嫪丽切拉函数

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:

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

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

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。

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

• 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.

• Ronald M. Aarts. . MathWorld.