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arXiv · 2502.03276

High-precision numerical evaluation of Lauricella functions

M. A. Bezuglov·B. A. Kniehl·A. I. Onishchenko·O. L. Veretin
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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.

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M. A. Bezuglov, B. A. Kniehl, A. I. Onishchenko, O. L. Veretin. 2025-02-05. High-precision numerical evaluation of Lauricella functions. https://arxiv.org/abs/2502.03276

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