High-precision numerical evaluation of Lauricella functions

We present a method for high-precision numerical evaluations of Lauricella functions whose indices are linearly dependent on some parameter ε 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 ε dependence of the result is reconstructed from the evaluations of given Lauricella functions at specific numerical values of ε , which, in addition, allows for efficient parallel implementation. The method has been implemented in the PrecisionLauricella package, written in Wolfram Mathematica language.