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Let's consider the Lauricella function of third kind, denote as $F_{C}(a,b;c_1,...,c_n;x_1,...x_n)$ in MathWorld. Is anyone aware of an algorithm that allows to numerically evaluate such a function for the case $n=3$ ?

I tried to have a look at the literature on the topic, but I did not find anything (but I'm not an expert :-)).

Many thanks

J. M.'s missing motivation
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Alessandro Pini
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    Hi, have you looked at this answer? J.M has a thorough analysis on these functions https://mathematica.stackexchange.com/questions/154434/evaluating-lauricella-functions-in-mathematica –  Jan 31 '20 at 15:46
  • Thanks, I read that answer. Unfortunately it discusses only the case of a Lauricella function of fourth kind (at least that's my level of understanding). – Alessandro Pini Jan 31 '20 at 15:51
  • I am not an expert on the matter, but if I am not confused myself too much, I think that the approach there should also work for the case n=3. For the three variate case, do you know any results against which I can check and let you know? –  Jan 31 '20 at 15:58
  • Oh I see, then I will try. Unfortunately I'm not aware of any known results :-( – Alessandro Pini Jan 31 '20 at 16:05
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    What I did in that other answer is indeed only applicable to $F_D$; for $F_C$, a more general procedure like Cuyt's transformation (used here) would need to be employed. – J. M.'s missing motivation Jan 31 '20 at 16:06
  • I see, many thanks :-) – Alessandro Pini Jan 31 '20 at 16:20

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