Generating functions of Legendre polynomials: a tribute to Fred Brafman. (English) Zbl 1282.33015

Wan, James; Zudilin, Wadim

J. Approx. Theory 170, 198-213 (2013).

This is a reprint of an article (with the same title and content) by these authors, that appeared in [ibid. 164, No. 4, 488–503 (2012; Zbl 1242.33018)].

Reviewer: Marcel G. de Bruin (Haarlem)

MSC:

33C45	Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.)
11B83	Special sequences and polynomials
65Q30	Numerical aspects of recurrence relations

Keywords:

Legendre polynomial; Brafman’s generating function; hypergeometric series; Clausen’s identity; Apeéry-like sequence; modular function; \(\pi\)

Citations:

Zbl 1242.33018

References:

[1]	Almkvist, G.; van Straten, D.; Zudilin, W., Generalizations of Clausen’s formula and algebraic transformations of Calabi-Yau differential equations, Proc. Edinb. Math. Soc., 54, 2, 273-295 (2011) · Zbl 1223.33007
[2]	Bailey, W. N., (Generalized Hypergeometric Series. Generalized Hypergeometric Series, Cambridge Math. Tracts, vol. 32 (1935), Cambridge Univ. Press: Cambridge Univ. Press Cambridge), second reprinted ed., Stechert-Hafner, New York, London, 1964 · JFM 61.0406.01
[3]	Baruah, N. D.; Berndt, B. C., Eisenstein series and Ramanujan-type series for \(1 / \pi \), Ramanujan J., 23, 1-3, 17-44 (2010) · Zbl 1204.33005
[4]	Borwein, J. M.; Straub, A.; Wan, J.; Zudilin, W., With an appendix by D. Zagier, Densities of short uniform random walks, Canad. J. Math., 64, 961-990 (2012) · Zbl 1296.33011
[5]	Brafman, F., Generating functions of Jacobi and related polynomials, Proc. Amer. Math. Soc., 2, 942-949 (1951) · Zbl 0044.07602
[6]	Brafman, F., Generating functions and associated Legendre polynomials, Q. J. Math. (Oxford), 10, 2, 156-160 (1959) · Zbl 0087.28203
[7]	Chan, H. H.; Cooper, S., Rational analogues of Ramanujan’s series for \(1 / \pi \), Math. Proc. Cambridge Philos. Soc., 153, 361-383 (2012) · Zbl 1268.11165
[8]	Chan, H. H.; Tanigawa, Y.; Yang, Y.; Zudilin, W., New analogues of Clausen’s identities arising from the theory of modular forms, Adv. Math., 228, 2, 1294-1314 (2011) · Zbl 1234.33009
[10]	Clausen, T., Ueber die Fälle wenn die Reihe \(y = 1 + \frac{\alpha \beta}{1 \cdot \gamma} x + \cdots\) ein quadrat von der Form \(y = 1 + \frac{\alpha^\prime \beta^\prime \gamma^\prime}{1 \cdot \delta^\prime \epsilon^\prime} x + \cdots\) hat, J. Math., 3, 89-95 (1828)
[11]	Ramanujan, S., Modular equations and approximations to \(\pi \), Q. J. Math. (Oxford), 45, 350-372 (1914) · JFM 45.1249.01
[12]	Srivastava, H. M., An equivalence theorem on generating functions, Proc. Amer. Math. Soc., 52, 159-165 (1975) · Zbl 0311.33014
[14]	Zagier, D., Integral solutions of Apéry-like recurrence equations, (Groups and Symmetries: From Neolithic Scots to John McKay. Groups and Symmetries: From Neolithic Scots to John McKay, CRM Proc. Lecture Notes, vol. 47 (2009), Amer. Math. Soc.: Amer. Math. Soc. Providence, RI), 349-366 · Zbl 1244.11042