**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雙变数函数，定义如下

阿佩尔函数——F1

  

{\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}~,}

  

{\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}~,}

  

{\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}~,}

  

{\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{\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。

归递关系
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{\displaystyle (a-b\_{1}-b\_{2})F\_{1}(a,b\_{1},b\_{2},c;x,y)-a\,F\_{1}(a+1,b\_{1},b\_{2},c;x,y)+b\_{1}F\_{1}(a,b\_{1}+1,b\_{2},c;x,y)+b\_{2}F\_{1}(a,b\_{1},b\_{2}+1,c;x,y)=0~,}
{\displaystyle c\,F\_{1}(a,b\_{1},b\_{2},c;x,y)-(c-a)F\_{1}(a,b\_{1},b\_{2},c+1;x,y)-a\,F\_{1}(a+1,b\_{1},b\_{2},c+1;x,y)=0~,}
{\displaystyle c\,F\_{1}(a,b\_{1},b\_{2},c;x,y)+c(x-1)F\_{1}(a,b\_{1}+1,b\_{2},c;x,y)-(c-a)x\,F\_{1}(a,b\_{1}+1,b\_{2},c+1;x,y)=0~,}
{\displaystyle c\,F\_{1}(a,b\_{1},b\_{2},c;x,y)+c(y-1)F\_{1}(a,b\_{1},b\_{2}+1,c;x,y)-(c-a)y\,F\_{1}(a,b\_{1},b\_{2}+1,c+1;x,y)=0~.}
其它式子可從這四個關係導出。

  

{\displaystyle c\,F\_{3}(a\_{1},a\_{2},b\_{1},b\_{2},c;x,y)+(a\_{1}+a\_{2}-c)F\_{3}(a\_{1},a\_{2},b\_{1},b\_{2},c+1;x,y)-a\_{1}F\_{3}(a\_{1}+1,a\_{2},b\_{1},b\_{2},c+1;x,y)-a\_{2}F\_{3}(a\_{1},a\_{2}+1,b\_{1},b\_{2},c+1;x,y)=0~,}
{\displaystyle c\,F\_{3}(a\_{1},a\_{2},b\_{1},b\_{2},c;x,y)-c\,F\_{3}(a\_{1}+1,a\_{2},b\_{1},b\_{2},c;x,y)+b\_{1}x\,F\_{3}(a\_{1}+1,a\_{2},b\_{1}+1,b\_{2},c+1;x,y)=0~,}
{\displaystyle c\,F\_{3}(a\_{1},a\_{2},b\_{1},b\_{2},c;x,y)-c\,F\_{3}(a\_{1},a\_{2}+1,b\_{1},b\_{2},c;x,y)+b\_{2}y\,F\_{3}(a\_{1},a\_{2}+1,b\_{1},b\_{2}+1,c+1;x,y)=0~,}
{\displaystyle c\,F\_{3}(a\_{1},a\_{2},b\_{1},b\_{2},c;x,y)-c\,F\_{3}(a\_{1},a\_{2},b\_{1}+1,b\_{2},c;x,y)+a\_{1}x\,F\_{3}(a\_{1}+1,a\_{2},b\_{1}+1,b\_{2},c+1;x,y)=0~,}
{\displaystyle c\,F\_{3}(a\_{1},a\_{2},b\_{1},b\_{2},c;x,y)-c\,F\_{3}(a\_{1},a\_{2},b\_{1},b\_{2}+1,c;x,y)+a\_{2}y\,F\_{3}(a\_{1},a\_{2}+1,b\_{1},b\_{2}+1,c+1;x,y)=0~.}

导数与微分方程
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{\displaystyle {\frac {\partial }{\partial x}}F\_{1}(a,b\_{1},b\_{2},c;x,y)={\frac {ab\_{1}}{c}}F\_{1}(a+1,b\_{1}+1,b\_{2},c+1;x,y)~,}
{\displaystyle {\frac {\partial }{\partial y}}F\_{1}(a,b\_{1},b\_{2},c;x,y)={\frac {ab\_{2}}{c}}F\_{1}(a+1,b\_{1},b\_{2}+1,c+1;x,y)~.}

  

