嫪麗切拉函數 （Lauricella functions）是1893年義大利 數學家 Giuseppe Lauricella 超幾何函數 。
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}}}
其中 |x 1 | + |x 2 | + |x 3 | < 1
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}}}
其中 |x 1 | < 1, |x 2 | < 1, |x 3 | < 1
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}}}
其中|x 1 |½ + |x 2 |½ + |x 3 |½ < 1
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}}}
其中 |x 1 | < 1, |x 2 | < 1, |x 3 | < 1.
其中階乘冪 (q )i
(
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)}}~,}
x 1 , x 2 , x 3 等變數擴展到其他數值.
Lauricella指出，另外還有十個三元超幾何函數： F E F F F T Saran 1954 ）.
n 元推廣
編輯
嫪麗切拉n變量函數
F
A
(
n
)
{\displaystyle F_{A}^{(n)}}
F
A
(
n
)
(
a
;
b
1
,
…
,
b
n
;
c
1
,
…
,
c
n
;
z
1
,
…
,
z
n
)
=
∑
k
1
=
0
∞
…
∑
k
n
=
0
∞
(
a
)
k
1
+
…
+
k
n
(
b
1
)
k
1
…
(
b
n
)
k
n
(
c
1
)
k
1
…
(
c
n
)
k
n
z
1
k
1
…
z
n
k
n
k
1
!
…
k
n
!
;
/
|
z
1
|
+
…
+
|
z
n
|
<
1
{\displaystyle F_{A}^{(n)}\left(a;b_{1},\ldots ,b_{n};c_{1},\ldots ,c_{n};z_{1},\ldots ,z_{n}\right)=\sum _{k_{1}=0}^{\infty }\ldots \sum _{k_{n}=0}^{\infty }{\frac {(a)_{k_{1}+\ldots +k_{n}}\left(b_{1}\right)_{k_{1}}\ldots \left(b_{n}\right)_{k_{n}}}{\left(c_{1}\right)_{k_{1}}\ldots \left(c_{n}\right)_{k_{n}}}}{\frac {z_{1}^{k_{1}}\ldots z_{n}^{k_{n}}}{k_{1}!\ldots k_{n}!}};/\left|z_{1}\right|+\ldots +\left|z_{n}\right|<1}
嫪麗切拉n變量函數
F
B
(
n
)
{\displaystyle F_{B}^{(n)}}
F
B
(
n
)
(
a
1
,
…
,
a
n
;
b
1
,
…
,
b
n
;
c
;
z
1
,
…
,
z
n
)
=
∑
k
1
=
0
∞
…
∑
k
n
=
0
∞
(
a
1
)
k
1
…
(
a
n
)
k
n
(
b
1
)
k
1
…
(
b
n
)
k
n
(
c
)
k
1
+
…
k
n
z
1
k
1
…
z
n
k
n
k
1
!
…
k
n
!
;
/
max
(
|
z
1
|
,
…
,
|
z
n
|
)
<
1
{\displaystyle F_{B}^{(n)}\left(a_{1},\ldots ,a_{n};b_{1},\ldots ,b_{n};c;z_{1},\ldots ,z_{n}\right)=\sum _{k_{1}=0}^{\infty }\ldots \sum _{k_{n}=0}^{\infty }{\frac {\left(a_{1}\right)_{k_{1}}\ldots \left(a_{n}\right)_{k_{n}}\left(b_{1}\right)_{k_{1}}\ldots \left(b_{n}\right)_{k_{n}}}{\left(c\right)_{k_{1}+\dots k_{n}}}}{\frac {z_{1}^{k_{1}}\ldots z_{n}^{k_{n}}}{k_{1}!\ldots k_{n}!}};/\max(\left|z_{1}\right|,\dots ,\left|z_{n}\right|)<1}
嫪麗切拉n變量函數
F
C
(
n
)
{\displaystyle F_{C}^{(n)}}
F
C
(
n
)
(
a
;
b
;
c
1
,
…
,
c
n
;
z
1
,
…
,
z
n
)
=
∑
k
1
=
0
∞
…
∑
k
n
=
0
∞
(
a
)
k
1
+
…
+
k
n
(
b
)
k
1
+
…
+
k
n
(
c
1
)
k
1
…
(
c
n
)
k
n
z
1
k
1
…
z
n
k
n
k
1
!
