Lauricella hypergeometric series

In 1893 Giuseppe Lauricella defined and studied four hypergeometric series FA, FB, FC, FD of three variables. They are :



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|1/2 + |x2|1/2 + |x3|1/2 < 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 of q, i.e.


 * $$(q)_i = q\,(q+1) \cdots (q+i-1) = \frac{\Gamma(q+i)}{\Gamma(q)}~,$$

where the second equality is true for all complex $$q$$ except $$q=0,-1,-2,\ldots$$. 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. 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



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+\cdots+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
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, \qquad \operatorname{Re} c > \operatorname{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:



\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), \qquad |\operatorname{Re} \phi| < \frac{\pi}{2} ~. $$

Finite-sum solutions of FD
Case 1 : $$a>c$$, $$a-c$$ a positive integer

One can relate FD to the Carlson R function $$R_n$$ via

$$ F_D(a,\overline{b},c,\overline{z})=R_{a-c}(\overline{b^*}, \overline{z^*}) \cdot \prod_i (z_i^*)^{b_i^*} = \frac{\Gamma(a-c+1)\Gamma(b^*)}{\Gamma(a-c+b^*)} \cdot D_{a-c}(\overline{b^*}, \overline{z^*}) \cdot \prod_i (z_i^*)^{b_i^*} $$

with the iterative sum

$$D_n(\overline{b^*}, \overline{z^*})=\frac{1}{n} \sum_{k=1}^{n} \left(\sum_{i=1}^{N} b_i^* \cdot (z_i^*)^k\right) \cdot D_{k-i}$$ and $$D_0=1$$

where it can be exploited that the Carlson R function with $$n>0$$ has an exact representation (see for more information).

The vectors are defined as

$$\overline{b^*}=[\overline{b}, c-\sum_i b_i]$$

$$\overline{z^*}=[\frac{1}{1-z_1}, \ldots, \frac{1}{1-z_{N-1}}, 1]$$

where the length of $$\overline{z}$$ and $$\overline{b}$$ is $$N-1$$, while the vectors $$\overline{z^*}$$ and $$\overline{b^*}$$ have length $$N$$.

Case 2: $$c>a$$, $$c-a$$ 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 for more information.