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.


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
Full Text: DOI


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