Lauricella hypergeometric series

Template:Short description In 1893 Giuseppe Lauricella defined and studied four hypergeometric series F_A, F_B, F_C, F_D of three variables. They are Template:Harv:

$F_A^{(3)}(a,b_1,b_2,b_3,c_1,c_2,c_3;x_1,x_2,x_3)=\sum _{i_1,i_2,i_3=0}^{\mathrm{\infty }}\frac{(a{)}_{i_1+i_2+i_3}(b_1{)}_{i_1}(b_2{)}_{i_2}(b_3{)}_{i_3}}{(c_1{)}_{i_1}(c_2{)}_{i_2}(c_3{)}_{i_3}{i}_1!i_2!i_3!}{x}_1^{i_1}{x}_2^{i_2}{x}_3^{i_3}$

for |x₁| + |x₂| + |x₃| < 1 and

$F_B^{(3)}(a_1,a_2,a_3,b_1,b_2,b_3,c;x_1,x_2,x_3)=\sum _{i_1,i_2,i_3=0}^{\mathrm{\infty }}\frac{(a_1{)}_{i_1}(a_2{)}_{i_2}(a_3{)}_{i_3}(b_1{)}_{i_1}(b_2{)}_{i_2}(b_3{)}_{i_3}}{(c{)}_{i_1+i_2+i_3}{i}_1!i_2!i_3!}{x}_1^{i_1}{x}_2^{i_2}{x}_3^{i_3}$

for |x₁| < 1, |x₂| < 1, |x₃| < 1 and

$F_C^{(3)}(a,b,c_1,c_2,c_3;x_1,x_2,x_3)=\sum _{i_1,i_2,i_3=0}^{\mathrm{\infty }}\frac{(a{)}_{i_1+i_2+i_3}(b{)}_{i_1+i_2+i_3}}{(c_1{)}_{i_1}(c_2{)}_{i_2}(c_3{)}_{i_3}{i}_1!i_2!i_3!}{x}_1^{i_1}{x}_2^{i_2}{x}_3^{i_3}$

for |x₁|^1/2 + |x₂|^1/2 + |x₃|^1/2 < 1 and

$F_D^{(3)}(a,b_1,b_2,b_3,c;x_1,x_2,x_3)=\sum _{i_1,i_2,i_3=0}^{\mathrm{\infty }}\frac{(a{)}_{i_1+i_2+i_3}(b_1{)}_{i_1}(b_2{)}_{i_2}(b_3{)}_{i_3}}{(c{)}_{i_1+i_2+i_3}{i}_1!i_2!i_3!}{x}_1^{i_1}{x}_2^{i_2}{x}_3^{i_3}$

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

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

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

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 Template:Harv. 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

$F_A^{(n)}(a,b_1,\dots ,b_n,c_1,\dots ,c_n;x_1,\dots ,x_n)=\sum _{i_1,\dots ,i_n=0}^{\mathrm{\infty }}\frac{(a{)}_{i_1+\cdots +i_n}(b_1{)}_{i_1}\cdots (b_n{)}_{i_n}}{(c_1{)}_{i_1}\cdots (c_n{)}_{i_n}{i}_1!\cdots i_n!}{x}_1^{i_1}\cdots x_n^{i_n},$

where |x₁| + ... + |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:

$F_A^{(2)}\equiv F_2,F_B^{(2)}\equiv F_3,F_C^{(2)}\equiv F_4,F_D^{(2)}\equiv F_1.$

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 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:

$F_D^{(n)}(a,b_1,\dots ,b_n,c;x_1,\dots ,x_n)=\frac{\mathrm{\Gamma }(c)}{\mathrm{\Gamma }(a)\mathrm{\Gamma }(c-a)}\underset{0}{\overset{1}{\int }}{t}^{a-1}(1-t{)}^{c-a-1}(1-x_{1}t{)}^{-b_1}\cdots (1-x_{n}t{)}^{-b_n}\mathrm{d}t,\mathrm{Re}c>\mathrm{Re}a>0.$

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 F_D with three variables:

$\mathrm{\Pi }(n,\varphi ,k)=\underset{0}{\overset{\varphi }{\int }}\frac{\mathrm{d}\theta }{(1-n{\sin }^2\theta )\sqrt{1-k^2{\sin }^2\theta }}=\sin (\varphi )F_D^{(3)}(\frac{1}{2},1,\frac{1}{2},\frac{1}{2},\frac{3}{2};n{\sin }^2\varphi ,{\sin }^2\varphi ,k^2{\sin }^2\varphi ),|\mathrm{Re}\varphi |<\frac{\pi }{2}.$

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

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

with the iterative sum

$D_n(\overline{b^{*}},\overline{z^{*}})=\frac{1}{n}\sum _{k=1}^n(\sum _{i=1}^N{b}_i^{*}\cdot (z_i^{*}{)}^k)\cdot D_{k-i}$ 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

$\overline{b^{*}}=[\bar{b},c-\sum _i{b}_i]$

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

where the length of $\bar{z}$ and $\bar{b}$ 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.

References

• Template:Cite book (see p. 114)
• Template:Cite book
• Template:Cite journal
• Template:Cite journal (corrigendum 1956 in Ganita 7, p. 65)
• Template:Cite book (there is a 2008 paperback with Template:Isbn)
• Template:Cite book (there is another edition with Template:Isbn)

• Template:Mathworld

Template:Series (mathematics)