PrecisionLauricella: Package for numerical computation of Lauricella functions depending on a parameter

Abstract

We introduce the PrecisionLauricella package, a computational tool developed in Wolfram Mathematica for high-precision numerical evaluations of the Laurent expansion coefficients of Lauricella functions whose parameters depend linearly on a small regulator, ε. In practical multi-loop calculations, Lauricella functions are required only as series around $\epsilon =0$, and PrecisionLauricella is designed specifically to deliver such coefficients with arbitrary precision. The package leverages a method based on analytic 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 ε-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 PrecisionLauricella package.

Program summary

Program Title: PrecisionLauricella

CPC Library link to program files: https://doi.org/10.17632/6f958yz2dr.1

Developer's repository link: https://bitbucket.org/BezuglovMaxim/precisionlauricella-package/src/main/

Licensing provisions: GPLv3

Programming language: Wolfram Mathematica

Supplementary material: PrecisionLauricella_Examples.nb

Nature of problem: Lauricella functions, generalizations of hypergeometric functions, appearing in physics and mathematics, including Feynman integrals and string theory. When their indices depend linearly on a small parameter ε, their numerical evaluation becomes challenging due to the complexity of high-dimensional series and singularities. Traditional methods, like hypergeometric re-expansion or Mellin–Barnes integrals, often lack efficiency and precision.

Managing multi-dimensional sums exacerbates computational costs, especially for high-precision requirements, making these approaches unsuitable for many practical applications. Thus, there is a pressing need for efficient, scalable methods capable of maintaining numerical accuracy and effectively handling parameter dependencies.

Solution method: Our method uses the Frobenius approach to achieve analytic continuations of Lauricella functions through generalized power series. Representing the functions as one-dimensional series simplifies high-precision numerical evaluations compared to traditional methods relying on multi-dimensional expansions or Mellin–Barnes integrals.

We further optimize calculations by reconstructing ε dependencies from evaluations at specific values, enabling efficient parallelization and reducing computational costs.

A comprehensive mathematical exposition of the method is provided in our previous work [1].

References

• [1]
  M. Bezuglov, B. Kniehl, A. Onishchenko, O. Veretin, High-precision numerical evaluation of Lauricella functions, arXiv:2502.03276, 2 2025.