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):

$F_{A}^{(3)}(a,b_{1},b_{2},b_{3},c_{1},c_{2},c_{3};x_{1},x_{2},x_{3})=\sum _{i_{1},i_{2},i_{3}=0}^{\infty }{\frac {(a)_{i_{1}+i_{2}+i_{3}}(b_{1})_{i_{1}}(b_{2})_{i_{2}}(b_{3})_{i_{3}}}{(c_{1})_{i_{1}}(c_{2})_{i_{2}}(c_{3})_{i_{3}}\,i_{1}!\,i_{2}!\,i_{3}!}}\,x_{1}^{i_{1}}x_{2}^{i_{2}}x_{3}^{i_{3}}$

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 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 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

$F_{A}^{(n)}(a,b_{1},\ldots ,b_{n},c_{1},\ldots ,c_{n};x_{1},\ldots ,x_{n})=\sum _{i_{1},\ldots ,i_{n}=0}^{\infty }{\frac {(a)_{i_{1}+\cdots +i_{n}}(b_{1})_{i_{1}}\cdots (b_{n})_{i_{n}}}{(c_{1})_{i_{1}}\cdots (c_{n})_{i_{n}}\,i_{1}!\cdots \,i_{n}!}}\,x_{1}^{i_{1}}\cdots x_{n}^{i_{n}}~,$

where |x₁| + ... + |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:

$F_{A}^{(2)}\equiv F_{2},\quad F_{B}^{(2)}\equiv F_{3},\quad F_{C}^{(2)}\equiv F_{4},\quad F_{D}^{(2)}\equiv F_{1}.$

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 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:

$F_{D}^{(n)}(a,b_{1},\ldots ,b_{n},c;x_{1},\ldots ,x_{n})={\frac {\Gamma (c)}{\Gamma (a)\Gamma (c-a)}}\int _{0}^{1}t^{a-1}(1-t)^{c-a-1}(1-x_{1}t)^{-b_{1}}\cdots (1-x_{n}t)^{-b_{n}}\,\mathrm {d} t,\qquad \operatorname {Re} c>\operatorname {Re} a>0~.$

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 F_D 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 F_D

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

One can relate F_D to the Carlson R function $R_{n}$ via

$F_{D}(a,{\overline {b}},c,{\overline {z}})=R_{a-c}({\overline {b^{*}}},{\overline {z^{*}}})\cdot \prod _{i}(z_{i}^{*})^{b_{i}^{*}}={\frac {\Gamma (a-c+1)\Gamma (b^{*})}{\Gamma (a-c+b^{*})}}\cdot D_{a-c}({\overline {b^{*}}},{\overline {z^{*}}})\cdot \prod _{i}(z_{i}^{*})^{b_{i}^{*}}$

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.

References

• Appell, Paul; Kampé de Fériet, Joseph (1926). Fonctions hypergéométriques et hypersphériques; Polynômes d'Hermite (in French). Paris: Gauthier–Villars. JFM 52.0361.13. (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. MR 0422713.
• Lauricella, Giuseppe (1893). "Sulle funzioni ipergeometriche a più variabili". Rendiconti del Circolo Matematico di Palermo (in Italian). 7 (S1): 111–158. doi:10.1007/BF03012437. JFM 25.0756.01. S2CID 122316343.
• Saran, Shanti (1954). "Hypergeometric Functions of Three Variables". Ganita. 5 (1): 77–91. ISSN 0046-5402. MR 0087777. Zbl 0058.29602. (corrigendum 1956 in Ganita 7, p. 65)
• Slater, Lucy Joan (1966). Generalized hypergeometric functions. Cambridge, UK: Cambridge University Press. ISBN 0-521-06483-X. MR 0201688. (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. MR 0834385. (there is another edition with ISBN 0-85312-602-X)

• Ronald M. Aarts. "Lauricella Functions". MathWorld.