嫪麗切拉函數

嫪麗切拉函數（Lauricella functions）是1893年義大利數學家Giuseppe Lauricella首先研究的三元超幾何函數。

$$F_A^{(3)}(a,b_1,b_2,b_3,c_1,c_2,c_3;x_1,x_2,x_3) = \sum_{i_1,i_2,i_3=0}^{\infty} \frac{(a)_{i_1+i_2+i_3} (b_1)_{i_1} (b_2)_{i_2} (b_3)_{i_3}} {(c_1)_{i_1} (c_2)_{i_2} (c_3)_{i_3} \,i_1! \,i_2! \,i_3!} \,x_1^{i_1}x_2^{i_2}x_3^{i_3}$$

$$F_B^{(3)}(a_1,a_2,a_3,b_1,b_2,b_3,c;x_1,x_2,x_3) = \sum_{i_1,i_2,i_3=0}^{\infty} \frac{(a_1)_{i_1} (a_2)_{i_2} (a_3)_{i_3} (b_1)_{i_1} (b_2)_{i_2} (b_3)_{i_3}} {(c)_{i_1+i_2+i_3} \,i_1! \,i_2! \,i_3!} \,x_1^{i_1}x_2^{i_2}x_3^{i_3}$$

$$F_C^{(3)}(a,b,c_1,c_2,c_3;x_1,x_2,x_3) = \sum_{i_1,i_2,i_3=0}^{\infty} \frac{(a)_{i_1+i_2+i_3} (b)_{i_1+i_2+i_3}} {(c_1)_{i_1} (c_2)_{i_2} (c_3)_{i_3} \,i_1! \,i_2! \,i_3!} \,x_1^{i_1}x_2^{i_2}x_3^{i_3}$$

$$F_D^{(3)}(a,b_1,b_2,b_3,c;x_1,x_2,x_3) = \sum_{i_1,i_2,i_3=0}^{\infty} \frac{(a)_{i_1+i_2+i_3} (b_1)_{i_1} (b_2)_{i_2} (b_3)_{i_3}} {(c)_{i_1+i_2+i_3} \,i_1! \,i_2! \,i_3!} \,x_1^{i_1}x_2^{i_2}x_3^{i_3}$$

其中階乘冪 (q)i 為：

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

通過解析延拓，可將 x1, x2, x3等變數擴展到其他數值.

Lauricella指出，另外還有十個三元超幾何函數： FE, FF, ..., FT Saran.

## n 元推廣

嫪麗切拉n變量函數$F_{A}^{(n)}$：$$F_{A}^{(n)}\left(a;b_{1}, \ldots, b_{n} ; c_{1}, \ldots, c_{n} ; z_{1}, \ldots, z_{n}\right)=\sum_{k_{1}=0}^{\infty} \ldots \sum_{k_{n}=0}^{\infty} \frac{(a)_{k_{1}+\ldots+k_{n}}\left(b_{1}\right)_{k_{1}} \ldots\left(b_{n}\right)_{k_{n}}}{\left(c_{1}\right)_{k_{1}} \ldots\left(c_{n}\right)_{k_{n}}} \frac{z_{1}^{k_{1}} \ldots z_{n}^{k_{n}}}{k_{1} ! \ldots k_{n} !};/\left|z_{1}\right|+\ldots+\left|z_{n}\right|<1$$

嫪麗切拉n變量函數$F_{B}^{(n)}$：$$F_{B}^{(n)}\left(a_1,\ldots,a_n;b_{1}, \ldots, b_{n} ;c; z_{1}, \ldots, z_{n}\right)=\sum_{k_{1}=0}^{\infty} \ldots \sum_{k_{n}=0}^{\infty} \frac{\left(a_{1}\right)_{k_{1}} \ldots\left(a_{n}\right)_{k_{n}}\left(b_{1}\right)_{k_{1}} \ldots\left(b_{n}\right)_{k_{n}}}{\left(c\right)_{k_{1}+\dots k_n} } \frac{z_{1}^{k_{1}} \ldots z_{n}^{k_{n}}}{k_{1} ! \ldots k_{n} !};/\max(\left|z_{1}\right|,\dots,\left|z_{n}\right|)<1$$

