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

:
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) = \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) = \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) = \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}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) ~.


:
\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} - a b_1 \right) F_1(x,y) = 0 ~,


:
\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} - a b_2 \right) 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) ~.


:
\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 ~,


:
\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
:
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 \real \,c > \real \,a > 0 ~.

## 特例
:
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 |\real \,\phi| 

:
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 |\real \,\phi| 

:
\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
Q阿佩尔函数

## 参考文献

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