…
k
n
!
;
/
|
z
1
|
+
…
+
|
z
n
|
<
1
{\displaystyle F_{C}^{(n)}\left(a;b;c_{1},\ldots ,c_{n};z_{1},\ldots ,z_{n}\right)=\sum _{k_{1}=0}^{\infty }\ldots \sum _{k_{n}=0}^{\infty }{\frac {(a)_{k_{1}+\ldots +k_{n}}(b)_{k_{1}+\ldots +k_{n}}}{\left(c_{1}\right)_{k_{1}}\ldots \left(c_{n}\right)_{k_{n}}}}{\frac {z_{1}^{k_{1}}\ldots z_{n}^{k_{n}}}{k_{1}!\ldots k_{n}!}};/{\sqrt {\left|z_{1}\right|}}+\ldots +{\sqrt {\left|z_{n}\right|}}<1}
嫪麗切拉n變量函數
F
D
(
n
)
{\displaystyle F_{D}^{(n)}}
F
D
(
n
)
(
a
;
b
1
,
…
,
b
n
;
c
;
z
1
,
…
,
z
n
)
=
∑
k
1
=
0
∞
…
∑
k
n
=
0
∞
(
a
)
k
1
+
…
k
n
(
b
1
)
k
1
…
(
b
n
)
k
n
(
c
)
k
1
+
…
k
n
z
1
k
1
…
z
n
k
n
k
1
!
…
k
n
!
;
/
max
(
|
z
1
|
,
…
,
|
z
n
|
)
<
1
{\displaystyle F_{D}^{(n)}\left(a;b_{1},\ldots ,b_{n};c;z_{1},\ldots ,z_{n}\right)=\sum _{k_{1}=0}^{\infty }\ldots \sum _{k_{n}=0}^{\infty }{\frac {\left(a\right)_{k_{1}+\dots k_{n}}\left(b_{1}\right)_{k_{1}}\ldots \left(b_{n}\right)_{k_{n}}}{\left(c\right)_{k_{1}+\dots k_{n}}}}{\frac {z_{1}^{k_{1}}\ldots z_{n}^{k_{n}}}{k_{1}!\ldots k_{n}!}};/\max(\left|z_{1}\right|,\dots ,\left|z_{n}\right|)<1}
當 n = 2,時 the Lauricella 超幾何函數化為二元阿佩爾函數 :
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}.}
當 n = 1, a則化為超幾何函數 :
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).}
F D
編輯
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
,
ℜ
c
>
ℜ
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,\quad \Re \,c>\Re \,a>0~.}
第三類不完全橢圓積分 可以通過三元的嫪麗切拉函數表示。
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
)
,
|
ℜ
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 ),\quad |\Re \,\phi |<{\frac {\pi }{2}}~.}
參考文獻
編輯
Appell, Paul ; Kampé de Fériet, Joseph . Fonctions hypergéométriques et hypersphériques; Polynômes d'Hermite. Paris: Gauthier–Villars. 1926. JFM 52.0361.13 （法語） . Exton, Harold. Multiple hypergeometric functions and applications. Mathematics and its applications. Chichester, UK: Halsted Press, Ellis Horwood Ltd. 1976. ISBN 0-470-15190-0MR 0422713 Lauricella, Giuseppe . Sulle funzioni ipergeometriche a più variabili. Rendiconti del Circolo Matematico di Palermo . 1893, 7 (S1): 111–158. JFM 25.0756.01 doi:10.1007/BF03012437 （義大利語） . Saran, Shanti. Hypergeometric Functions of Three Variables. Ganita. 1954, 5 (1): 77–91. ISSN 0046-5402 MR 0087777 Zbl 0058.29602 Ganita 7 , p. 65)Slater, Lucy Joan . Generalized hypergeometric functions . Cambridge, UK: Cambridge University Press. 1966. ISBN 0-521-06483-XMR 0201688 ISBN 978-0-521-09061-2 )Srivastava, Hari M.; Karlsson, Per W. Multiple Gaussian hypergeometric series. Mathematics and its applications. Chichester, UK: Halsted Press, Ellis Horwood Ltd. 1985. ISBN 0-470-20100-2MR 0834385 ISBN 0-85312-602-X )Erdélyi, A. "Hypergeometric Functions of Two Variables." Acta Math. 83, 131-164, 1950. 外部連結
編輯
