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
)
=
∑
i
1
,
i
2
,
i
3
=
0
∞
(
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
{\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}}}
for |x 1 | + |x 2 | + |x 3 | < 1 and
F
B
(
3
)
(
a
1
,
a
2
,
a
3
,
b
1
,
b
2
,
b
3
,
c
;
x
1
,
x
2
,
x
3
)
=
∑
i
1
,
i
2
,
i
3
=
0
∞
(
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
{\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}}}
for |x 1 | < 1, |x 2 | < 1, |x 3 | < 1 and
F
C
(
3
)
(
a
,
b
,
c
1
,
c
2
,
c
3
;
x
1
,
x
2
,
x
3
)
=
∑
i
1
,
i
2
,
i
3
=
0
∞
(
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
{\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}}}
for |x 1 |1/2 + |x 2 |1/2 + |x 3 |1/2 < 1 and
F
D
(
3
)
(
a
,
b
1
,
b
2
,
b
3
,
c
;
x
1
,
x
2
,
x
3
)
=
∑
i
1
,
i
2
,
i
3
=
0
∞
(
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
{\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}}}
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
)
⋯
(
q
+
i
−
1
)
=
Γ
(
q
+
i
)
Γ
(
q
)
,
{\displaystyle (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
{\displaystyle q}
except
q
=
0
,
−
1
,
−
2
,
…
{\displaystyle 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 (Saran 1954 ). There are therefore a total of 14 Lauricella–Saran hypergeometric functions.
Generalization to n variables [ edit ]
These functions can be straightforwardly extended to n variables. One writes for example
F
A
(
n
)
(
a
,
b
1
,
…
,
b
n
,
c
1
,
…
,
c
n
;
x
1
,
…
,
x
n
)
=
∑
i
1
,
…
,
i
n
=
0
∞
(
a
)
i
1
+
⋯
+
i
n
(
b
1
)
i
1
⋯
(
b
n
)
i
n
(
c
1
)
i
1
⋯
(
c
n
)
i
n
i
1
!
⋯
i
n
!
x
1
i
1
⋯
x
n
i
n
,
{\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}}~,}
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
(
2
)
≡
F
2
,
F
B
(
2
)
≡
F
3
,
F
C
(
2
)
≡
F
4
,
F
D
(
2
)
≡
F
1
.
{\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}.}
When n = 1, all four functions reduce to the Gauss hypergeometric function :
F
A
(
1
)
(
a
,
b
,
c
;
x
)
≡
F
B
(
1
)
(
a
,
b
,
c
;
x
)
≡
F
C
(
1
)
(
a
,
b
,
c
;
x
)
≡
F
D
(
1
)
(
a
,
b
,
c
;
x
)
≡
2
F
1
(
a
,
b
;
c
;
x
)
.
{\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).}
Integral representation of F D [ edit ]
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
(
n
)
(
a
,
b
1
,
…
,
b
n
,
c
;
x
1
,
…
,
x
n
)
=
Γ
(
c
)
Γ
(
a
)
Γ
(
c
−
a
)
∫
0
1
t
a
−
1
(
1
−
t
)
c
−
a
−
1
(
1
−
x
1
t
)
−
b
1
⋯
(
1
−
x
n
t
)
−
b
n
d
t
,
Re
c
>
Re
a
>
0
.
{\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_{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:
Π
(
n
,
ϕ
,
k
)
=
∫
0
ϕ
d
θ
(
1
−
n
sin
2
θ
)
1
−
k
2
sin
2
θ
=
sin
(
ϕ
)
F
D
(
3
)
(
1
2
,
1
,
1
2
,
1
2
,
3
2
;
n
sin
2
ϕ
,
sin
2
ϕ
,
k
2
sin
2
ϕ
)
,
|
Re
ϕ
|
<
π
2
.
{\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 |<{\frac {\pi }{2}}~.}
Finite-sum solutions of F D [ edit ]
Case 1 :
a
>
c
{\displaystyle a>c}
,
a
−
c
{\displaystyle a-c}
a positive integer
One can relate F D to the Carlson R function
R
n
{\displaystyle R_{n}}
via
F
D
(
a
,
b
¯
,
c
,
z
¯
)
=
R
a
−
c
(
b
∗
¯
,
z
∗
¯
)
⋅
∏
i
(
z
i
∗
)
b
i
∗
=
Γ
(
a
−
c
+
1
)
Γ
(
b
∗
)
Γ
(
a
−
c
+
b
∗
)
⋅
D
a
−
c
(
b
∗
¯
,
z
∗
¯
)
⋅
∏
i
(
z
i
∗
)
b
i
∗
{\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}^{*}}}
with the iterative sum
D
n
(
b
∗
¯
,
z
∗
¯
)
=
1
n
∑
k
=
1
n
(
∑
i
=
1
N
b
i
∗
⋅
(
z
i
∗
)
k
)
⋅
D
k
−
i
{\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}}
and
D
0
=
1
{\displaystyle D_{0}=1}
where it can be exploited that the Carlson R function with
n
>
0
{\displaystyle n>0}
has an exact representation (see [ 1] for more information).
The vectors are defined as
b
∗
¯
=
[
b
¯
,
c
−
∑
i
b
i
]
{\displaystyle {\overline {b^{*}}}=[{\overline {b}},c-\sum _{i}b_{i}]}
z
∗
¯
=
[
1
1
−
z
1
,
…
,
1
1
−
z
N
−
1
,
1
]
{\displaystyle {\overline {z^{*}}}=[{\frac {1}{1-z_{1}}},\ldots ,{\frac {1}{1-z_{N-1}}},1]}
where the length of
z
¯
{\displaystyle {\overline {z}}}
and
b
¯
{\displaystyle {\overline {b}}}
is
N
−
1
{\displaystyle N-1}
, while the vectors
z
∗
¯
{\displaystyle {\overline {z^{*}}}}
and
b
∗
¯
{\displaystyle {\overline {b^{*}}}}
have length
N
{\displaystyle N}
.
Case 2:
c
>
a
{\displaystyle c>a}
,
c
−
a
{\displaystyle 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 [ 2] for more information.
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 )