嫪麗切拉n變量函數$F_{C}^{(n)}$：$$F_{C}^{(n)}\left(a;b; c_{1}, \ldots, c_{n} ; z_{1}, \ldots, z_{n}\right)=\sum_{k_{1}=0}^{\infty} \ldots \sum_{k_{n}=0}^{\infty} \frac{(a)_{k_{1}+\ldots+k_{n}}(b)_{k_{1}+\ldots+k_{n}}}{\left(c_{1}\right)_{k_{1}} \ldots\left(c_{n}\right)_{k_{n}}} \frac{z_{1}^{k_{1}} \ldots z_{n}^{k_{n}}}{k_{1} ! \ldots k_{n} !};/ \sqrt{\left|z_{1}\right|}+\ldots+\sqrt{\left|z_{n}\right|}<1$$

嫪麗切拉n變量函數$F_{D}^{(n)}$：$$F_{D}^{(n)}\left(a;b_{1}, \ldots, b_{n} ;c; z_{1}, \ldots, z_{n}\right)=\sum_{k_{1}=0}^{\infty} \ldots \sum_{k_{n}=0}^{\infty} \frac{\left(a\right)_{k_{1}+\dots k_n}\left(b_{1}\right)_{k_{1}} \ldots\left(b_{n}\right)_{k_{n}}}{\left(c\right)_{k_{1}+\dots k_n} } \frac{z_{1}^{k_{1}} \ldots z_{n}^{k_{n}}}{k_{1} ! \ldots k_{n} !};/\max(\left|z_{1}\right|,\dots,\left|z_{n}\right|)<1$$

當 n = 2,時 the Lauricella 超幾何函數化為二元阿佩爾函數 :

$$F_A^{(2)} \equiv F_2 ,\quad F_B^{(2)} \equiv F_3 ,\quad F_C^{(2)} \equiv F_4 ,\quad F_D^{(2)} \equiv F_1.$$

當 n = 1, a則化為超幾何函數:

$$F_A^{(1)}(a,b,c;x) \equiv F_B^{(1)}(a,b,c;x) \equiv F_C^{(1)}(a,b,c;x) \equiv F_D^{(1)}(a,b,c;x) \equiv {_2}F_1(a,b;c;x).$$

## FD積分式

$$F_D^{(n)}(a, b_1,\ldots,b_n, c; x_1,\ldots,x_n) = \frac{\Gamma(c)} {\Gamma(a) \Gamma(c-a)} \int_0^1 t^{a-1} (1-t)^{c-a-1} (1-x_1t)^{-b_1} \cdots (1-x_nt)^{-b_n} \,\mathrm{d}t, \quad \real \,c > \real \,a > 0 ~.$$

第三類不完全橢圓積分可以通過三元的嫪麗切拉函數表示。

$$\Pi(n,\phi,k) = \int_0^{\phi} \frac{\mathrm{d} \theta} {(1 - n \sin^2 \theta) \sqrt{1 - k^2 \sin^2 \theta}} = \sin \phi \,F_D^{(3)}(\tfrac 1 2, 1, \tfrac 1 2, \tfrac 1 2, \tfrac 3 2; n \sin^2 \phi, \sin^2 \phi, k^2 \sin^2 \phi), \quad |\real \,\phi| < \frac \pi 2 ~.$$

## 參考文獻

- (see p. 114)
- (corrigendum 1956 in Ganita 7, p. 65)
- (there is a 2008 paperback with ISBN 978-0-521-09061-2)
- (there is another edition with ISBN 0-85312-602-X)
- Erdélyi, A. "Hypergeometric Functions of Two Variables." Acta Math. 83, 131-164, 1950.