**嫪丽切拉函数**（Lauricella functions）是1893年意大利数学家Giuseppe Lauricella（英语：）首先研究的三元超几何函数。

{\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 

{\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 

{\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 

{\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. 

其中阶乘幂 (*q*)*i* 为： 

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

  

通过解析延拓，可将 *x*1, *x*2, *x*3等变数扩展到其他数值.

Lauricella指出，另外还有十个三元超几何函数： *F**E*, *F**F*, ... *F**T* （Saran 1954）.

n 元推广
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嫪丽切拉n变量函数{\displaystyle F\_{A}^{(n)}}
{\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}
嫪丽切拉n变量函数{\displaystyle F\_{B}^{(n)}}
{\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}
嫪丽切拉n变量函数{\displaystyle F\_{C}^{(n)}}
{\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}
嫪丽切拉n变量函数{\displaystyle F\_{D}^{(n)}}
{\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}
当 *n* = 2,时 the Lauricella 超几何函数化为二元阿佩尔函数 :

{\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}.}
当 *n* = 1, a则化为超几何函数:

{\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*积分式
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{\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~.}
第三类不完全椭圆积分可以通过三元的嫪丽切拉函数表示。

{\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}}~.}

参考文献
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* Appell, Paul; Kampé de Fériet, Joseph.. Paris: Gauthier–Villars. 1926. JFM 52.0361.13 （法语）.  (see p. 114)
* Exton, Harold.. Mathematics and its applications. Chichester, UK: Halsted Press, Ellis Horwood Ltd. 1976. ISBN 0-470-15190-0. MR 0422713.
* Lauricella, Giuseppe.. Rendiconti del Circolo Matematico di Palermo. 1893, **7** (S1): 111–158. JFM 25.0756.01. doi:10.1007/BF03012437 （意大利语）.
* Saran, Shanti.. Ganita. 1954, **5** (1): 77–91. ISSN 0046-5402. MR 0087777. Zbl 0058.29602.  (corrigendum 1956 in *Ganita* **7**, p. 65)
* Slater, Lucy Joan.. 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.. 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.

外部链接
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* Ronald M. Aarts.. MathWorld.