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Quadratic iterations to ${\pi}$ associated with elliptic functions to the cubic and septic base


Authors: Heng Huat Chan, Kok Seng Chua and Patrick Solé
Journal: Trans. Amer. Math. Soc. 355 (2003), 1505-1520
MSC (2000): Primary 11Y60, 33C05, 33E05, 11F03
DOI: https://doi.org/10.1090/S0002-9947-02-03192-6
Published electronically: December 2, 2002
MathSciNet review: 1946402
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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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Additional Information

Heng Huat Chan
Affiliation: Department of Mathematics, National University of Singapore, Singapore 117543, Republic of Singapore
Email: chanhh@math.nus.edu.sg

Kok Seng Chua
Affiliation: Department of Mathematics, National University of Singapore, Singapore 117543, Republic of Singapore
Email: matv2@nus.edu.sg

Patrick Solé
Affiliation: CNRS-I3S, ESSI, Route des Colles, 06 903 Sophia Antipolis, France
Email: ps@essi.fr

DOI: https://doi.org/10.1090/S0002-9947-02-03192-6
Received by editor(s): January 15, 2002
Received by editor(s) in revised form: August 21, 2002
Published electronically: December 2, 2002
Additional Notes: The first author was funded by National University of Singapore Academic Research Fund, Project Number R14000027112
Article copyright: © Copyright 2002 American Mathematical Society

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