Numerical Methods for the Computation of the Confluent and Gauss Hypergeometric Functions

TL;DR

This paper reviews various numerical techniques for accurately and efficiently computing confluent and Gauss hypergeometric functions across different parameter regimes, providing practical guidelines for method selection.

Contribution

It offers a comprehensive comparison of methods and practical recommendations for computing hypergeometric functions in various scenarios.

Findings

Taylor and asymptotic series are effective in certain regimes

Gauss-Jacobi quadrature is suitable for specific parameter ranges

Recurrence relations and differential equation solutions are also evaluated

Abstract

The two most commonly used hypergeometric functions are the confluent hypergeometric function and the Gauss hypergeometric function. We review the available techniques for accurate, fast, and reliable computation of these two hypergeometric functions in different parameter and variable regimes. The methods that we investigate include Taylor and asymptotic series computations, Gauss-Jacobi quadrature, numerical solution of differential equations, recurrence relations, and others. We discuss the results of numerical experiments used to determine the best methods, in practice, for each parameter and variable regime considered. We provide 'roadmaps' with our recommendation for which methods should be used in each situation.