an:00870299
Zbl 0843.33012
Berndt, Bruce C.; Bhargava, S.; Garvan, Frank G.
Ramanujan's theories of elliptic functions to alternative bases
EN
Trans. Am. Math. Soc. 347, No. 11, 4163-4244 (1995).
00032805
1995
j
33E05 30D10 33C05 11F27
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)], \textit{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 \textit{J. M. Borwein}and \textit{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-Bia??a)
JFM 45.1249.01; Zbl 0903.11001