Quadratic iterations to 𝜋 associated with elliptic functions to the cubic and septic base

Chan, Heng; Chua, Kok; Solé, Patrick

doi:10.1090/S0002-9947-02-03192-6

Quadratic iterations to ${\pi }$ associated with elliptic functions to the cubic and septic base
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by Heng Huat Chan, Kok Seng Chua and Patrick Solé PDF

Trans. Amer. Math. Soc. 355 (2003), 1505-1520 Request permission

Abstract:

In this paper, properties of the functions $A_d(q)$, $B_d(q)$ and $C_d(q)$ are derived. Specializing at $d=1$ and $2$, we construct two new quadratic iterations to $\pi$. These are analogues of previous iterations discovered by the Borweins (1987), J. M. Borwein and F. G. Garvan (1997), and H. H. Chan (2002). Two new transformations of the hypergeometric series $_2F_1(1/3,1/6;1;z)$ are also derived.

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