阿佩尔函数

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

$F_1(a,b_1,b_2,c;x,y)=\sum _{m,n=0}^{\mathrm{\infty }}\frac{(a{)}_{m+n}(b_1{)}_m(b_2{)}_n}{(c{)}_{m+n}m!n!}{x}^m{y}^n,$

$F_2(a,b_1,b_2,c_1,c_2;x,y)=\sum _{m,n=0}^{\mathrm{\infty }}\frac{(a{)}_{m+n}(b_1{)}_m(b_2{)}_n}{(c_1{)}_m(c_2{)}_{n}m!n!}{x}^m{y}^n,$

$F_3(a_1,a_2,b_1,b_2,c;x,y)=\sum _{m,n=0}^{\mathrm{\infty }}\frac{(a_1{)}_m(a_2{)}_n(b_1{)}_m(b_2{)}_n}{(c{)}_{m+n}m!n!}{x}^m{y}^n,$

$F_4(a,b,c_1,c_2;x,y)=\sum _{m,n=0}^{\mathrm{\infty }}\frac{(a{)}_{m+n}(b{)}_{m+n}}{(c_1{)}_m(c_2{)}_{n}m!n!}{x}^m{y}^n.$

其中的符号$:(a{)}_{m+n}$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

$(a-b_1-b_2)F_1(a,b_1,b_2,c;x,y)-a{F}_1(a+1,b_1,b_2,c;x,y)+b_1{F}_1(a,b_1+1,b_2,c;x,y)+b_2{F}_1(a,b_1,b_2+1,c;x,y)=0,$

$c{F}_1(a,b_1,b_2,c;x,y)-(c-a)F_1(a,b_1,b_2,c+1;x,y)-a{F}_1(a+1,b_1,b_2,c+1;x,y)=0,$

$c{F}_1(a,b_1,b_2,c;x,y)+c(x-1)F_1(a,b_1+1,b_2,c;x,y)-(c-a)x{F}_1(a,b_1+1,b_2,c+1;x,y)=0,$

$c{F}_1(a,b_1,b_2,c;x,y)+c(y-1)F_1(a,b_1,b_2+1,c;x,y)-(c-a)y{F}_1(a,b_1,b_2+1,c+1;x,y)=0.$

其它式子可從這四個關係導出。

$c{F}_3(a_1,a_2,b_1,b_2,c;x,y)+(a_1+a_2-c)F_3(a_1,a_2,b_1,b_2,c+1;x,y)-a_1{F}_3(a_1+1,a_2,b_1,b_2,c+1;x,y)-a_2{F}_3(a_1,a_2+1,b_1,b_2,c+1;x,y)=0,$

$c{F}_3(a_1,a_2,b_1,b_2,c;x,y)-c{F}_3(a_1+1,a_2,b_1,b_2,c;x,y)+b_{1}x{F}_3(a_1+1,a_2,b_1+1,b_2,c+1;x,y)=0,$

$c{F}_3(a_1,a_2,b_1,b_2,c;x,y)-c{F}_3(a_1,a_2+1,b_1,b_2,c;x,y)+b_{2}y{F}_3(a_1,a_2+1,b_1,b_2+1,c+1;x,y)=0,$

$c{F}_3(a_1,a_2,b_1,b_2,c;x,y)-c{F}_3(a_1,a_2,b_1+1,b_2,c;x,y)+a_{1}x{F}_3(a_1+1,a_2,b_1+1,b_2,c+1;x,y)=0,$

$c{F}_3(a_1,a_2,b_1,b_2,c;x,y)-c{F}_3(a_1,a_2,b_1,b_2+1,c;x,y)+a_{2}y{F}_3(a_1,a_2+1,b_1,b_2+1,c+1;x,y)=0.$

导数与微分方程

$\frac{\partial }{\partial x}{F}_1(a,b_1,b_2,c;x,y)=\frac{a{b}_1}{c}{F}_1(a+1,b_1+1,b_2,c+1;x,y),$

$\frac{\partial }{\partial y}{F}_1(a,b_1,b_2,c;x,y)=\frac{a{b}_2}{c}{F}_1(a+1,b_1,b_2+1,c+1;x,y).$

