Legendre polynomials and Ramanujan-type series for \(1/\pi\). (English) Zbl 1357.11123

Chan, Heng Huat; Wan, James; Zudilin, Wadim

Isr. J. Math. 194, Part A, 183-207 (2013).

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

Keywords:

identities; generating function; Legendre polynomials; product of two Gaussian hypergeometric functions; Ramanujan-type series

Citations:

Zbl 0044.07602

References:

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