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Legendre polynomials and Ramanujan-type series for $$1/\pi$$. (English) Zbl 1357.11123
Summary: We resolve a family of recently observed identities involving $$1/\pi$$ using the theory of modular forms and hypergeometric series. In particular, we resort to a formula of F. Brafman [Proc. Am. Math. Soc. 2, 942–949 (1951; Zbl 0044.07602)] 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.

##### MSC:
 11Y60 Evaluation of number-theoretic constants 33C45 Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.) 11F27 Theta series; Weil representation; theta correspondences
Zbl 0044.07602
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##### References:
 [1] 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. · Zbl 0011.02303 [2] N. D. Baruah and B. C. Berndt, Eisenstein series and Ramanujan-type series for 1/{$$\pi$$}, The Ramanujan Journal 23 (2010), 17–44. · Zbl 1204.33005 [3] B. C. Berndt, Ramanujan’s Notebooks, Part III, Springer-Verlag, New York, 1991. · Zbl 0733.11001 [4] B. C. Berndt, Ramanujan’s Notebooks, Part V, Springer-Verlag, New York, 1997. · Zbl 0886.11001 [5] 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. · Zbl 0843.33012 [6] B. C. Berndt and H. H. Chan, Eisenstein series and approximations to {$$\pi$$}, Illinois Journal of Mathematics 45 (2001), 75–90. · Zbl 0998.33003 [7] 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. · Zbl 1005.33009 [8] J. M. Borwein and P. B. Borwein, Pi and the AGM: A Study in Analytic Number Theory and Computational Complexity, Wiley, New York, 1987. · Zbl 0611.10001 [9] 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 · Zbl 1233.60024 [10] F. Brafman, Generating functions of Jacobi and related polynomials, Proceedings of the American Mathematical Society 2 (1951), 942–949. · Zbl 0044.07602 [11] H. H. Chan, Ramanujan’s elliptic functions to alternative bases and approximations to {$$\pi$$}, in Number Theory for the Millennium, I, Urbana, IL, 2000, A K Peters, Natick, MA, 2002, pp. 197–213. · Zbl 1140.11358 [12] 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. · Zbl 0647.10002 [13] S. Cooper, Inversion formulas for elliptic functions, Proceedings of the London Mathematical Society 99 (2009), 461–483. · Zbl 1248.11031 [14] J. Guillera and W. Zudilin, ”Divergent” Ramanujan-type supercongruences, Proceedings of the American Mathematical Society 140 (2012), 765–777. · Zbl 1276.11027 [15] S. Ramanujan, Modular equations and approximations to {$$\pi$$}, The Quarterly Journal of Mathematics 45 (1914), 350–372. · JFM 45.1249.01 [16] Z.-W. Sun, List of conjectural series for powers of {$$\pi$$} and other constants, preprint, arXiv: 1102.5649v21 [math.CA], May 23, 2011. [17] J. Wan and W. Zudilin, Generating functions of Legendre polynomials: a tribute to Fred Brafman, Journal of Approximation Theory 164 (2012), 488–503. · Zbl 1242.33018
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