The Gamma Function via Interpolation

TL;DR

This paper introduces a novel interpolation-based computational framework for evaluating the gamma function across the complex plane, leveraging rational functions and adaptive algorithms for improved efficiency.

Contribution

It generalizes classical gamma function approximation methods by incorporating interpolation and adaptive rational approximation techniques.

Findings

The framework generalizes Lanczos and Spouge methods as interpolatory.

Adaptive AAA algorithm yields near-optimal rational approximations.

Approximations are competitive with Stirling's formula in efficiency.

Abstract

A new computational framework for evaluation of the gamma function $\Gamma(z)$ over the complex plane is developed. The algorithm is based on interpolation by rational functions, and generalizes the classical methods of Lanczos \cite{Lanczos} and Spouge \cite{Spouge} (which we show are also interpolatory). This framework utilizes the exact poles of the gamma function. By relaxing this condition and allowing the poles to vary, a near-optimal rational approximation is possible, which is demonstrated using the adaptive Antoulous Anderson (AAA) algorithm, developed in \cite{AAA,AAA_2020}. The resulting approximations are competitive with Stirling's formula in terms of overall efficiency.