Chebychev interpolations of the Gamma and Polygamma Functions and their analytical properties

TL;DR

This paper presents Chebyshev polynomial approximations for the Gamma and Polygamma functions over [1, infinity], achieving high precision with minimal coefficients and providing insights into their analytical properties.

Contribution

It introduces efficient Chebyshev approximations for the Gamma and Polygamma functions with high accuracy and analyzes their analytical properties over a continuous interval.

Findings

Achieved 100-digit precision with about three coefficients per decimal.

Provided Chebyshev approximations valid on [1, infinity].

Analyzed the functions' analytical properties.

Abstract

Chebychev approximations are given for the Gamma and the Polygamma functions in only one contiguous intervall [1..inf] with a definable maximal relative error. The approximations need about three coefficients per decimal until a checked precision of 100 decimal digits.