/\pi \). Adv. Math. 186, 396–410 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    H.H. Chan, S. Cooper, Rational analogues of Ramanujan’s series for \(1/\pi \). Math. Proc. Camb. Phil. Soc. 153, 361–383 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    H.H. Chan, Y. Tanigawa, Y. Yang, W. Zudilin, New analogues of Clausen’s identities arising from the theory of modular forms. Adv. Math. 228, 1294–1314 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    H.H. Chan, J.G. Wan, W. Zudilin, Legendre polynomials and Ramanujan-type series for \(1/\pi \). Isr. J. Math. 194, 183–207 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    H.H. Chan, W. Zudilin, New representations for Apéry-like sequences. Mathematika 56, 107–117 (2010)Google Scholar
  8. 8.
    S. Cooper, Level \(10\) analogues of Ramanujan’s series for \(1/\pi \). J. Ramanujan Math. Soc. 27, 59–76 (2012)MathSciNetzbMATHGoogle Scholar
  9. 9.
    S. Cooper, D. Ye, Level \(14\) and \(15\) analogues of Ramanujan’s elliptic functions to alternative bases. Trans. Am. Math. Soc. 368, 7883–7910 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    N. Fine, Basic Hypergeometric Series and Applications (American Mathematical Society, Providence, RI, 1988)CrossRefzbMATHGoogle Scholar
  11. 11.
    J. Guillera, A family of Ramanujan–Orr formulas for \(1/\pi \). Integral Transform. Spec. Funct. 26, 531–538 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    J. Guillera, W. Zudilin, Ramanujan-type formulae for \(1/\pi \): the art of translation, in The Legacy of Srinivasa Ramanujan, eds. by B.C. Berndt, D. Prasad. Ramanujan Mathematical Society Lecture Notes Series, vol. 20 (2013), pp. 181–195Google Scholar
  13. 13.
    M. Petkovsek, H. Wilf, D. Zeilberger, \(A = B\) (A.K. Peters, Wellesley, 1996)Google Scholar
  14. 14.
    S. Ramanujan, Modular equations and approximations to \(\pi \). Quart. J. Math (Oxford) 45, 350–372 (1914). Reprinted in [15, pp. 23–39] (2000)zbMATHGoogle Scholar
  15. 15.
    S. Ramanujan, Collected Papers, 3rd printing (American Mathematical Society Chelsea, Providence, RI, 2000)Google Scholar
  16. 16.
    M. Rogers, New \({}_5F_4\) hypergeometric transformations, three-variable Mahler measures, and formulas for \(1/\pi \). Ramanujan J. 18, 327–340 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    M. Rogers, A. Straub, A solution of Sun’s $520 challenge concerning \(520/\pi \). Int. J. Number Theory 9, 1273–1288 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    N.J.A. Sloane, The On-Line Encyclopedia of Integer Sequences (2015), published electronically at
  19. 19.
    Z.-W. Sun, List of conjectural series for powers of \(\pi \) and other constants (2014), 33 pp.
  20. 20.
    J.G. Wan, Random walks, elliptic integrals and related constants, Ph.D. Thesis (University of Newcastle, NSW, Australia, 2013)Google Scholar
  21. 21.
    J.G. Wan, Series for \(1/\pi \) using Legendre’s relation. Integral Transform. Spec. Funct. 25, 1–14 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    J.G. Wan, W. Zudilin, Generating functions of Legendre polynomials: a tribute to Fred Brafman. J. Approx. Theory 164, 488–503 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    D. Ye, Private communication to S. Cooper (11 November, 2013)Google Scholar
  24. 24.
    W. Zudilin, A generating function of the squares of Legendre polynomials. Bull. Aust. Math. Soc. 89, 125–131 (2014)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.Institute of Natural and Mathematical Sciences, Massey University — Albany, Private Bag 102904AucklandNew Zealand
  2. 2.Engineering Systems and Design, Singapore University of Technology and DesignSingaporeSingapore
  3. 3.School of Mathematical and Physical Sciences, The University of NewcastleCallaghan NSWAustralia

Personalised recommendations