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Zbl 0843.33012
Berndt, Bruce C.; Bhargava, S.; Garvan, Frank G.
Ramanujan's theories of elliptic functions to alternative bases. (English)
[J] Trans. Am. Math. Soc. 347, No. 11, 4163-4244 (1995). ISSN 0002-9947; ISSN 1088-6850/e

Authors' abstract: In his famous paper on modular equations and approximations to $\pi$ [Q. J. Math. (Oxford) 45, 350--372 (1914; JFM 45.1249.01)], {\it S. Ramanujan} offered several series representations of $1/\pi$, which he claims are derived from corresponding theories in which the classical base $q$ is replaced by one of three other bases. The formulas for $1/\pi$ were only recently proved by {\it J. M. Borwein}and {\it P. B. Borwein} in 1987 [Pi and the AGM, New York, NY: Wiley (1987; Zbl 0903.11001)], but these corresponding theories have never been heretofore developed. However, on six pages of his notebooks, Ramanujan gives approximately 50 results without proofs in these theories. The purpose of this paper is to prove all of these claims, and several other results are established as well.
[J.Matkowski (Bielsko-Biała)]

MSC 2000:: *33E05 Elliptic functions and integrals
30D10 Representations of entire functions by series and integrals
33C05 Classical hypergeometric functions
11F27 Theta series; Weil representation

Citations: JFM 45.1249.01; Zbl 0903.11001

Cited in: Zbl 1098.33015 Zbl 1050.33015 Zbl 1110.11300 Zbl 1125.11317 Zbl 0949.33002 Zbl 0911.11024 Zbl 0849.68063

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