Uppsättning av fyra hypergeometriska serier
I matematik är Appell- serier en uppsättning av fyra hypergeometriska serier F 1 , F 2 , F 3 , F 4 av två variabler som introducerades av Paul Appell ( 1880 ) och som generaliserar Gauss hypergeometriska serie 2 F 1 av en variabel. Appell etablerade uppsättningen av partiella differentialekvationer där dessa funktioner är lösningar, och hittade olika reduktionsformler och uttryck för dessa serier i form av hypergeometriska serier av en variabel.
Definitioner
Appell-serien F 1 är definierad för | x | < 1, | y | < 1 av dubbelserien
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)=\summa _{m,n=0}^{\infty }{\frac {(a)_{m+n}(b_{1})_{m}(b_{2})_{n}}{(c)_{m+n}\,m!\,n!} }\,x^{m}y^{n}~,}
där
( q
)
n
{\displaystyle (q)_{n}}
Pochhammer-symbolen . För andra värden på x och y kan funktionen F 1 definieras genom analytisk fortsättning . Det kan man visa
F
1
( a ,
b
1
,
b
2
; c ; x , y ) =
∑
r =
0
∞
( a
)
r
(
b
1
)
r
(
b
2
)
r
( c − a
)
r
( c + r − 1
)
r
( c
)
2 r
r !
x
r
y
r
2
F
1
(
a + r ,
b
1
+ r ; c + 2 r ; x
)
2
F
1
(
a + r ,
b
2
+ r ; c + 2 r ; y
)
.
{\displaystyle F_{1}(a,b_{1},b_{2};c;x,y)=\summa _{r=0}^{\infty }{\frac {(a)_{r }(b_{1})_{r}(b_{2})_{r}(ca)_{r}}{(c+r-1)_{r}(c)_{2r}r! }}\,x^{r}y^{r}{}_{2}F_{1}\left(a+r,b_{1}+r;c+2r;x\right){}_{ 2}F_{1}\left(a+r,b_{2}+r;c+2r;y\right)~.}
På liknande sätt är funktionen F 2 definierad för | x | + | y | < 1 av serien
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)=\summa _{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}}
och det kan man visa
F
2
( a ,
b
1
,
b
2
;
c
1
,
c
2
; x , y ) =
∑
r =
0
∞
( a
)
r
(
b
1
)
r
(
b
2
)
r
(
c
1
)
r
(
c
2
)
r
r !
x
r
y
r
2
F
1
(
a + r ,
b
1
+ r ;
c
1
+ r ; x
)
2
F
1
(
a + r ,
b
2
+ r ;
c
2
+ r ; y
)
.
{\displaystyle F_{2}(a,b_{1},b_{2};c_{1},c_{2};x,y)=\summa _{r=0}^{\infty }{\ frac {(a)_{r}(b_{1})_{r}(b_{2})_{r}}{(c_{1})_{r}(c_{2})_{r }r!}}\,x^{r}y^{r}{}_{2}F_{1}\left(a+r,b_{1}+r;c_{1}+r;x\ höger){}_{2}F_{1}\left(a+r,b_{2}+r;c_{2}+r;y\right)~.}
Även funktionen F 3 för | x | < 1, | y | < 1 kan definieras av serien
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)=\summa _{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}~,}
och funktionen F 4 för | x | ½ + | y | ½ < 1 av serien
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)=\summa _{m,n=0}^{\infty }{\frac {(a)_ {m+n}(b)_{m+n}}{(c_{1})_{m}(c_{2})_{n}\,m!\,n!}}\,x^ {m}y^{n}~.}
Återkommande relationer
Liksom den Gauss hypergeometriska serien 2 F 1 , innebär Appell dubbelserier återkommande relationer mellan sammanhängande funktioner. Till exempel ges en grundläggande uppsättning av sådana relationer för Appells F 1 av:
0
( 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 ) = ,
{\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~,}
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 ) = ,
{\displaystyle c\, F_{1}(a,b_{1},b_{2},c;x,y)-(ca)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~,}
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 ) = ,
{\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)-(ca)x\,F_{1}(a,b_{1}+1,b_{2},c +1;x,y)=0~,}
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 ) = .
