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
)
=
∑
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}~.}
:
(
a
)
m
+
n
{\displaystyle :(a)_{m+n}}
阶乘幂
阿佩尔函数是嫪丽切拉函数 和Kampé_de_Fériet函数 的特例。
(
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 (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
,
{\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~,}
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(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
.
{\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[ 1]
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)+(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
,
{\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~,}
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}+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
,
{\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~,}
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
.
{\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~.}
∂
∂
x
F
1
(
a
,
b
1
,
b
2
,
c
;
x
,
y
)
=
a
b
1
c
F
1
(
a
+
1
,
b
1
+
1
,
b
2
,
c
+
1
;
x
,
y
)
,
{\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)~,}
∂
∂
y
F
1
(
a
,
b
1
,
b
2
,
c
;
x
,
y
)
=
a
b
2
c
F
1
(
a
+
1
,
b
1
,
b
2
+
1
,
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)~.}
(
x
(
1
−
x
)
∂
2
∂
x
2
+
y
(
1
−
x
)
∂
2
∂
x
∂
y
+
[
c
−
(
a
+
b
1
+
1
)
x
]
∂
∂
x
−
b
1
y
∂
∂
y
−
a
b
1
)
F
1
(
x
,
y
)
=
0
,
{\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~,}
(
y
(
1
−
y
)
∂
2
∂
y
2
+
x
(
1
−
y
)
∂
2
∂
x
∂
y
+
[
c
−
(
a
+
b
2
+
1
)
y
]
∂
∂
y
−
b
2
x
∂
∂
x
−
a
b
2
)
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~.}
∂
∂
x
F
3
(
a
1
,
a
2
,
b
1
,
b
2
,
c
;
x
,
y
)
=
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 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)~,}
∂
∂
y
F
3
(
a
1
,
a
2
,
b
1
,
b
2
,
c
;
x
,
y
)
=
a
2
b
2
c
F
3
(
a
1
,
a
2
+
1
,
b
1
,
b
2
+
1
,
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)~.}
(
x
(
1
−
x
)
∂
2
∂
x
2
+
y
∂
2
∂
x
∂
y
+
[
c
−
(
a
1
+
b
1
+
1
)
x
]
∂
∂
x
−
a
1
b
1
)
F
3
(
x
,
y
)
=
0
,
{\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~,}
(
y
(
1
−
y
)
∂
2
∂
y
2
+
x
∂
2
∂
x
∂
y
+
[
c
−
(
a
2
+
b
2
+
1
)
y
]
∂
∂
y
−
a
2
b
2
)
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~.}
F
1
(
a
,
b
1
,
b
2
,
c
;
x
,
y
)
=
Γ
(
c
)
Γ
(
a
)
Γ
(
c
−
a
)
∫
0
1
t
a
−
1
(
1
−
t
)
c
−
a
−
1
(
1
−
x
t
)
−
b
1
(
1
−
y
t
)
−
b
2
d
t
,
ℜ
c
>
ℜ
a
>
0
.
{\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~.}
F
(
ϕ
,
k
)
=
∫
0
ϕ
d
θ
1
−
k
2
sin
2
θ
=
sin
ϕ
F
1
(
1
2
,
1
2
,
1
2
,
3
2
;
sin
2
ϕ
,
k
2
sin
2
ϕ
)
,
|
ℜ
ϕ
|
<
π
2
,
{\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}}~,}
E
(
ϕ
,
k
)
=
∫
0
ϕ
1
−
k
2
sin
2
θ
d
θ
=
sin
ϕ
F
1
(
1
2
,
1
2
,
−
1
2
,
3
2
;
sin
2
ϕ
,
k
2
sin
2
ϕ
)
,
|
ℜ
ϕ
|
<
π
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}}~,}
Π
(
n
,
k
)
=
∫
0
π
/
2
d
θ
(
1
−
n
sin
2
θ
)
1
−
k
2
sin
2
θ
=
π
2
F
1
(
1
2
,
1
,
1
2
,
1
;
n
,
k
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})~.}
^ 例如：
(
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
)
{\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)}
Appell, Paul . Sur les séries hypergéométriques de deux variables et sur des équations différentielles linéaires aux dérivées partielles. Comptes rendus hebdomadaires des séances de l'Académie des sciences. 1880, 90 : 296–298 and 731–735. JFM 12.0296.01 （法语） . 3 (α,α',β,β',γ; x,y)" in C. R. Acad. Sci. 90 , pp. 977–980)Appell, Paul. Sur les fonctions hypergéométriques de deux variables . Journal de Mathématiques Pures et Appliquées . (3ème série). 1882, 8 : 173–216 [2015-04-04 ] . （原始内容 存档于2013-04-12） （法语） . Appell, Paul; Kampé de Fériet, Joseph . Fonctions hypergéométriques et hypersphériques; Polynômes d'Hermite. Paris: Gauthier–Villars. 1926. JFM 52.0361.13 （法语） . Askey, R. A.; Daalhuis, Adri B. Olde, 阿佩尔函数 , Olver, Frank W. J. ; Lozier, Daniel M.; Boisvert, Ronald F.; Clark, Charles W. (编), NIST Handbook of Mathematical Functions , Cambridge University Press, 2010, ISBN 978-0521192255MR 2723248 Bateman, H. ; Erdélyi, A. Higher Transcendental Functions, Vol. I (PDF) . New York: McGraw–Hill. 1953 [2015-04-04 ] . （原始内容存档 (PDF) 于2011-08-11）. Gradshteyn, Izrail' Solomonovich; Ryzhik, Iosif Moiseevich. Tablitsy integralov, summ, ryadov i proizvedeniy [Tables of integrals, sums, series and products] 5th. Moscow: Nauka. 1971 （俄语） . Humbert, Pierre . Sur les fonctions hypercylindriques. Comptes rendus hebdomadaires des séances de l'Académie des sciences. 1920, 171 : 490–492. JFM 47.0348.01 （法语） . Lauricella, Giuseppe . Sulle funzioni ipergeometriche a più variabili. Rendiconti del Circolo Matematico di Palermo . 1893, 7 : 111–158. JFM 25.0756.01 doi:10.1007/BF03012437 （意大利语） . Picard, Émile . Sur une extension aux fonctions de deux variables du problème de Riemann relativ aux fonctions hypergéométriques . Annales scientifiques de l'École Normale Supérieure. (2ème série). 1881, 10 : 305–322 [2015-04-04 ] . JFM 13.0389.01 存档 于2015-01-21） （法语） . C. R. Acad. Sci. 90 (1880), pp. 1119–1121 and 1267–1269)Slater, Lucy Joan . Generalized hypergeometric functions . Cambridge, UK: Cambridge University Press. 1966. ISBN 0-521-06483-XMR 0201688 ISBN 978-0-521-09061-2 ) 
![{\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~,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7ecd9e46eb277f753c463528965f37c14dc16100)
![{\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~.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8137c25373d0eec94366e92e2292229f5e0199bc)
![{\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~,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fd61d488a75f9b17046a890e58cb1ee15d731110)
![{\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~.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/af44ba7394aa6a7152069fbb91aceaec237cd2f9)
