Lauricella hypergeometric series

In 1893 Giuseppe Lauricella defined and studied four hypergeometric series F_A, F_B, F_C, F_D of three variables. They are (Lauricella 1893):

[math]\displaystyle{ 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} }[/math]

for |x₁| + |x₂| + |x₃| < 1 and

[math]\displaystyle{ 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} }[/math]

for |x₁| < 1, |x₂| < 1, |x₃| < 1 and

[math]\displaystyle{ 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} }[/math]

for |x₁|^½ + |x₂|^½ + |x₃|^½ < 1 and

[math]\displaystyle{ 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} }[/math]

for |x₁| < 1, |x₂| < 1, |x₃| < 1. Here the Pochhammer symbol (q)_i indicates the i-th rising factorial of q, i.e.

[math]\displaystyle{ (q)_i = q\,(q+1) \cdots (q+i-1) = \frac{\Gamma(q+i)}{\Gamma(q)}~, }[/math]

where the second equality is true for all complex [math]\displaystyle{ q }[/math] except [math]\displaystyle{ q=0,-1,-2,\ldots }[/math].

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 F_E, F_F, ..., F_T 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

[math]\displaystyle{ 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} ~, }[/math]

where |x₁| + ... + |x_n| < 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:

[math]\displaystyle{ 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. }[/math]

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

[math]\displaystyle{ 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). }[/math]

Integral representation of F_D

In analogy with Appell's function F₁, Lauricella's F_D can be written as a one-dimensional Euler-type integral for any number n of variables:

[math]\displaystyle{ 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 \gt \operatorname{Re} a \gt 0 ~. }[/math]

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 F_D with three variables:

[math]\displaystyle{ \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| \lt \frac{\pi}{2} ~. }[/math]

Finite-sum solutions of F_D

Case 1 : [math]\displaystyle{ a\gt c }[/math], [math]\displaystyle{ a-c }[/math] a positive integer

One can relate F_D to the Carlson R function [math]\displaystyle{ R_n }[/math] via

[math]\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^*} }[/math]

with the iterative sum

[math]\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} }[/math] and [math]\displaystyle{ D_0=1 }[/math]

where it can be exploited that the Carlson R function with [math]\displaystyle{ n\gt 0 }[/math] has an exact representation (see for more information).

The vectors are defined as

[math]\displaystyle{ \overline{b^*}=[\overline{b}, c-\sum_i b_i] }[/math]

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

where the length of [math]\displaystyle{ \overline{z} }[/math] and [math]\displaystyle{ \overline{b} }[/math] is [math]\displaystyle{ N-1 }[/math], while the vectors [math]\displaystyle{ \overline{z^*} }[/math] and [math]\displaystyle{ \overline{b^*} }[/math] have length [math]\displaystyle{ N }[/math].

Case 2: [math]\displaystyle{ c\gt a }[/math], [math]\displaystyle{ c-a }[/math] 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.

References

• Appell, Paul; Kampé de Fériet, Joseph (1926) (in French). Fonctions hypergéométriques et hypersphériques; Polynômes d'Hermite. Paris: Gauthier–Villars.  (see p. 114)
• Exton, Harold (1976). Multiple hypergeometric functions and applications. Mathematics and its applications. Chichester, UK: Halsted Press, Ellis Horwood Ltd.. ISBN 0-470-15190-0. 
• Lauricella, Giuseppe (1893). "Sulle funzioni ipergeometriche a più variabili" (in Italian). Rendiconti del Circolo Matematico di Palermo 7 (S1): 111–158. doi:10.1007/BF03012437. 
• Saran, Shanti (1954). "Hypergeometric Functions of Three Variables". Ganita 5 (1): 77–91. ISSN 0046-5402.  (corrigendum 1956 in Ganita 7, p. 65)
• Slater, Lucy Joan (1966). Generalized hypergeometric functions. Cambridge, UK: Cambridge University Press. ISBN 0-521-06483-X. https://archive.org/details/generalizedhyper0000unse.  (there is a 2008 paperback with ISBN 978-0-521-09061-2)
• Srivastava, Hari M.; Karlsson, Per W. (1985). Multiple Gaussian hypergeometric series. Mathematics and its applications. Chichester, UK: Halsted Press, Ellis Horwood Ltd.. ISBN 0-470-20100-2.  (there is another edition with ISBN 0-85312-602-X)

• Ronald M. Aarts. "Lauricella Functions". http://mathworld.wolfram.com/LauricellaFunctions.html.