查看“︁嫪丽切拉函数”︁的源代码

'''嫪丽切拉函数'''（Lauricella functions）是1893年[[意大利]][[数学家]]{{le|Giuseppe Lauricella}}首先研究的三元[[超几何函数]]。

:<math>
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}
</math>

其中 |''x''<sub>1</sub>| + |''x''<sub>2</sub>| + |''x''<sub>3</sub>| < 1  

:<math>
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}
</math>

其中 |''x''<sub>1</sub>| < 1, |''x''<sub>2</sub>| < 1, |''x''<sub>3</sub>| < 1  

:<math>
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}
</math>

其中|''x''<sub>1</sub>|<sup>½</sup> + |''x''<sub>2</sub>|<sup>½</sup> + |''x''<sub>3</sub>|<sup>½</sup> < 1 

:<math>
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}
</math> 

其中 |''x''<sub>1</sub>| < 1, |''x''<sub>2</sub>| < 1, |''x''<sub>3</sub>| < 1. 

其中[[阶乘幂]] (''q'')<sub>''i''</sub> 为： 

:<math>(q)_i = q\,(q+1) \cdots (q+i-1) = \frac{\Gamma(q+i)}{\Gamma(q)}~,</math>

               
通过解析延拓，可将 ''x''<sub>1</sub>, ''x''<sub>2</sub>, ''x''<sub>3</sub>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''<sub>''E''</sub>, ''F''<sub>''F''</sub>, ..., ''F''<sub>''T''</sub> {{harv|Saran|1954}}.

==n 元推广==
; 嫪丽切拉n变量函数<math>F_{A}^{(n)}</math>
: <math>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</math>
; 嫪丽切拉n变量函数<math>F_{B}^{(n)}</math>
: <math>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</math>
; 嫪丽切拉n变量函数<math>F_{C}^{(n)}</math>
: <math>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</math>
; 嫪丽切拉n变量函数<math>F_{D}^{(n)}</math>
: <math>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</math>
当 ''n'' = 2,时 the Lauricella 超几何函数化为二元[[阿佩尔函数]] :

:<math>
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.
</math>  

当 ''n'' = 1, a则化为[[超几何函数]]:

:<math>
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).
</math>

== ''F''<sub>''D''</sub>积分式==

:<math>
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_1t)^{-b_1} \cdots (1-x_nt)^{-b_n} \,\mathrm{d}t, \quad \real \,c > \real \,a > 0 ~.
</math>
[[椭圆积分|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。
:<math>
\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 |\real \,\phi| < \frac \pi 2 ~.
</math>

==参考文献==

* {{cite book | last1= Appell | first1= Paul | author1-link= Paul Émile Appell | last2= Kampé de Fériet | first2= Joseph | author2-link= Joseph Kampé de Fériet | title= Fonctions hypergéométriques et hypersphériques; Polynômes d'Hermite | language= fr | location= Paris | publisher= Gauthier–Villars | year= 1926 | jfm= 52.0361.13 | ref= harv}} (see p. 114)
* {{cite book | last= Exton | first= Harold | title= Multiple hypergeometric functions and applications | location= Chichester, UK | publisher= Halsted Press, Ellis Horwood Ltd. | year= 1976 | series= Mathematics and its applications | isbn= 0-470-15190-0 | mr= 0422713 | ref= harv}}
* {{cite journal | last= Lauricella | first= Giuseppe | authorlink= Giuseppe Lauricella | title= Sulle funzioni ipergeometriche a più variabili | language= it | journal= [[Rendiconti del Circolo Matematico di Palermo]] | year= 1893 | volume= 7 | issue= S1 | pages= 111–158 | doi= 10.1007/BF03012437 | jfm= 25.0756.01 | ref= harv}}
* {{cite journal | last= Saran | first= Shanti | title= Hypergeometric Functions of Three Variables | journal= Ganita | year= 1954 | volume= 5 | issue= 1 | pages= 77–91 | issn= 0046-5402 | mr= 0087777 | zbl= 0058.29602 | ref= harv}} (corrigendum 1956 in ''Ganita'' '''7''', p. 65)
* {{cite book | last= Slater | first= Lucy Joan | authorlink= Lucy Joan Slater | title= Generalized hypergeometric functions | url= https://archive.org/details/generalizedhyper0000unse_g0b6 | location= Cambridge, UK | publisher= Cambridge University Press | year= 1966 | isbn= 0-521-06483-X | mr= 0201688 | ref= harv}} (there is a 2008 paperback with ISBN 978-0-521-09061-2)
* {{cite book | last1= Srivastava | first1= Hari M. | last2= Karlsson | first2= Per W. | title= Multiple Gaussian hypergeometric series | location= Chichester, UK | publisher= Halsted Press, Ellis Horwood Ltd. | year= 1985 | series= Mathematics and its applications | isbn= 0-470-20100-2 | mr= 0834385 | ref= harv}} (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==

* {{mathworld | urlname= LauricellaFunctions | title= Lauricella Functions | author= Ronald M. Aarts}}


[[Category:超几何函数]]
[[Category:特殊函数]]