×

zbMATH — the first resource for mathematics

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
Citations:
Zbl 0044.07602
PDF BibTeX XML Cite
Full Text: DOI
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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.