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).