嫪丽切拉函数(Lauricella functions)是1893年意大利数学家Giuseppe Lauricella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}}}](//wikimedia.org/api/rest_v1/media/math/render/svg/6516974ef41c6534478772181e0a57c2aa14191f)
其中 |x1| + |x2| + |x3| < 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}}}](//wikimedia.org/api/rest_v1/media/math/render/svg/4cb6e4a4075066d31f60e8a9967e61be45d8a2ee)
其中 |x1| < 1, |x2| < 1, |x3| < 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}}}](//wikimedia.org/api/rest_v1/media/math/render/svg/52e85ecdd983cf51a2a140a8a3053f2ed2dc2da4)
其中|x1|½ + |x2|½ + |x3|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}}}](//wikimedia.org/api/rest_v1/media/math/render/svg/dd5c751797994c65c7fe2e53eea6b9b1f3b608fb)
其中 |x1| < 1, |x2| < 1, |x3| < 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)}}~,}](//wikimedia.org/api/rest_v1/media/math/render/svg/bf42e78bbb97fadd3204748f850323cecb0fa3f3)
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).
n 元推广
- 嫪丽切拉n变量函数
![{\displaystyle F_{A}^{(n)}}](//wikimedia.org/api/rest_v1/media/math/render/svg/52dd79e57a23d49637c97781c0a4293cff731701)
![{\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}](//wikimedia.org/api/rest_v1/media/math/render/svg/46b32cf46762bf7948bf57f97495afd4fc204b5e)
- 嫪丽切拉n变量函数
![{\displaystyle F_{B}^{(n)}}](//wikimedia.org/api/rest_v1/media/math/render/svg/b1becabde0127fe429a6d1332bf584494de920d3)
![{\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}](//wikimedia.org/api/rest_v1/media/math/render/svg/ed720d5fc9a6d7a8431a7f0f14fa2fbe9d71c9ee)
- 嫪丽切拉n变量函数
![{\displaystyle F_{C}^{(n)}}](//wikimedia.org/api/rest_v1/media/math/render/svg/43e2edaf41124439b3d93e1f66c4d7c1cf8eed7d)
![{\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}](//wikimedia.org/api/rest_v1/media/math/render/svg/6e2c65be57d5d1e08af764168fb71dfdf7f39b76)
- 嫪丽切拉n变量函数
![{\displaystyle F_{D}^{(n)}}](//wikimedia.org/api/rest_v1/media/math/render/svg/1ac23ed70d085fd2c2343232f2d2eb27c56633f9)
![{\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}](//wikimedia.org/api/rest_v1/media/math/render/svg/49c215e99af16a2fba5e41c22f552d23cb28bf59)
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}.}](//wikimedia.org/api/rest_v1/media/math/render/svg/41265719d83c4dff31afb591f243a149df332ed3)
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).}](//wikimedia.org/api/rest_v1/media/math/render/svg/d752c02a014a4cc1753b51dba976cba821e6c9cf)
FD积分式
![{\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~.}](//wikimedia.org/api/rest_v1/media/math/render/svg/a2f587c48fa219006cfddee63549cf364b052f74)
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}}~.}](//wikimedia.org/api/rest_v1/media/math/render/svg/51e54bd9d41d1dd7eee7a2832709ee34bdea259c)
参考文献
- 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.
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