{\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)-(ca)y\,F_{1}(a,b_{1},b_{2}+1,c+1;x,y)=0~.}
Varje annan relation som är giltig för F 1 kan härledas från dessa fyra.
På samma sätt följer alla återkommande relationer för Appells F 3 från denna uppsättning av fem:
0
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 ) = ,
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}
0
a2
}
c (
a1
c
, , , b_{2} , c+
F3
1
b1
;
,
b2
,
; 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 ) = ,
{\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~,}
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 ) = ,
{\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~,}
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 ) = ,
{\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~,}
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 ) = .
{\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~.}
Derivater och differentialekvationer
För Appells F 1 resulterar följande derivator från definitionen av en dubbel serie:
∂
n
∂
x
n
F
1
( a ,
b
1
,
b
2
, c ; x , y ) =
( a )
n
(
b
1
)
n
( c )
n
F
1
( a + n ,
b
1
+ n ,
b
2 )
, c + n ; x , y )
{\displaystyle {\frac {\partial ^{n}}{\partial x^{n}}}F_{1}(a,b_{1},b_{2}, c;x,y)={\frac {\left(a\right)_{n}\left(b_{1}\right)_{n}}{\left(c\right)_{n}} }F_{1}(a+n,b_{1}+n,b_{2},c+n;x,y)}
∂
n
∂
y
n
F
1
( a ,
b
1
,
b
2
, c ; x , y ) =
( a )
n
(
b
2
)
n
( c )
n
F
1
( a + n ,
b
1
,
b
2
+ n , c + n ; x , y )
{\displaystyle {\frac {\partial ^ {n}}{\partial y^{n}}}F_{1}(a,b_{1},b_{2},c;x,y)={\frac {\left(a\right)_ {n}\left(b_{2}\right)_{n}}{\left(c\right)_{n}}}F_{1}(a+n,b_{1},b_{2} +n,c+n;x,y)}
Från dess definition har Appells F 1 vidare befunnits uppfylla följande system av andra ordningens differentialekvationer :
x ( 1 − x )
∂
2
F
1
( x , y )
∂
x
2
+ y ( 1 − x )
∂
2
F
1
( x , y )
∂ x ∂ y
+ [ c − ( a +
b
1
+ 1 ) x ]
∂
F
1
( x , y )
∂ x
−
b
1
y
∂
F
1
( x , y )
∂ y
− a
b
1
F
1
( x , y ) =
0
{\displaystyle x(1-x){\frac {\partial ^{2}F_{1}(x,y)}{\partial x^{2}}}+y(1-x){\frac {\partial ^{2}F_{1}(x ,y)}{\partial x\partial y}}+[c-(a+b_{1}+1)x]{\frac {\partial F_{1}(x,y)}{\partial x} }-b_{1}y{\frac {\partial F_{1}(x,y)}{\partial y}}-ab_{1}F_{1}(x,y)=0}
y ( 1 − y )
∂
2
F
1
( x , y )
∂
y
2
+ x ( 1 − y )
∂
2
F
1
( x , y )
∂ x ∂ y
+ [ c − ( a +
b
2
+ 1 ) y ]
∂
F
1
( x , y )
∂ y
−
b
2
x
∂
F
1
( x , y )
∂ x
− a
b
2
F
1
( x , y ) =
0
