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A GENERATING FUNCTION OF THE SQUARES OF LEGENDRE POLYNOMIALS

  • WADIM ZUDILIN (a1)
Abstract

We relate a one-parametric generating function for the squares of Legendre polynomials to an arithmetic hypergeometric series whose parametrisation by a level 7 modular function was recently given by Cooper. By using this modular parametrisation we resolve a subfamily of identities involving $1/ \pi $ which was experimentally observed by Sun.

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Copyright
Corresponding author
wadim.zudilin@newcastle.edu.au
References
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[1]Bailey, W. N., Generalized Hypergeometric Series, Cambridge Mathematical Tracts 32 (Cambridge University Press, Cambridge, 1935).
[2]Bailey, W. N., ‘The generating function of Jacobi polynomials’, J. Lond. Math. Soc. 13 (1938), 812.
[3]Brafman, F., ‘Generating functions of Jacobi and related polynomials’, Proc. Amer. Math. Soc. 2 (1951), 942949.
[4]Chan, H. H. and Cooper, S., ‘Eisenstein series and theta functions to the septic base’, J. Number Theory 128 (2008), 680699.
[5]Chan, H. H. and Cooper, S., ‘Rational analogues of Ramanujan’s series for $1/ \pi $’, Math. Proc. Cambridge Philos. Soc. 153 (2012), 361383.
[6]Chan, H. H., Tanigawa, Y., Yang, Y. and Zudilin, W., ‘New analogues of Clausen’s identities arising from the theory of modular forms’, Adv. Math. 228 (2011), 12941314.
[7]Chan, H. H., Wan, J. and Zudilin, W., ‘Legendre polynomials and Ramanujan-type series for $1/ \pi $’, Israel J. Math. (2013), to appear, doi:10.1007/s11856-012-0081-5.
[8]Chan, H. H. and Zudilin, W., ‘New representations for Apéry-like sequences’, Mathematika 56 (2010), 107117.
[9]Cooper, S., ‘Sporadic sequences, modular forms and new series for $1/ \pi $’, Ramanujan J. 29 (2012), 163183.
[10]Maximon, L. C., ‘A generating function for the product of two Legendre polynomials’, Norske Vid. Selsk. Forh. Trondheim 29 (1956), 8286.
[11]Rogers, M. and Straub, A., ‘A solution of Sun’s $520 challenge concerning $520/ \pi $’, Preprint, arXiv:1210.2373, 2012.
[12]Sloane, N. J. A., The On-Line Encyclopedia of Integer Sequences, published electronically at http://www.research.att.com/~njas/sequences/, 2012.
[13]Sun, Z.-W., ‘List of conjectural series for powers of $\pi $ and other constants’, Preprint, arXiv:1102.5649v37, 24 January 2012.
[14]Wan, J. and Zudilin, W., ‘Generating functions of Legendre polynomials: a tribute to Fred Brafman’, J. Approx. Theory 164 (2012), 488503.
[15]Zudilin, W., ‘Lost in translation’, Proceedings of the Waterloo Workshop in Computer Algebra (W80) (May 2011), In Honour of Herbert S. Wilf, Springer Proceedings in Mathematics and Statistics (eds. I. Kotsireas and E. V. Zima) (Springer, New York, 2013), to appear.
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Bulletin of the Australian Mathematical Society
  • ISSN: 0004-9727
  • EISSN: 1755-1633
  • URL: /core/journals/bulletin-of-the-australian-mathematical-society
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