Computing hypergeometric functions rigorously

TL;DR

This paper introduces an efficient implementation of hypergeometric functions in arbitrary-precision interval arithmetic, supporting a wide range of special functions with high accuracy and performance, enabling rigorous numerical computations.

Contribution

The authors develop a comprehensive, high-performance implementation of hypergeometric functions supporting complex parameters, extending to many related special functions, with rigorous error bounds.

Findings

Supports complex parameters and arguments for various hypergeometric functions.

Achieves performance comparable or superior to existing software, often much faster.

Enables rigorous interval computations and accurate floating-point approximations.

Abstract

We present an efficient implementation of hypergeometric functions in arbitrary-precision interval arithmetic. The functions ${}_0F_1$, ${}_1F_1$, ${}_2F_1$ and ${}_2F_0$ (or the Kummer $U$-function) are supported for unrestricted complex parameters and argument, and by extension, we cover exponential and trigonometric integrals, error functions, Fresnel integrals, incomplete gamma and beta functions, Bessel functions, Airy functions, Legendre functions, Jacobi polynomials, complete elliptic integrals, and other special functions. The output can be used directly for interval computations or to generate provably correct floating-point approximations in any format. Performance is competitive with earlier arbitrary-precision software, and sometimes orders of magnitude faster. We also partially cover the generalized hypergeometric function ${}_pF_q$ and computation of high-order parameter derivatives.