{\displaystyle y(1-y){\frac {\partial ^ {2}F_{1}(x,y)}{\partial y^{2}}}+x(1-y){\frac {\partial ^{2}F_{1}(x,y)} {\partial x\partial y}}+[c-(a+b_{2}+1)y]{\frac {\partial F_{1}(x,y)}{\partial y}}-b_{ 2}x{\frac {\partial F_{1}(x,y)}{\partial x}}-ab_{2}F_{1}(x,y)=0}
Ett system partiella differentialekvationer för F 2 är
x ( 1 − x )
∂
2
F
2
( x , y )
∂
x
2
− x y
∂
2
F
2
( x , y )
∂ x ∂ y
+ [
c
1
− ( a +
b
1
+ 1 ) x ]
∂
F
2
( x , y )
∂ x
−
b
1
y
∂
F
2
( x , y )
∂ y
− a
b
1
F
2
( x , y ) =
0
{\displaystyle x(1-x){\frac {\partial ^{2}F_{2}(x,y)}{\partial x^{2}}}-xy{\frac {\partial ^{2}F_{2}(x,y)}{\partial x \partial y}}+[c_{1}-(a+b_{1}+1)x]{\frac {\partial F_{2}(x,y)}{\partial x}}-b_{1 }y{\frac {\partial F_{2}(x,y)}{\partial y}}-ab_{1}F_{2}(x,y)=0}
y ( 1 − y )
∂
2
F
2
( x , y )
∂
y
2
− x y
∂
2
F
2
( x , y )
∂ x ∂ y
+ [
c
2
− ( a +
b
2
+ 1 ) y ]
∂
F
2
( x , y )
∂ y
−
b
2
x
∂
F
2
( x , y )
∂ x
− a
b
2
F
2
( x , y ) =
0
{\displaystyle y(1-y){\frac {\partial ^{2}F_{2}( x,y)}{\partial y^{2}}}-xy{\frac {\partial ^{2}F_{2}(x,y)}{\partial x\partial y}}+[c_{ 2}-(a+b_{2}+1)y]{\frac {\partial F_{2}(x,y)}{\partial y}}-b_{2}x{\frac {\partial F_ {2}(x,y)}{\partial x}}-ab_{2}F_{2}(x,y)=0}
Systemet har en lösning
F
2
( x , y ) =
C
1
F
2
( a ,
b
1
,
b
2
,
c
1
,
c
2
; x , y ) +
C
2
x
1 −
c
1
F
2
( a −
c
1
+ 1 ,
b
1
−
c
1
+ 1 ,
b
2
, 2 −
c
1
,
c
2
; x , y ) +
C
3
y
1 −
c
2
F
2
( a −
c
2
+ 1 ,
b
1
,
b
2
−
c
2
+ 1 ,
c
1
, 2 −
c
2
; x , y ) +
C
4
x
1 −
c
1
y
1 −
c
2
F
2
( a −
c
1
−
c
2
+ 2 ,
b
1
−
c
1
+ 1 ,
b
2
−
c
2
+ 1 , 2 −
c
1
, 2 −
c
2
; x , y )
{\displaystyle F_{2}(x,y)=C_{1}F_{2}(a,b_{1},b_{ 2},c_{1},c_{2};x,y)+C_{2}x^{1-c_{1}}F_{2}(a-c_{1}+1,b_{1} -c_{1}+1,b_{2},2-c_{1},c_{2};x,y)+C_{3}y^{1-c_{2}}F_{2}(a -c_{2}+1,b_{1},b_{2}-c_{2}+1,c_{1},2-c_{2};x,y)+C_{4}x^{1 -c_{1}}y^{1-c_{2}}F_{2}(a-c_{1}-c_{2}+2,b_{1}-c_{1}+1,b_{2 }-c_{2}+1,2-c_{1},2-c_{2};x,y)}
På liknande sätt, för F 3 resulterar följande derivator från definitionen:
∂
∂ 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)}
Och för F 3 erhålls följande system av differentialekvationer:
x ( 1 − x )
∂
2
F
3
( x , y )
∂
x
2
+ y
∂
2
F
3
( x , y )
∂ x ∂ y
+ [ c − (
a
1
+
b
1
+ 1 ) x ]
∂
F
3
( x , y )
∂ x
−
a
1
b
1
F
3
( x , y ) =
0
{\displaystyle x(1-x){\frac {\partial ^{2}F_{3}(x,y)} {\partial x^{2}}}+y{\frac {\partial ^{2}F_{3}(x,y)}{\partial x\partial y}}+[c-(a_{1} +b_{1}+1)x]{\frac {\partial F_{3}(x,y)}{\partial x}}-a_{1}b_{1}F_{3}(x,y) =0}
y ( 1 − y )
∂
2
F
3
( x , y )
