In 1893 Giuseppe Lauricella defined and studied four hypergeometric series FA, FB, FC, FD of three variables.
They are (Lauricella 1893): for |x1| + |x2| + |x3| < 1 and for |x1| < 1, |x2| < 1, |x3| < 1 and for |x1|1/2 + |x2|1/2 + |x3|1/2 < 1 and 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
q
{\displaystyle q}
.
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.
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.
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 F1, Lauricella's FD 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 Π is a special case of Lauricella's function FD with three variables: Case 1 :
{\displaystyle a-c}
a positive integer One can relate FD to the Carlson R function
{\displaystyle F_{D}(a,{\overline {b}},c,{\overline {z}})=R_{a-c}({\overline {b^{*}}},{\overline {z^{*}}})\cdot \prod _{i}(z_{i}^{*})^{b_{i}^{*}}={\frac {\Gamma (a-c+1)\Gamma (b^{*})}{\Gamma (a-c+b^{*})}}\cdot D_{a-c}({\overline {b^{*}}},{\overline {z^{*}}})\cdot \prod _{i}(z_{i}^{*})^{b_{i}^{*}}}
with the iterative sum
{\displaystyle D_{n}({\overline {b^{*}}},{\overline {z^{*}}})={\frac {1}{n}}\sum _{k=1}^{n}\left(\sum _{i=1}^{N}b_{i}^{*}\cdot (z_{i}^{*})^{k}\right)\cdot D_{k-i}}
where it can be exploited that the Carlson R function with
has an exact representation (see [1] for more information).
The vectors are defined as
where the length of
, while the vectors
have length
Case 2:
{\displaystyle c-a}
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 [2] for more information.