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

Abstract

We introduce the $\text{PrecisionLauricella}$ package, a computational tool developed in Wolfram Mathematica for high-precision numerical evaluations of Lauricella functions with indices linearly dependent on a parameter, $\epsilon$. 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 $\epsilon$-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 $\text{PrecisionLauricella}$ package.

Comments: 21 pages, 6 figures. arXiv admin note: text overlap with arXiv:2502.03276