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Self-replication and Borwein-like algorithms

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Abstract

Using a self-replicating method, we generalize with a free parameter some Borwein algorithms for the number \(\pi \). This generalization includes values of the gamma function like \(\Gamma (1/3)\), \(\Gamma (1/4)\), and of course \(\Gamma (1/2)=\sqrt{\pi }\). In addition, we give new rapid algorithms for the perimeter of an ellipse.

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References

  1. Almkvist, G., Berndt, B.: Gauss, Landen, Ramanujan, the arithmetic-geometric mean, ellipses, \(\pi \), and the ladies diary. Am. Math. Mon. 95, 585–608 (1988)

    MathSciNet  MATH  Google Scholar 

  2. Arndt, J.: Matters Computational (Ideas, Algorithms, Source Code). Springer, Berlin (2011)

    MATH  Google Scholar 

  3. Bailey, D., Borwein, J.: Pi: The Next Generation. Springer, Basel (2016)

    Book  Google Scholar 

  4. Berggren, L., Borwein, J., Borwein, P.: Pi: A Source Book. Springer, New York (1997)

    Book  Google Scholar 

  5. Borwein, J., Bailey, D.: Mathematics by Experiment (Plausible Reasoning in the 21st Century). A. K. Peters, Ltd., Wellesley (2008)

    MATH  Google Scholar 

  6. Borwein, J., Borwein, P.J.: Pi and the AGM: A Study in Analytic Number Theory and Computational Complexity. Canadian Mathematical Society Series of Monographs and Advanced Texts. Wiley, New York (1987)

    MATH  Google Scholar 

  7. Borwein, J., Borwein, P.: Ramanujan, modular equations, and approximations to Pi or how to compute one billion digits of Pi. Am. Math. Mon. 96, 201–219 (1989)

    Article  MathSciNet  Google Scholar 

  8. Borwein, J., Borwein, P.: A cubic counterpart of Jacobi’s identity and the AGM. Trans. Am. Math. Soc. 343, 691–701 (1991)

    MathSciNet  MATH  Google Scholar 

  9. Borwein, J., Garvan, F.: Approximations to \(\pi \) via the Dedekind eta function. CMS Conf. Proc. 20, 89–115 (1997)

    MathSciNet  MATH  Google Scholar 

  10. Cooper, S.: Ramanujans Theta Functions. Springer, New York (2017)

    Book  Google Scholar 

  11. Cooper, S., Guillera, J., Straub, A., Zudilin, W.: Crouching AGM, Hidden Modularity. (Submitted)

  12. Guillera, J.: Easy proofs of some Borwein algorithms for \(\pi \). Am. Math. Mon. 115, 850–854 (2008)

    Article  MathSciNet  Google Scholar 

  13. Guillera, J.: New proofs of Borwein-type algorithms for Pi. Integral Transforms and Special functions. (Published online: June 29, 2016)

  14. Ramanujan, S.: Modular equations and approximations to \(\pi \). Q. J. Math. 45, 350–372 (1914)

    MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, University of Zaragoza, 50009, Saragossa, Spain

    Jesús Guillera

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  1. Jesús Guillera
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Correspondence to Jesús Guillera.

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To the memory of Jonathan Borwein

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Guillera, J. Self-replication and Borwein-like algorithms. Ramanujan J 47, 447–455 (2018). https://doi.org/10.1007/s11139-017-9927-0

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  • DOI: https://doi.org/10.1007/s11139-017-9927-0

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