∂
y
2
+ x
∂
2
F
3
( x , y )
∂ x ∂ y
+ [ c − (
a
2
+
b
2
+ 1 ) y ]
∂
F
3
( x , y )
∂ y
−
a
2
b
2
F
3
( x , y ) =
0
{\displaystyle y(1-y){\frac {\partial ^{2}F_{3}(x, y)}{\partial y^{2}}}+x{\frac {\partial ^{2}F_{3}(x,y)}{\partial x\partial y}}+[c-(a_ {2}+b_{2}+1)y]{\frac {\partial F_{3}(x,y)}{\partial y}}-a_{2}b_{2}F_{3}(x ,y)=0}
Ett system partiella differentialekvationer för F 4 är
x ( 1 − x )
∂
2
F
4
( x , y )
∂
x
2
−
y
2
∂
2
F
4
( x , y )
∂
y
2
− 2 x y
∂
2
F
4
( x , y )
∂ y x ∂
+ [
c
1
− ( a + b + 1 ) x ]
∂
F
4
( x , y )
∂ x
− ( a + b + 1 ) y
∂
F
4
( x , y )
∂ y
− a b
F
4
( x , y ) =
0
{\displaystyle x(1-x){\frac {\partial ^{2}F_{4}(x,y)}{\partial x^{2}}}-y^{2}{ \frac {\partial ^{2}F_{4}(x,y)}{\partial y^{2}}}-2xy{\frac {\partial ^{2}F_{4}(x,y) }{\partial x\partial y}}+[c_{1}-(a+b+1)x]{\frac {\partial F_{4}(x,y)}{\partial x}}-( a+b+1)y{\frac {\partial F_{4}(x,y)}{\partial y}}-abF_{4}(x,y)=0}
y ( 1 − y )
∂
2
F
4
( x , y )
∂
y
2
−
x
2
∂
2
F
4
( x , y )
∂
x
2
− 2 x y
∂
2
F
4
( x , y )
∂ x ∂ y
+ [
c
2
− ( a + b + 1 ) y ]
∂
F
4
( x , y )
∂ y
− ( a + b + 1 ) x
∂
F
4
( x , y )
∂ x
− a b
F
4
( x , y ) =
0
{\displaystyle y (1-y){\frac {\partial ^{2}F_{4}(x,y)}{\partial y^{2}}}-x^{2}{\frac {\partial ^{2 }F_{4}(x,y)}{\partial x^{2}}}-2xy{\frac {\partial ^{2}F_{4}(x,y)}{\partial x\partial y }}+[c_{2}-(a+b+1)y]{\frac {\partial F_{4}(x,y)}{\partial y}}-(a+b+1)x{ \frac {\partial F_{4}(x,y)}{\partial x}}-abF_{4}(x,y)=0}
Systemet har en lösning
F
4
( x , y ) =
C
1
F
4
( a , b ,
c
1
,
c
2
; x , y ) +
C
2
x
1 −
c
1
F
4
( a −
c
1
+ 1 , b −
c
1
+ 1 , 2 −
c
1
,
c
2
; x , y ) +
C
3
y
1 −
c
2
F
4
( a −
c
2
+ 1 , b −
c
2
+ 1 ,
c
1
, 2 −
c
2
; x , y ) +
C
4
x
1 −
c
1
y
1 −
c
2
F
4
( 2 + a −
c
1
−
c
2
, 2 + b −
c
1
−
c
2
, 2 −
c
1
, 2 −
c
2
; x , y )
{\displaystyle F_{4}(x,y)=C_{1}F_{4}(a,b,c_{1},c_{2};x,y)+C_{2}x^{1 -c_{1}}F_{4}(a-c_{1}+1,b-c_{1}+1,2-c_{1},c_{2};x,y)+C_{3} y^{1-c_{2}}F_{4}(a-c_{2}+1,b-c_{2}+1,c_{1},2-c_{2};x,y)+ C_{4}x^{1-c_{1}}y^{1-c_{2}}F_{4}(2+a-c_{1}-c_{2},2+b-c_{1 }-c_{2},2-c_{1},2-c_{2};x,y)}
Integral representationer
De fyra funktionerna som definieras av Appells dubbla serier kan representeras i termer av dubbla integraler som endast involverar elementära funktioner ( Gradshteyn et al. 2015, §9.184). Emellertid Émile Picard ( 1881 ) att Appells F 1 också kan skrivas som en endimensionell integral av Euler -typ :
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 > .
