阿佩尔函数维基百科,自由的 encyclopedia 阿佩尔函数是法国数学家(Paul Apell)在1880年为推广高斯超几何函数而创建的一组双变数函数,定义如下 阿佩尔函数——F1 F 1 ( a , b 1 , b 2 , c ; x , y ) = ∑ m , n = 0 ∞ ( a ) m + n ( b 1 ) m ( b 2 ) n ( c ) m + n m ! n ! x m y n , {\displaystyle F_{1}(a,b_{1},b_{2},c;x,y)=\sum _{m,n=0}^{\infty }{\frac {(a)_{m+n}(b_{1})_{m}(b_{2})_{n}}{(c)_{m+n}\,m!\,n!}}\,x^{m}y^{n}~,} F 2 ( a , b 1 , b 2 , c 1 , c 2 ; x , y ) = ∑ m , n = 0 ∞ ( a ) m + n ( b 1 ) m ( b 2 ) n ( c 1 ) m ( c 2 ) n m ! n ! x m y n , {\displaystyle F_{2}(a,b_{1},b_{2},c_{1},c_{2};x,y)=\sum _{m,n=0}^{\infty }{\frac {(a)_{m+n}(b_{1})_{m}(b_{2})_{n}}{(c_{1})_{m}(c_{2})_{n}\,m!\,n!}}\,x^{m}y^{n}~,} F 3 ( a 1 , a 2 , b 1 , b 2 , c ; x , y ) = ∑ m , n = 0 ∞ ( a 1 ) m ( a 2 ) n ( b 1 ) m ( b 2 ) n ( c ) m + n m ! n ! x m y n , {\displaystyle F_{3}(a_{1},a_{2},b_{1},b_{2},c;x,y)=\sum _{m,n=0}^{\infty }{\frac {(a_{1})_{m}(a_{2})_{n}(b_{1})_{m}(b_{2})_{n}}{(c)_{m+n}\,m!\,n!}}\,x^{m}y^{n}~,} F 4 ( a , b , c 1 , c 2 ; x , y ) = ∑ m , n = 0 ∞ ( a ) m + n ( b ) m + n ( c 1 ) m ( c 2 ) n m ! n ! x m y n . {\displaystyle F_{4}(a,b,c_{1},c_{2};x,y)=\sum _{m,n=0}^{\infty }{\frac {(a)_{m+n}(b)_{m+n}}{(c_{1})_{m}(c_{2})_{n}\,m!\,n!}}\,x^{m}y^{n}~.} 其中的符号 : ( a ) m + n {\displaystyle :(a)_{m+n}} 是阶乘幂 阿佩尔函数是嫪丽切拉函数和Kampé_de_Fériet函数的特例。
阿佩尔函数是法国数学家(Paul Apell)在1880年为推广高斯超几何函数而创建的一组双变数函数,定义如下 阿佩尔函数——F1 F 1 ( a , b 1 , b 2 , c ; x , y ) = ∑ m , n = 0 ∞ ( a ) m + n ( b 1 ) m ( b 2 ) n ( c ) m + n m ! n ! x m y n , {\displaystyle F_{1}(a,b_{1},b_{2},c;x,y)=\sum _{m,n=0}^{\infty }{\frac {(a)_{m+n}(b_{1})_{m}(b_{2})_{n}}{(c)_{m+n}\,m!\,n!}}\,x^{m}y^{n}~,} F 2 ( a , b 1 , b 2 , c 1 , c 2 ; x , y ) = ∑ m , n = 0 ∞ ( a ) m + n ( b 1 ) m ( b 2 ) n ( c 1 ) m ( c 2 ) n m ! n ! x m y n , {\displaystyle F_{2}(a,b_{1},b_{2},c_{1},c_{2};x,y)=\sum _{m,n=0}^{\infty }{\frac {(a)_{m+n}(b_{1})_{m}(b_{2})_{n}}{(c_{1})_{m}(c_{2})_{n}\,m!\,n!}}\,x^{m}y^{n}~,} F 3 ( a 1 , a 2 , b 1 , b 2 , c ; x , y ) = ∑ m , n = 0 ∞ ( a 1 ) m ( a 2 ) n ( b 1 ) m ( b 2 ) n ( c ) m + n m ! n ! x m y n , {\displaystyle F_{3}(a_{1},a_{2},b_{1},b_{2},c;x,y)=\sum _{m,n=0}^{\infty }{\frac {(a_{1})_{m}(a_{2})_{n}(b_{1})_{m}(b_{2})_{n}}{(c)_{m+n}\,m!\,n!}}\,x^{m}y^{n}~,} F 4 ( a , b , c 1 , c 2 ; x , y ) = ∑ m , n = 0 ∞ ( a ) m + n ( b ) m + n ( c 1 ) m ( c 2 ) n m ! n ! x m y n . {\displaystyle F_{4}(a,b,c_{1},c_{2};x,y)=\sum _{m,n=0}^{\infty }{\frac {(a)_{m+n}(b)_{m+n}}{(c_{1})_{m}(c_{2})_{n}\,m!\,n!}}\,x^{m}y^{n}~.} 其中的符号 : ( a ) m + n {\displaystyle :(a)_{m+n}} 是阶乘幂 阿佩尔函数是嫪丽切拉函数和Kampé_de_Fériet函数的特例。