In 1893 Giuseppe Lauricella defined and studied four hypergeometric series FA, FB, FC, FD of three variables. They are (Lauricella 1893):
F A ( 3 ) ( a , b 1 , b 2 , b 3 , c 1 , c 2 , c 3 ; x 1 , x 2 , x 3 ) = ∑ i 1 , i 2 , i 3 = 0 ∞ ( a ) i 1 + i 2 + i 3 ( b 1 ) i 1 ( b 2 ) i 2 ( b 3 ) i 3 ( c 1 ) i 1 ( c 2 ) i 2 ( c 3 ) i 3 i 1 ! i 2 ! i 3 ! x 1 i 1 x 2 i 2 x 3 i 3 for |x1| + |x2| + |x3| < 1 and
F B ( 3 ) ( a 1 , a 2 , a 3 , b 1 , b 2 , b 3 , c ; x 1 , x 2 , x 3 ) = ∑ i 1 , i 2 , i 3 = 0 ∞ ( a 1 ) i 1 ( a 2 ) i 2 ( a 3 ) i 3 ( b 1 ) i 1 ( b 2 ) i 2 ( b 3 ) i 3 ( c ) i 1 + i 2 + i 3 i 1 ! i 2 ! i 3 ! x 1 i 1 x 2 i 2 x 3 i 3 for |x1| < 1, |x2| < 1, |x3| < 1 and
F C ( 3 ) ( a , b , c 1 , c 2 , c 3 ; x 1 , x 2 , x 3 ) = ∑ i 1 , i 2 , i 3 = 0 ∞ ( a ) i 1 + i 2 + i 3 ( b ) i 1 + i 2 + i 3 ( c 1 ) i 1 ( c 2 ) i 2 ( c 3 ) i 3 i 1 ! i 2 ! i 3 ! x 1 i 1 x 2 i 2 x 3 i 3 for |x1|½ + |x2|½ + |x3|½ < 1 and
F D ( 3 ) ( a , b 1 , b 2 , b 3 , c ; x 1 , x 2 , x 3 ) = ∑ i 1 , i 2 , i 3 = 0 ∞ ( a ) i 1 + i 2 + i 3 ( b 1 ) i 1 ( b 2 ) i 2 ( b 3 ) i 3 ( c ) i 1 + i 2 + i 3 i 1 ! i 2 ! i 3 ! x 1 i 1 x 2 i 2 x 3 i 3 for |x1| < 1, |x2| < 1, |x3| < 1. Here the Pochhammer symbol (q)i indicates the i-th rising factorial of q, i.e.
( q ) i = q ( q + 1 ) ⋯ ( q + i − 1 ) = Γ ( q + i ) Γ ( q ) , where the second equality is true for all complex q except q = 0 , − 1 , − 2 , … .
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
F A ( n ) ( a , b 1 , … , b n , c 1 , … , c n ; x 1 , … , x n ) = ∑ i 1 , … , i n = 0 ∞ ( a ) i 1 + … + i n ( b 1 ) i 1 ⋯ ( b n ) i n ( c 1 ) i 1 ⋯ ( c n ) i n i 1 ! ⋯ i n ! x 1 i 1 ⋯ x n i n , 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:
F A ( 2 ) ≡ F 2 , F B ( 2 ) ≡ F 3 , F C ( 2 ) ≡ F 4 , F D ( 2 ) ≡ F 1 . When n = 1, all four functions reduce to the Gauss hypergeometric function:
F A ( 1 ) ( a , b , c ; x ) ≡ F B ( 1 ) ( a , b , c ; x ) ≡ F C ( 1 ) ( a , b , c ; x ) ≡ F D ( 1 ) ( a , b , c ; x ) ≡ 2 F 1 ( a , b ; c ; x ) . 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:
F D ( n ) ( a , b 1 , … , b n , c ; x 1 , … , x n ) = Γ ( c ) Γ ( a ) Γ ( c − a ) ∫ 0 1 t a − 1 ( 1 − t ) c − a − 1 ( 1 − x 1 t ) − b 1 ⋯ ( 1 − x n t ) − b n d t , Re c > Re a > 0 . 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:
Π ( n , ϕ , k ) = ∫ 0 ϕ d θ ( 1 − n sin 2 θ ) 1 − k 2 sin 2 θ = sin ( ϕ ) F D ( 3 ) ( 1 2 , 1 , 1 2 , 1 2 , 3 2 ; n sin 2 ϕ , sin 2 ϕ , k 2 sin 2 ϕ ) , | Re ϕ | < π 2 . 