# Self-replication and Borwein-like algorithms

## TL;DR

This paper introduces a self-replicating approach to generalize Borwein algorithms for pi, incorporating Gamma function values and providing new fast algorithms for ellipse perimeter calculations.

## Contribution

It presents a novel self-replicating method that extends Borwein algorithms to include Gamma function values and develops new rapid algorithms for ellipse perimeter.

## Key findings

- Generalized Borwein algorithms for pi using a self-replicating method.
- Included Gamma function values such as Γ(1/3), Γ(1/4), and √π.
- Developed new fast algorithms for calculating the perimeter of an ellipse.

## 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.