In 1893 Giuseppe Lauricella defined and studied four hypergeometric series F<sub>A</sub>, F<sub>B</sub>, F<sub>C</sub>, F<sub>D</sub> of three variables. They are :
for |x<sub>1</sub>| + |x<sub>2</sub>| + |x<sub>3</sub>| < 1 and
for |x<sub>1</sub>| < 1, |x<sub>2</sub>| < 1, |x<sub>3</sub>| < 1 and
for |x<sub>1</sub>|<sup>1/2</sup> + |x<sub>2</sub>|<sup>1/2</sup> + |x<sub>3</sub>|<sup>1/2</sup> < 1 and
for |x<sub>1</sub>| < 1, |x<sub>2</sub>| < 1, |x<sub>3</sub>| < 1. Here the Pochhammer symbol (q)<sub>i</sub> 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 x<sub>1</sub>, x<sub>2</sub>, x<sub>3</sub> by means of analytic continuation.
Lauricella also indicated the existence of ten other hypergeometric functions of three variables. These were named F<sub>E</sub>, F<sub>F</sub>, ..., F<sub>T</sub> and studied by Shanti Saran in 1954 . There are therefore a total of 14 LauricellaâÂÂSaran hypergeometric functions.
These functions can be straightforwardly extended to n variables. One writes for example
where |x<sub>1</sub>| + ... + |x<sub>n</sub>| < 1. These generalized series too are sometimes referred to as Lauricella functions.
When n = 2, the Lauricella functions correspond to the Appell hypergeometric series of two variables:
When n = 1, all four functions reduce to the Gauss hypergeometric function:
In analogy with Appell's function F<sub>1</sub>, Lauricella's F<sub>D</sub> can be written as a one-dimensional Euler-type integral for any number n of variables:
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 ÃÂ is a special case of Lauricella's function F<sub>D</sub> with three variables:
Case 1 : , a positive integer
One can relate F<sub>D</sub> to the Carlson R function via
with the iterative sum
and
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.