Abstract
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.
To Krishna Alladi,
on his smooth transition from the sixth decade
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References
G.E. Andrews, R. Askey, R. Roy, Special Functions (Cambridge University Press, Cambridge, 1999)
J.M. Borwein, P.B. Borwein, Pi and the AGM: A Study in Analytic Number Theory and Computational Complexity (Wiley, New York, 1987)
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)
H.H. Chan, S. Cooper, Rational analogues of Ramanujan’s series for \(1/\pi \). Math. Proc. Camb. Phil. Soc. 153, 361–383 (2012)
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)
H.H. Chan, J.G. Wan, W. Zudilin, Legendre polynomials and Ramanujan-type series for \(1/\pi \). Isr. J. Math. 194, 183–207 (2013)
H.H. Chan, W. Zudilin, New representations for Apéry-like sequences. Mathematika 56, 107–117 (2010)
S. Cooper, Level \(10\) analogues of Ramanujan’s series for \(1/\pi \). J. Ramanujan Math. Soc. 27, 59–76 (2012)
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)
N. Fine, Basic Hypergeometric Series and Applications (American Mathematical Society, Providence, RI, 1988)
J. Guillera, A family of Ramanujan–Orr formulas for \(1/\pi \). Integral Transform. Spec. Funct. 26, 531–538 (2015)
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. 181–195
M. Petkovsek, H. Wilf, D. Zeilberger, \(A = B\) (A.K. Peters, Wellesley, 1996)
S. Ramanujan, Modular equations and approximations to \(\pi \). Quart. J. Math (Oxford) 45, 350–372 (1914). Reprinted in [15, pp. 23–39] (2000)
S. Ramanujan, Collected Papers, 3rd printing (American Mathematical Society Chelsea, Providence, RI, 2000)
M. Rogers, New \({}_5F_4\) hypergeometric transformations, three-variable Mahler measures, and formulas for \(1/\pi \). Ramanujan J. 18, 327–340 (2009)
M. Rogers, A. Straub, A solution of Sun’s $520 challenge concerning \(520/\pi \). Int. J. Number Theory 9, 1273–1288 (2013)
N.J.A. Sloane, The On-Line Encyclopedia of Integer Sequences (2015), published electronically at https://oeis.org
Z.-W. Sun, List of conjectural series for powers of \(\pi \) and other constants (2014), 33 pp. http://arxiv.org/abs/1102.5649v47
J.G. Wan, Random walks, elliptic integrals and related constants, Ph.D. Thesis (University of Newcastle, NSW, Australia, 2013)
J.G. Wan, Series for \(1/\pi \) using Legendre’s relation. Integral Transform. Spec. Funct. 25, 1–14 (2014)
J.G. Wan, W. Zudilin, Generating functions of Legendre polynomials: a tribute to Fred Brafman. J. Approx. Theory 164, 488–503 (2012)
D. Ye, Private communication to S. Cooper (11 November, 2013)
W. Zudilin, A generating function of the squares of Legendre polynomials. Bull. Aust. Math. Soc. 89, 125–131 (2014)
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We thank the referee for helpful comments and suggestions.
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Cooper, S., Wan, J.G., Zudilin, W. (2017). Holonomic Alchemy and Series for \(1/\pi \) . In: Andrews, G., Garvan, F. (eds) Analytic Number Theory, Modular Forms and q-Hypergeometric Series. ALLADI60 2016. Springer Proceedings in Mathematics & Statistics, vol 221. Springer, Cham. https://doi.org/10.1007/978-3-319-68376-8_12
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DOI: https://doi.org/10.1007/978-3-319-68376-8_12
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