Abstract
We resolve a family of recently observed identities involving 1/π using the theory of modular forms and hypergeometric series. In particular, we resort to a formula of Brafman which relates a generating function of the Legendre polynomials to a product of two Gaussian hypergeometric functions. Using our methods, we also derive some new Ramanujan-type series.
References
W. N. Bailey, Generalized Hypergeometric Series, Cambridge Tracts in Mathematics and Mathematical Physics 32, Cambridge University Press, Cambridge, 1935; 2nd reprinted edition, Stechert-Hafner, New York-London, 1964.
N. D. Baruah and B. C. Berndt, Eisenstein series and Ramanujan-type series for 1/π, The Ramanujan Journal 23 (2010), 17–44.
B. C. Berndt, Ramanujan’s Notebooks, Part III, Springer-Verlag, New York, 1991.
B. C. Berndt, Ramanujan’s Notebooks, Part V, Springer-Verlag, New York, 1997.
B. C. Berndt, S. Bhargava and F. G. Garvan, Ramanujan’s theories of elliptic functions to alternative bases, Transactions of the American Mathematical Society 347 (1995), 4163–4244.
B. C. Berndt and H. H. Chan, Eisenstein series and approximations to π, Illinois Journal of Mathematics 45 (2001), 75–90.
B. C. Berndt, H. H. Chan and W.-C. Liaw, On Ramanujan’s quartic theory of elliptic functions, Journal of Number Theory 88 (2001), 129–156.
J. M. Borwein and P. B. Borwein, Pi and the AGM: A Study in Analytic Number Theory and Computational Complexity, Wiley, New York, 1987.
J. M. Borwein, D. Nuyens, A. Straub and J. Wan, Some arithmetic properties of short random walk integrals, The Ramanujan Journal 26 (2011), 109–132
F. Brafman, Generating functions of Jacobi and related polynomials, Proceedings of the American Mathematical Society 2 (1951), 942–949.
H. H. Chan, Ramanujan’s elliptic functions to alternative bases and approximations to π, in Number Theory for the Millennium, I, Urbana, IL, 2000, A K Peters, Natick, MA, 2002, pp. 197–213.
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.
S. Cooper, Inversion formulas for elliptic functions, Proceedings of the London Mathematical Society 99 (2009), 461–483.
J. Guillera and W. Zudilin, “Divergent” Ramanujan-type supercongruences, Proceedings of the American Mathematical Society 140 (2012), 765–777.
S. Ramanujan, Modular equations and approximations to π, The Quarterly Journal of Mathematics 45 (1914), 350–372.
Z.-W. Sun, List of conjectural series for powers of π and other constants, preprint, arXiv: 1102.5649v21 [math.CA], May 23, 2011.
J. Wan and W. Zudilin, Generating functions of Legendre polynomials: a tribute to Fred Brafman, Journal of Approximation Theory 164 (2012), 488–503.
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On Jon Borwein’s 60th birthday
The third author is supported by Australian Research Council grant DP110104419.
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Chan, H.H., Wan, J. & Zudilin, W. Legendre polynomials and Ramanujan-type series for 1/π . Isr. J. Math. 194, 183–207 (2013). https://doi.org/10.1007/s11856-012-0081-5
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DOI: https://doi.org/10.1007/s11856-012-0081-5