$\texttt{PrecisionLauricella}$: package for numerical computation of Lauricella functions depending on a parameter

TL;DR

The paper presents PrecisionLauricella, a Mathematica package for high-precision numerical evaluation of Lauricella functions with parameter-dependent indices, using an efficient analytical continuation method based on Frobenius series.

Contribution

It introduces a novel one-dimensional approach for computing Lauricella functions that improves efficiency and accuracy over traditional multi-dimensional series or Mellin--Barnes methods.

Findings

Enables high-precision evaluations of Lauricella functions.

Offers an efficient alternative to traditional multi-dimensional methods.

Facilitates parallel computation through epsilon-dependent reconstruction.

Abstract

We introduce the $\texttt{PrecisionLauricella}$ package, a computational tool developed in Wolfram Mathematica for high-precision numerical evaluations of Lauricella functions with indices linearly dependent on a parameter, $\varepsilon$. The package leverages a method based on analytical continuation via Frobenius generalized power series, providing an efficient and accurate alternative to conventional approaches relying on multi-dimensional series expansions or Mellin--Barnes representations. This one-dimensional approach is particularly advantageous for high-precision calculations and facilitates further optimization through $\varepsilon$-dependent reconstruction from evaluations at specific numerical values, enabling efficient parallelization. The underlying mathematical framework for this method has been detailed in our previous work, while the current paper focuses on the design, implementation, and practical applications of the $\texttt{PrecisionLauricella}$ package.