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 :

:$F_A^(a,b_1,b_2,b_3,c_1,c_2,c_3;x_1,x_2,x_3) = \sum_^ \frac \,x_1^x_2^x_3^$

for , ''x''₁, + , ''x''₂, + , ''x''₃, < 1 and

:$F_B^(a_1,a_2,a_3,b_1,b_2,b_3,c;x_1,x_2,x_3) = \sum_^ \frac \,x_1^x_2^x_3^$

for , ''x''₁, < 1, , ''x''₂, < 1, , ''x''₃, < 1 and

:$F_C^(a,b,c_1,c_2,c_3;x_1,x_2,x_3) = \sum_^ \frac \,x_1^x_2^x_3^$

for , ''x''₁, ^½ + , ''x''₂, ^½ + , ''x''₃, ^½ < 1 and

:$F_D^(a,b_1,b_2,b_3,c;x_1,x_2,x_3) = \sum_^ \frac \,x_1^x_2^x_3^$

for , ''x''₁, < 1, , ''x''₂, < 1, , ''x''₃, < 1. Here the Pochhammer symbol (''q'')_''i'' indicates the ''i''-th rising factorial of ''q'', i.e.

:$(q)_i = q\,(q+1) \cdots (q+i-1) = \frac~,$

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 ''x''₁, ''x''₂, ''x''₃ 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 . 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

:$F_A^(a, b_1,\ldots,b_n, c_1,\ldots,c_n; x_1,\ldots,x_n) = \sum_^ \frac \,x_1^ \cdots x_n^ ~,$

where , ''x''₁, + ... + , ''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:

:$F_A^ \equiv F_2 ,\quad F_B^ \equiv F_3 ,\quad F_C^ \equiv F_4 ,\quad F_D^ \equiv F_1.$

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

:$F_A^(a,b,c;x) \equiv F_B^(a,b,c;x) \equiv F_C^(a,b,c;x) \equiv F_D^(a,b,c;x) \equiv F_1(a,b;c;x).$

Integral representation of ''F''_''D''

In analogy with Appell's function ''F''₁, Lauricella's ''F''_''D'' can be written as a one-dimensional Euler-type integral for any number ''n'' of variables:

:$F_D^(a, b_1,\ldots,b_n, c; x_1,\ldots,x_n) = \frac \int_0^1 t^ (1-t)^ (1-x_1t)^ \cdots (1-x_nt)^ \,\mathrmt, \qquad \operatorname c > \operatorname a > 0 ~.$

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 ''F''_''D'' with three variables:

:$\Pi(n,\phi,k) = \int_0^ \frac = \sin (\phi) \,F_D^(\tfrac 1 2, 1, \tfrac 1 2, \tfrac 1 2, \tfrac 3 2; n \sin^2 \phi, \sin^2 \phi, k^2 \sin^2 \phi), \qquad , \operatorname \phi, < \frac ~.$

Finite-sum solutions of ''F''_''D''

Case 1 : $a>c$, $a-c$ a positive integer

One can relate ''F''_''D'' to the Carlson R function $R_n$ via

$F_D(a,\overline,c,\overline)=R_(\overline, \overline) \cdot \prod_i (z_i^*)^ = \frac \cdot D_(\overline, \overline) \cdot \prod_i (z_i^*)^$

with the iterative sum

$D_n(\overline, \overline)=\frac \sum_^ \left(\sum_^ b_i^* \cdot (z_i^*)^k\right) \cdot D_$ 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= overline, c-\sum_i b_i/math>

\overline= frac, \ldots, \frac, 1/math>

where the length of $\overline$ and $\overline$ is $N-1$, while the vectors $\overline$ and $\overline$ 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.

References

* (see p. 114)
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* (corrigendum 1956 in ''Ganita'' 7, p. 65)
* (there is a 2008 paperback with )
* (there is another edition with )

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{{series (mathematics)
Hypergeometric functions
Mathematical series