Lauricella超几何函数

**嫪麗切拉函數**（Lauricella functions）是 1893 年義大利數學家 Giuseppe Lauricella 首先研究的三元超幾何函數。

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

其中 |_x_1| + |_x_2| + |_x_3| < 1 

　
: $F_{B}^{(3)}(a_{1},a_{2},a_{3},b_{1},b_{2},b_{3},c;x_{1},x_{2},x_{3})=\sum _{i_{1},i_{2},i_{3}=0}^{\infty }{\frac {(a_{1})_{i_{1}}(a_{2})_{i_{2}}(a_{3})_{i_{3}}(b_{1})_{i_{1}}(b_{2})_{i_{2}}(b_{3})_{i_{3}}}{(c)_{i_{1}+i_{2}+i_{3}}\,i_{1}!\,i_{2}!\,i_{3}!}}\,x_{1}^{i_{1}}x_{2}^{i_{2}}x_{3}^{i_{3}}$

其中 |_x_1| < 1, |_x_2| < 1, |_x_3| < 1 

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

其中 |_x_1|½ + |_x_2|½ + |_x_3|½ < 1 

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

其中 |_x_1| < 1, |_x_2| < 1, |_x_3| < 1. 

其中階乘冪 (_q_)i 為：

　
: $(q)_{i}=q\,(q+1)\cdots (q+i-1)={\frac {\Gamma (q+i)}{\Gamma (q)}}~,$

 
 通過解析延拓，可將 _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

**嫪麗切拉 n 變量函數 $F_{A}^{(n)}$**
: $F_{A}^{(n)}\left(a;b_{1},\ldots ,b_{n};c_{1},\ldots ,c_{n};z_{1},\ldots ,z_{n}\right)=\sum _{k_{1}=0}^{\infty }\ldots \sum _{k_{n}=0}^{\infty }{\frac {(a)_{k_{1}+\ldots +k_{n}}\left(b_{1}\right)_{k_{1}}\ldots \left(b_{n}\right)_{k_{n}}}{\left(c_{1}\right)_{k_{1}}\ldots \left(c_{n}\right)_{k_{n}}}}{\frac {z_{1}^{k_{1}}\ldots z_{n}^{k_{n}}}{k_{1}!\ldots k_{n}!}};/\left|z_{1}\right|+\ldots +\left|z_{n}\right|<1$

**嫪麗切拉 n 變量函數 $F_{B}^{(n)}$**
: $F_{B}^{(n)}\left(a_{1},\ldots ,a_{n};b_{1},\ldots ,b_{n};c;z_{1},\ldots ,z_{n}\right)=\sum _{k_{1}=0}^{\infty }\ldots \sum _{k_{n}=0}^{\infty }{\frac {\left(a_{1}\right)_{k_{1}}\ldots \left(a_{n}\right)_{k_{n}}\left(b_{1}\right)_{k_{1}}\ldots \left(b_{n}\right)_{k_{n}}}{\left(c\right)_{k_{1}+\dots k_{n}}}}{\frac {z_{1}^{k_{1}}\ldots z_{n}^{k_{n}}}{k_{1}!\ldots k_{n}!}};/\max(\left|z_{1}\right|,\dots ,\left|z_{n}\right|)<1$

**嫪麗切拉 n 變量函數 $F_{C}^{(n)}$**
: $F_{C}^{(n)}\left(a;b;c_{1},\ldots ,c_{n};z_{1},\ldots ,z_{n}\right)=\sum _{k_{1}=0}^{\infty }\ldots \sum _{k_{n}=0}^{\infty }{\frac {(a)_{k_{1}+\ldots +k_{n}}(b)_{k_{1}+\ldots +k_{n}}}{\left(c_{1}\right)_{k_{1}}\ldots \left(c_{n}\right)_{k_{n}}}}{\frac {z_{1}^{k_{1}}\ldots z_{n}^{k_{n}}}{k_{1}!\ldots k_{n}!}};/{\sqrt {\left|z_{1}\right|}}+\ldots +{\sqrt {\left|z_{n}\right|}}<1$

**嫪麗切拉 n 變量函數 $F_{D}^{(n)}$**
: $F_{D}^{(n)}\left(a;b_{1},\ldots ,b_{n};c;z_{1},\ldots ,z_{n}\right)=\sum _{k_{1}=0}^{\infty }\ldots \sum _{k_{n}=0}^{\infty }{\frac {\left(a\right)_{k_{1}+\dots k_{n}}\left(b_{1}\right)_{k_{1}}\ldots \left(b_{n}\right)_{k_{n}}}{\left(c\right)_{k_{1}+\dots k_{n}}}}{\frac {z_{1}^{k_{1}}\ldots z_{n}^{k_{n}}}{k_{1}!\ldots k_{n}!}};/\max(\left|z_{1}\right|,\dots ,\left|z_{n}\right|)<1$

當 _n_ = 2, 時 the Lauricella 超幾何函數化為二元阿佩爾函數 :

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

當 _n_ = 1, a 則化為超幾何函數:

　
: $F_{A}^{(1)}(a,b,c;x)\equiv F_{B}^{(1)}(a,b,c;x)\equiv F_{C}^{(1)}(a,b,c;x)\equiv F_{D}^{(1)}(a,b,c;x)\equiv {_{2}}F_{1}(a,b;c;x).$

## _F_D 積分式

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

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。

　
: $\Pi (n,\phi ,k)=\int _{0}^{\phi }{\frac {\mathrm {d} \theta }{(1-n\sin ^{2}\theta ){\sqrt {1-k^{2}\sin ^{2}\theta }}}}=\sin \phi \,F_{D}^{(3)}({\tfrac {1}{2}},1,{\tfrac {1}{2}},{\tfrac {1}{2}},{\tfrac {3}{2}};n\sin ^{2}\phi ,\sin ^{2}\phi ,k^{2}\sin ^{2}\phi ),\quad |\Re \,\phi |<{\frac {\pi }{2}}~.$

## 參考文獻

* Appell, Paul; Kampé de Fériet, Joseph. Fonctions hypergéométriques et hypersphériques; Polynômes d'Hermite. Paris: Gauthier–Villars. 1926. JFM 52.0361.13 （法語）.  (see p. 114)
* Exton, Harold. Multiple hypergeometric functions and applications. Mathematics and its applications. Chichester, UK: Halsted Press, Ellis Horwood Ltd. 1976. ISBN 0-470-15190-0. MR 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.  (corrigendum 1956 in _Ganita_ **7**, p. 65)
* Slater, Lucy Joan. Generalized hypergeometric functions. Cambridge, UK: Cambridge University Press. 1966. 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. Multiple Gaussian hypergeometric series. Mathematics and its applications. Chichester, UK: Halsted Press, Ellis Horwood Ltd. 1985. ISBN 0-470-20100-2. MR 0834385.  (there is another edition with ISBN 0-85312-602-X)
* Erdélyi, A. "Hypergeometric Functions of Two Variables." Acta Math. 83, 131-164, 1950.

## 外部連結

* Ronald M. Aarts. Lauricella Functions. MathWorld.