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NEW REPRESENTATIONS FOR APÉRY-LIKE SEQUENCES

Part of: Discontinuous groups and automorphic forms

Published online by Cambridge University Press:  10 December 2009

Heng Huat Chan  and
Wadim Zudilin
Heng Huat Chan
Affiliation:
Department of Mathematics, National University of Singapore, 2 Science Drive 2, Singapore 117543 Max-Planck-Institut für Mathematik, Vivatsgasse 7, D-53111, Bonn, Germany (email: [email protected])
Wadim Zudilin
Affiliation:
School of Mathematical and Physical Sciences, University of Newcastle, Callaghan NSW 2308, Australia (email: [email protected])
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Abstract

We prove algebraic transformations for the generating series of three Apéry-like sequences. As application, we provide new binomial representations for the sequences. We also illustrate a method that derives three new series for 1/π from a classical Ramanujan’s series.

MSC classification

Secondary: 11F11: Holomorphic modular forms of integral weight 11F20: Dedekind eta function, Dedekind sums 33C20: Generalized hypergeometric series, $_pF_q$
Type
Research Article
Information
Mathematika , Volume 56 , Issue 1 , January 2010 , pp. 107 - 117
Copyright
Copyright © University College London 2010

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References

[1]Almkvist, G. and Zudilin, W., Differential equations, mirror maps and zeta values. In Mirror Symmetry V (AMS/IP Studies in Advanced Mathemtics 38) (eds N. Yui, S.-T. Yau and J. D. Lewis), International Press & American Mathematical Society (Providence, RI, 2007), 481–515.Google Scholar
[2]Apéry, R., Irrationalité de ζ(2) et ζ(3). Astérisque 61 (1979), 11–13.Google Scholar
[3]Bauer, G., Von den Coefficienten der Reihen von Kugelfunctionen einen Variablen. J. für Math. (Crelles J.) 56 (1859), 101–121.Google Scholar
[4]Berndt, B. C., Chan, H. H. and Huang, S.-S., Incomplete elliptic integrals in Ramanujan’s lost notebook. Contemp. Math. 254 (2000), 79–126.CrossRefGoogle Scholar
[5]Berndt, B. C., Chan, H. H. and Liaw, W.-L., On Ramanujan’s quartic theory of elliptic functions. J. Number Theory 88 (2001), 129–156.CrossRefGoogle Scholar
[6]Beukers, F., Irrationality proofs using modular forms, Journées arithmétiques (Besançon, 1985). Astérisque 147–148 (1987), 271–283.Google Scholar
[7]Chan, H. H., Chan, S. H. and Liu, Z., Domb’s numbers and Ramanujan–Sato type series for 1/π. Adv. Math. 186(2) (2004), 396–410.CrossRefGoogle Scholar
[8]Chan, H. H. and Verrill, H., The Apéry numbers, the Almkvist–Zudilin numbers and new series for 1/π. Math. Res. Lett. 16(3) (2009), 405–420.CrossRefGoogle Scholar
[9]Ramanujan, S., Modular equations and approximations to π. Q. J. Math. Oxford Ser. (2) 45 (1914), 350–372; Reprinted in Collected Papers of Srinivasa Ramanujan (eds G. H. Hardy, P. V. Sechu Aiyar and B. M. Wilson), Cambridge University Press (Cambridge, 1927; Chelsea Publ., New York, 1962), 23–39.Google Scholar
[10]Rogers, M. D., New 5F 4 hypergeometric transformations, three-variable Mahler measures, and formulas for 1/π. Ramanujan J. 18(3) (2009), 327–340.CrossRefGoogle Scholar
[11]Slater, L. J., Generalized Hypergeometric Functions, Cambridge University Press (Cambridge, 1966).Google Scholar
[12]Zudilin, W., Quadratic transformations and Guillera’s formulas for 1/π 2. Math. Notes 81(3) (2007), 297–301.CrossRefGoogle Scholar
[13]Zudilin, W., Ramanujan-type formulae for 1/π: a second wind? In Modular Forms and String Duality (Banff, June 2006) (Fields Inst. Commun. 54) (ed. N. Yui), American Mathematical Society (Providence, RI, 2008), 179–188.Google Scholar