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Accelerating the Hypergeometric Function with the Beta Integral to Derive New Infinite Series for $π$ and Values of the Gamma Function

Cetin Hakimoglu

Abstract

The beta integral is applied to accelerate the hypergeometric function $2 F 1\left\{1, B; C ; w\right\}$ to derive new infinite series for constants such as $π$ and values of the gamma function. A compendium of new infinite series is given. Ramanujan-like formulas for pi are also derived based on elementary inverse trigonometric functions, including a formula with rational values that adds 2.5 digits per terms, which makes the series much more compact than similar formulas in the existing literature.

Accelerating the Hypergeometric Function with the Beta Integral to Derive New Infinite Series for $π$ and Values of the Gamma Function

Abstract

The beta integral is applied to accelerate the hypergeometric function to derive new infinite series for constants such as and values of the gamma function. A compendium of new infinite series is given. Ramanujan-like formulas for pi are also derived based on elementary inverse trigonometric functions, including a formula with rational values that adds 2.5 digits per terms, which makes the series much more compact than similar formulas in the existing literature.
Paper Structure (6 sections, 61 equations)

This paper contains 6 sections, 61 equations.

Table of Contents

  1. Introduction
  2. Derivation of Algorithm
  3. Deriving Formulas for Pi
  4. Multivariable Values of P(x) and Q(x) for Infinite Series
  5. Hypergeometric Function Transformations for the Gamma Function
  6. Higher-Order Constants