嫪丽切拉函数

嫪丽切拉函数（Lauricella functions）是1893年意大利数学家Giuseppe Lauricella（英语：Giuseppe Lauricella）首先研究的三元超几何函数。 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 |x1| + |x2| + |x3| < 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 |x1| < 1, |x2| < 1, |x3| < 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|x1|½ + |x2|½ + |x3|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 |x1| < 1, |x2| < 1, |x3| < 1. 其中阶乘幂 (q)i 为： ( 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)}}~,} 通过解析延拓，可将 x1, x2, x3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： FE, FF, ..., FT (Saran 1954). 嫪丽切拉n变量函数 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} 嫪丽切拉n变量函数 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} 嫪丽切拉n变量函数 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} 嫪丽切拉n变量函数 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} 当 n = 2,时 the Lauricella 超几何函数化为二元阿佩尔函数 : 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}.} 当 n = 1, a则化为超几何函数: 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 ( 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~.} 第三类不完全椭圆积分可以通过三元的嫪丽切拉函数表示。 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}}~.} Appell, Paul; Kampé de Fériet, Joseph. Fonctions hypergéométriques et hypersphériques; Polynômes d'Hermite. Paris: Gauthier–Villars. 1926. JFM 52.0361.13 （法语）.  (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 （意大利语）.  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. Ronald M. Aarts. Lauricella Functions. MathWorld.