Self-replication and Borwein-like algorithms

Abstract

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