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

TL;DR

The paper tackles the need for high-precision numerical evaluation and Laurent expansion in $\varepsilon$ of Lauricella functions with indices linearly dependent on the regulator. It introduces PrecisionLauricella, a Mathematica package that employs analytic continuation based on Frobenius generalized power series to create one-dimensional series along a continuation path, avoiding expensive multi-dimensional or Mellin–Barnes representations. This approach enables accurate $\varepsilon$-dependent reconstructions from evaluations on an $\varepsilon$ grid and supports parallel computation. The authors demonstrate the package's robustness and efficiency, comparing it to existing hypergeometric tools and highlighting its potential for large-scale applications in quantum field theory and related areas of mathematical physics.

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.

Figures (6)

• Figure 1: Flowchart for the PrecisionLauricella package.
• Figure 2: On the left, one of the possible paths with complex singular points is shown. The singular points are shown in red, the end point in green, the staring point is at the origin, and shaded circles represent convergence regions. Horizontal dotted lines indicate branch cuts of singular points. On the right, the intersection graph of the expansion regions is displayed. The nodes in the intersection graph include expansion regions at regular points. The nodes and edges highlighted in red show the found path of analytic continuation. Note also, that the intersection graph may have several connected components.
• Figure 3: On the left, one of the possible paths with complex singular points is shown. The singular points are shown in red, the end point in green, the staring point is at the origin, and shaded circles represent convergence regions. Horizontal dotted lines indicate branch cuts of singular points. On the right, the intersection graph of the expansion regions is displayed. The nodes in the intersection graph include expansion regions at regular points. The nodes and edges highlighted in red show the found path of analytic continuation. Note also, that the intersection graph may have several connected components.
• Figure 4: Analytical continuation with option $\texttt{DeltaPrescription} \rightarrow -I$ on the left and option $\texttt{DeltaPrescription} \rightarrow +I$ on the right. The singular points are shown in red and the end point in green, the staring point is at the origin and shaded circles represent convergence regions. Horizontal dotted lines indicate branch cuts of singular points.
• Figure 5: Average time in seconds required to expand $F_1\left(\frac{1}{2};1,\varepsilon;\frac{3}{2};\frac{4}{3},\frac{7}{4}\right)$ using 16 parallel kernels on the left and 8 parallel kernels on the right. The horizontal axis shows the number of terms in $\varepsilon$, and the vertical axis shows the evaluation time in seconds. Solid lines represent the best linear fit.