{\displaystyle F_{1}(a,b_{1},b_{2},c;x,y)={\frac {\Gamma (c)}{\Gamma (a)\Gamma (ca)}} \int _{0}^{1}t^{a-1}(1-t)^{ca-1}(1-xt)^{-b_{1}}(1-yt)^{-b_ {2}}\,\mathrm {d} t,\quad \Re \,c>\Re \,a>0~.}
Denna representation kan verifieras med hjälp av Taylor-expansion av integranden, följt av termwise integration.
Speciella fall
Picards integralrepresentation antyder att de ofullständiga elliptiska integralerna F och E samt den kompletta elliptiska integralen Π är specialfall av Appells F 1 :
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
sin2
_
; 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})~.}
Relaterad serie
Appell, Paul (1880). "Sur les serie hypergéométriques de deux variables et sur des équations différentielles linéaires aux dérivées partielles". Comptes rendus hebdomadaires des séances de l'Académie des sciences (på franska). 90 : 296–298 och 731–735. JFM 12.0296.01 . (se även "Sur la série F 3 (α,α',β,β',γ; x,y)" i CR Acad. Sci. 90 , s. 977–980)
Appell, Paul (1882). "Sur les fonctions hypergéométriques deux variables" . Journal de Mathématiques Pures et Appliquées 8 : 173–216. Arkiverad från originalet den 12 april 2013.
Appell, Paul; Kampé de Fériet, Joseph (1926). Funktioner hypergéométriques et hypersphériques; Polynômes d'Hermite (på franska). Paris: Gauthier–Villars. JFM 52.0361.13 . (se sid. 14)
Askey, RA; Olde Daalhuis, AB (2010), "Appell series" , i Olver, Frank WJ ; Lozier, Daniel M.; Boisvert, Ronald F.; Clark, Charles W. (red.), NIST Handbook of Mathematical Functions ISBN 978-0-521-19225-5 MR 2723248
Burchnall, JL; Chaundy, TW (1940). "Utökningar av Appells dubbla hypergeometriska funktioner". QJ Math . Första serien. 11 : 249–270. doi : 10.1093/qmath/os-11.1.249 .
Erdélyi, A. (1953). Higher Transcendental Functions, Vol. Jag (PDF) . New York: McGraw-Hill. (se sid. 224)
Gradshteyn, Izrail Solomonovich ; Ryzhik, Iosif Moiseevich ; Geronimus, Yuri Veniaminovich ; Tseytlin, Michail Yulyevich ; Jeffrey, Alan (2015) [oktober 2014]. "9.18.". I Zwillinger, Daniel; Moll, Victor Hugo (red.). Tabell över integraler, serier och produkter Academic Press, Inc. ISBN 978-0-12-384933-5 LCCN 2014010276 .
Humbert, Pierre (1920). "Sur les fonctions hypercylindriques". Comptes rendus hebdomadaires des séances de l'Académie des sciences (på franska). 171 : 490–492. JFM 47.0348.01 .
Lauricella, Giuseppe (1893). "Sulle funzioni ipergeometriche a più variabili". Rendiconti del Circolo Matematico di Palermo 7 : 111-158. doi : 10.1007/BF03012437 . JFM 25.0756.01 . S2CID 122316343 .
