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**Ramanujan’s theories of elliptic functions to alternative bases.**
*(English)*
Zbl 0843.33012

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)], 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 J. M. Borweinand 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.

Reviewer: J.Matkowski (Bielsko-Biała)

### MSC:

33E05 | Elliptic functions and integrals |

30D10 | Representations of entire functions of one complex variable by series and integrals |

33C05 | Classical hypergeometric functions, \({}_2F_1\) |

11F27 | Theta series; Weil representation; theta correspondences |

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\textit{B. C. Berndt} et al., Trans. Am. Math. Soc. 347, No. 11, 4163--4244 (1995; Zbl 0843.33012)

### Digital Library of Mathematical Functions:

Ramanujan’s Cubic Transformation ‣ §15.8(v) Cubic Transformations ‣ §15.8 Transformations of Variable ‣ Properties ‣ Chapter 15 Hypergeometric Function§20.11(iii) Ramanujan’s Change of Base ‣ §20.11 Generalizations and Analogs ‣ Properties ‣ Chapter 20 Theta Functions