for |x1| < 1, |x2| < 1, |x3| < 1. Here the Pochhammer symbol (q)i indicates the i-th rising factorial of q, i.e.
where the second equality is true for all complex except .
These functions can be extended to other values of the variables x1, x2, x3 by means of analytic continuation.
Lauricella also indicated the existence of ten other hypergeometric functions of three variables. These were named FE, FF, ..., FT and studied by Shanti Saran in 1954 (Saran 1954). There are therefore a total of 14 Lauricella–Saran hypergeometric functions.
Generalization to n variablesEdit
These functions can be straightforwardly extended to n variables. One writes for example
where |x1| + ... + |xn| < 1. These generalized series too are sometimes referred to as Lauricella functions.
This representation can be easily verified by means of Taylor expansion of the integrand, followed by termwise integration. The representation implies that the incomplete elliptic integral Π is a special case of Lauricella's function FD with three variables:
where it can be exploited that the Carlson R function with has an exact representation (see  for more information).
The vectors are defined as
where the length of and is , while the vectors and have length .
Case 2: , a positive integer
In this case there is also a known analytic form, but it is rather complicated to write down and involves several steps.
See  for more information.
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^Tan, J.; Zhou, P. (2005). "On the finite sum representations of the Lauricella functions FD". Advances in Computational Mathematics. 23 (4): 333–351. doi:10.1007/s10444-004-1838-0. S2CID 7515235.
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