Picard, Émile (1881). "Sur une extension aux fonctions de deux variables du problème de Riemann relative aux fonctions hypergéométriques" . Annales Scientifiques de l'École Normale Supérieure . Série 2 (på franska). 10 : 305-322. doi : 10.24033/asens.203 JFM 13.0389.01 . (se även CR Acad. Sci. 90 (1880), s. 1119–1121 och 1267–1269)
Slater, Lucy Joan (1966). Generaliserade hypergeometriska funktioner ISBN 0-521-06483-X MR 0201688 . (det finns en pocketbok från 2008 med ISBN 978-0-521-09061-2 )
externa länkar
![{\displaystyle x(1-x){\frac {\partial ^{2}F_{1}(x,y)}{\partial x^{2}}}+y(1-x){\frac {\partial ^{2}F_{1}(x,y)}{\partial x\partial y}}+[c-(a+b_{1}+1)x]{\frac {\partial F_{1}(x,y)}{\partial x}}-b_{1}y{\frac {\partial F_{1}(x,y)}{\partial y}}-ab_{1}F_{1}(x,y)=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8706133888a0e338e4de551a4972366985207f9d)
![{\displaystyle y(1-y){\frac {\partial ^{2}F_{1}(x,y)}{\partial y^{2}}}+x(1-y){\frac {\partial ^{2}F_{1}(x,y)}{\partial x\partial y}}+[c-(a+b_{2}+1)y]{\frac {\partial F_{1}(x,y)}{\partial y}}-b_{2}x{\frac {\partial F_{1}(x,y)}{\partial x}}-ab_{2}F_{1}(x,y)=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1bff5fa0629d1a666987c3de41c19c844051e462)
![{\displaystyle x(1-x){\frac {\partial ^{2}F_{2}(x,y)}{\partial x^{2}}}-xy{\frac {\partial ^{2}F_{2}(x,y)}{\partial x\partial y}}+[c_{1}-(a+b_{1}+1)x]{\frac {\partial F_{2}(x,y)}{\partial x}}-b_{1}y{\frac {\partial F_{2}(x,y)}{\partial y}}-ab_{1}F_{2}(x,y)=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3d21a04bed4e7413e8920bbc7f497839ce1d825d)
![{\displaystyle y(1-y){\frac {\partial ^{2}F_{2}(x,y)}{\partial y^{2}}}-xy{\frac {\partial ^{2}F_{2}(x,y)}{\partial x\partial y}}+[c_{2}-(a+b_{2}+1)y]{\frac {\partial F_{2}(x,y)}{\partial y}}-b_{2}x{\frac {\partial F_{2}(x,y)}{\partial x}}-ab_{2}F_{2}(x,y)=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f65ead117e29aac6a889284b1053fefc246ebac8)
![{\displaystyle x(1-x){\frac {\partial ^{2}F_{3}(x,y)}{\partial x^{2}}}+y{\frac {\partial ^{2}F_{3}(x,y)}{\partial x\partial y}}+[c-(a_{1}+b_{1}+1)x]{\frac {\partial F_{3}(x,y)}{\partial x}}-a_{1}b_{1}F_{3}(x,y)=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/424d663fa4f5968774d6161e1e71b0b266e9e32d)
![{\displaystyle y(1-y){\frac {\partial ^{2}F_{3}(x,y)}{\partial y^{2}}}+x{\frac {\partial ^{2}F_{3}(x,y)}{\partial x\partial y}}+[c-(a_{2}+b_{2}+1)y]{\frac {\partial F_{3}(x,y)}{\partial y}}-a_{2}b_{2}F_{3}(x,y)=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6536613b1adff04079e9165521549cf56d2dabdf)
![{\displaystyle x(1-x){\frac {\partial ^{2}F_{4}(x,y)}{\partial x^{2}}}-y^{2}{\frac {\partial ^{2}F_{4}(x,y)}{\partial y^{2}}}-2xy{\frac {\partial ^{2}F_{4}(x,y)}{\partial x\partial y}}+[c_{1}-(a+b+1)x]{\frac {\partial F_{4}(x,y)}{\partial x}}-(a+b+1)y{\frac {\partial F_{4}(x,y)}{\partial y}}-abF_{4}(x,y)=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/84c1762747b2fa7376e0fb2c8021f60a750a0531)
![{\displaystyle y(1-y){\frac {\partial ^{2}F_{4}(x,y)}{\partial y^{2}}}-x^{2}{\frac {\partial ^{2}F_{4}(x,y)}{\partial x^{2}}}-2xy{\frac {\partial ^{2}F_{4}(x,y)}{\partial x\partial y}}+[c_{2}-(a+b+1)y]{\frac {\partial F_{4}(x,y)}{\partial y}}-(a+b+1)x{\frac {\partial F_{4}(x,y)}{\partial x}}-abF_{4}(x,y)=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d908adf62352b754c3f772f24b2a81e342a67ddc)
