Lauricella hypergeometric series

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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) = 
\sum_{i_1,i_2,i_3=0}^{\infty} \frac{(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) = 
\sum_{i_1,i_2,i_3=0}^{\infty} \frac{(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) = 
\sum_{i_1,i_2,i_3=0}^{\infty} \frac{(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) = 
\sum_{i_1,i_2,i_3=0}^{\infty} \frac{(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 power of q, i.e.

(q)_{i} = \frac{\Gamma(q+i)} {\Gamma(q)} = q\,(q+1) \cdots (q+i-1).

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[edit]

These functions can be straightforwardly extended to n variables. One writes for example

F_A^{(n)}(a, b_1,\ldots,b_n, c_1,\ldots,c_n; x_1,\ldots,x_n) = 
\sum_{i_1,\ldots,i_n=0}^{\infty} \frac{(a)_{i_1+\ldots+i_n} (b_1)_{i_1} \cdots (b_n)_{i_n}} {(c_1)_{i_1} \cdots (c_n)_{i_n} \,i_1! \cdots \,i_n!} \,x_1^{i_1} \cdots 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)} \equiv F_2 ,\quad F_B^{(2)} \equiv F_3 ,\quad F_C^{(2)} \equiv F_4 ,\quad F_D^{(2)} \equiv F_1.

When n = 1, all four functions reduce to the Gauss hypergeometric function:

F_A^{(1)}(a,b,c;x) \equiv F_B^{(1)}(a,b,c;x) \equiv F_C^{(1)}(a,b,c;x) \equiv F_D^{(1)}(a,b,c;x) \equiv {_2}F_1(a,b;c;x).

Integral representation of FD[edit]

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,\ldots,b_n, c; x_1,\ldots,x_n) = 
\frac{\Gamma(c)} {\Gamma(a) \Gamma(c-a)} \int_0^1 t^{a-1} (1-t)^{c-a-1} (1-x_1t)^{-b_1} \cdots (1-x_nt)^{-b_n} \,\mathrm{d}t, \quad \real \,c > \real \,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:

\Pi(n,\phi,k) = 
\int_0^{\phi} \frac{\mathrm{d} \theta} {(1 - n \sin^2 \theta) \sqrt{1 - k^2 \sin^2 \theta}} = 
\sin \phi \,F_D^{(3)}(\tfrac 1 2, 1, \tfrac 1 2, \tfrac 1 2, \tfrac 3 2; n \sin^2 \phi, \sin^2 \phi, k^2 \sin^2 \phi), \quad |\real \,\phi| < \frac \pi 2 ~.


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