The Ramanujan Journal

, Volume 47, Issue 2, pp 447–455 | Cite as

Self-replication and Borwein-like algorithms

  • Jesús GuilleraEmail author


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.


Hypergeometric series Algebraic transformations Borwein-type algorithms 

Mathematics Subject Classification

11Y60 33F05 33C75 33C05 33F10 


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Copyright information

© Springer Science+Business Media New York 2017

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of ZaragozaSaragossaSpain

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