{\displaystyle \left(x(1-x){\frac {\partial ^{2}}{\partial x^{2}}}+y(1-x){\frac {\partial ^{2}}{\partial x\partial y}}+[c-(a+b\_{1}+1)x]{\frac {\partial }{\partial x}}-b\_{1}y{\frac {\partial }{\partial y}}-ab\_{1}\right)F\_{1}(x,y)=0~,}
{\displaystyle \left(y(1-y){\frac {\partial ^{2}}{\partial y^{2}}}+x(1-y){\frac {\partial ^{2}}{\partial x\partial y}}+[c-(a+b\_{2}+1)y]{\frac {\partial }{\partial y}}-b\_{2}x{\frac {\partial }{\partial x}}-ab\_{2}\right)F\_{1}(x,y)=0~.}

  

{\displaystyle {\frac {\partial }{\partial x}}F\_{3}(a\_{1},a\_{2},b\_{1},b\_{2},c;x,y)={\frac {a\_{1}b\_{1}}{c}}F\_{3}(a\_{1}+1,a\_{2},b\_{1}+1,b\_{2},c+1;x,y)~,}
{\displaystyle {\frac {\partial }{\partial y}}F\_{3}(a\_{1},a\_{2},b\_{1},b\_{2},c;x,y)={\frac {a\_{2}b\_{2}}{c}}F\_{3}(a\_{1},a\_{2}+1,b\_{1},b\_{2}+1,c+1;x,y)~.}

  

{\displaystyle \left(x(1-x){\frac {\partial ^{2}}{\partial x^{2}}}+y{\frac {\partial ^{2}}{\partial x\partial y}}+[c-(a\_{1}+b\_{1}+1)x]{\frac {\partial }{\partial x}}-a\_{1}b\_{1}\right)F\_{3}(x,y)=0~,}
{\displaystyle \left(y(1-y){\frac {\partial ^{2}}{\partial y^{2}}}+x{\frac {\partial ^{2}}{\partial x\partial y}}+[c-(a\_{2}+b\_{2}+1)y]{\frac {\partial }{\partial y}}-a\_{2}b\_{2}\right)F\_{3}(x,y)=0~.}

积分关系
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{\displaystyle F\_{1}(a,b\_{1},b\_{2},c;x,y)={\frac {\Gamma (c)}{\Gamma (a)\Gamma (c-a)}}\int \_{0}^{1}t^{a-1}(1-t)^{c-a-1}(1-xt)^{-b\_{1}}(1-yt)^{-b\_{2}}\,\mathrm {d} t,\quad \Re \,c>\Re \,a>0~.}

特例
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{\displaystyle F(\phi ,k)=\int \_{0}^{\phi }{\frac {\mathrm {d} \theta }{\sqrt {1-k^{2}\sin ^{2}\theta }}}=\sin \phi \,F\_{1}({\tfrac {1}{2}},{\tfrac {1}{2}},{\tfrac {1}{2}},{\tfrac {3}{2}};\sin ^{2}\phi ,k^{2}\sin ^{2}\phi ),\quad |\Re \,\phi |<{\frac {\pi }{2}}~,}
{\displaystyle E(\phi ,k)=\int \_{0}^{\phi }{\sqrt {1-k^{2}\sin ^{2}\theta }}\,\mathrm {d} \theta =\sin \phi \,F\_{1}({\tfrac {1}{2}},{\tfrac {1}{2}},-{\tfrac {1}{2}},{\tfrac {3}{2}};\sin ^{2}\phi ,k^{2}\sin ^{2}\phi ),\quad |\Re \,\phi |<{\frac {\pi }{2}}~,}
{\displaystyle \Pi (n,k)=\int \_{0}^{\pi /2}{\frac {\mathrm {d} \theta }{(1-n\sin ^{2}\theta ){\sqrt {1-k^{2}\sin ^{2}\theta }}}}={\frac {\pi }{2}}\,F\_{1}({\tfrac {1}{2}},1,{\tfrac {1}{2}},1;n,k^{2})~.}

参见
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Q阿佩尔函数

参考文献
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1. ↑  例如：{\displaystyle (y-x)F\_{1}(a,b\_{1}+1,b\_{2}+1,c,x,y)=y\,F\_{1}(a,b\_{1},b\_{2}+1,c,x,y)-x\,F\_{1}(a,b\_{1}+1,b\_{2},c,x,y)}

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