High-precision numerical evaluation of Lauricella functions

TL;DR

This paper introduces a high-precision numerical method for evaluating Lauricella functions using Laurent series expansions and analytic continuations, optimized for parallel computation and implemented in Mathematica.

Contribution

The paper develops a novel approach for high-precision evaluation of Lauricella functions via Laurent series and analytic continuation, enabling efficient parallel computation.

Findings

Method achieves high-precision evaluations of Lauricella functions.

Implementation in Mathematica package facilitates practical computations.

Approach outperforms multi-dimensional sum methods in speed and accuracy.

Abstract

We present a method for high-precision numerical evaluations of Lauricella functions, whose indices are linearly dependent on some parameter $\varepsilon$, in terms of their Laurent series expansions at zero. This method is based on finding analytic continuations of these functions in terms of Frobenius generalized power series. Being one-dimensional, these series are much more suited for high-precision numerical evaluations than multi-dimensional sums arising in approaches to analytic continuations based on re-expansions of hypergeometric series or Mellin--Barnes integral representations. To accelerate the calculation procedure further, the $\varepsilon$ dependence of the result is reconstructed from the evaluations of given Lauricella functions at specific numerical values of $\varepsilon$, which, in addition, allows for efficient parallel implementation. The method has been implemented in the $\texttt{PrecisionLauricella}$ package, written in Wolfram Mathematica language.