$(x(1-x)\frac{{\partial }^2}{\partial x^2}+y(1-x)\frac{{\partial }^2}{\partial x\partial y}+[c-(a+b_1+1)x]\frac{\partial }{\partial x}-b_{1}y\frac{\partial }{\partial y}-a{b}_1)F_1(x,y)=0,$

$(y(1-y)\frac{{\partial }^2}{\partial y^2}+x(1-y)\frac{{\partial }^2}{\partial x\partial y}+[c-(a+b_2+1)y]\frac{\partial }{\partial y}-b_{2}x\frac{\partial }{\partial x}-a{b}_2)F_1(x,y)=0.$

$\frac{\partial }{\partial x}{F}_3(a_1,a_2,b_1,b_2,c;x,y)=\frac{a_1{b}_1}{c}{F}_3(a_1+1,a_2,b_1+1,b_2,c+1;x,y),$

$\frac{\partial }{\partial y}{F}_3(a_1,a_2,b_1,b_2,c;x,y)=\frac{a_2{b}_2}{c}{F}_3(a_1,a_2+1,b_1,b_2+1,c+1;x,y).$

$(x(1-x)\frac{{\partial }^2}{\partial x^2}+y\frac{{\partial }^2}{\partial x\partial y}+[c-(a_1+b_1+1)x]\frac{\partial }{\partial x}-a_1{b}_1)F_3(x,y)=0,$

$(y(1-y)\frac{{\partial }^2}{\partial y^2}+x\frac{{\partial }^2}{\partial x\partial y}+[c-(a_2+b_2+1)y]\frac{\partial }{\partial y}-a_2{b}_2)F_3(x,y)=0.$

积分关系

$F_1(a,b_1,b_2,c;x,y)=\frac{\mathrm{\Gamma }(c)}{\mathrm{\Gamma }(a)\mathrm{\Gamma }(c-a)}\underset{0}{\overset{1}{\int }}{t}^{a-1}(1-t{)}^{c-a-1}(1-xt{)}^{-b_1}(1-yt{)}^{-b_2}\mathrm{d}t,\mathrm{\Re }c>\mathrm{\Re }a>0.$

特例

$F(\varphi ,k)=\underset{0}{\overset{\varphi }{\int }}\frac{\mathrm{d}\theta }{\sqrt{1-k^2{\sin }^2\theta }}=\sin \varphi F_1(\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{3}{2};{\sin }^2\varphi ,k^2{\sin }^2\varphi ),|\mathrm{\Re }\varphi |<\frac{\pi }{2},$

$E(\varphi ,k)=\underset{0}{\overset{\varphi }{\int }}\sqrt{1-k^2{\sin }^2\theta }\mathrm{d}\theta =\sin \varphi F_1(\frac{1}{2},\frac{1}{2},-\frac{1}{2},\frac{3}{2};{\sin }^2\varphi ,k^2{\sin }^2\varphi ),|\mathrm{\Re }\varphi |<\frac{\pi }{2},$

$\mathrm{\Pi }(n,k)=\underset{0}{\overset{\pi /2}{\int }}\frac{\mathrm{d}\theta }{(1-n{\sin }^2\theta )\sqrt{1-k^2{\sin }^2\theta }}=\frac{\pi }{2}{F}_1(\frac{1}{2},1,\frac{1}{2},1;n,k^2).$

参见

Q阿佩尔函数

参考文献

• Template:Cite journal (see also "Sur la série F₃(α,α',β,β',γ; x,y)" in C. R. Acad. Sci. 90, pp. 977–980)
• Template:Cite journal
• Template:Cite book (see p. 14)
• Template:Dlmf
• Template:Cite book (see p. 224)
• Template:Cite book (see Chapter 9.18)
• Template:Cite journal
• Template:Cite journal
• Template:Cite journal (see also C. R. Acad. Sci. 90 (1880), pp. 1119–1121 and 1267–1269)
• Template:Cite book (there is a 2008 paperback with ISBN 978-0-521-09061-2)