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 variables
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.
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:
^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.
S2CID7515235.
Srivastava, Hari M.; Karlsson, Per W. (1985). Multiple Gaussian hypergeometric series. Mathematics and its applications. Chichester, UK: Halsted Press, Ellis Horwood Ltd.
ISBN0-470-20100-2.
MR0834385. (there is another edition with
ISBN0-85312-602-X)
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 variables
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.
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:
^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.
S2CID7515235.
Srivastava, Hari M.; Karlsson, Per W. (1985). Multiple Gaussian hypergeometric series. Mathematics and its applications. Chichester, UK: Halsted Press, Ellis Horwood Ltd.
ISBN0-470-20100-2.
MR0834385. (there is another edition with
ISBN0-85312-602-X)