2

Does anyone know how to calculate the Lauricella hypergeometric function of type A with multiple variables by using Matlab?

I saw in a paper that it's a function that can be computed directly by using a software supplied by Exton (2007). But I didn't find anything that is useful.

The Lauricella function of type A is given by

$F_{A}^{(n)}(a,b_{1},\ldots ,b_{n},c_{1},\ldots ,c_{n};x_{1},\ldots ,x_{n})=\sum _{i_{1},\ldots ,i_{n}=0}^{\infty }{\frac {(a)_{i_{1}+\ldots +i_{n}}(b_{1})_{i_{1}}\cdots (b_{n})_{i_{n}}}{(c_{1})_{i_{1}}\cdots (c_{n})_{i_{n}}\,i_{1}!\cdots \,i_{n}!}}\,x_{1}^{i_{1}}\cdots x_{n}^{i_{n}}~$

https://en.wikipedia.org/wiki/Lauricella_hypergeometric_series

Does anyone knows? A lot of thanks!

mingzhanzhang
  • 81

1 Answers1

1

See equation 35 in

Chamayou, Jean-Fran\c cois; Weso{\l}owski, Jacek, Lauricella and Humbert functions through probabilistic tools, Integral Transforms Spec. Funct. 20, No. 7-8, 529-538 (2009). ZBL1232.33022.

(available online)

This gives a one-dimensional integral representation, which is well-suited for numeric computation.

Igor Rivin
  • 95,560
  • Thanks. The Eq. (34) is a one-dimensional integral representation and is good for numerical computation. But, the integral runs from 0 to +$\infty$ and it seems not so efficient when the number of variables is large, such as 3 or more. I am wondering whether there exist methods that involves no integrals? – mingzhanzhang May 21 '17 at 04:57