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International Journal of Scientific and Research Publications, Volume 2, Issue 10, October 2012 45
ISSN 2250-3153
⎡
⎢
= ⎣
Γ(
p + q + 2) Γ(
p + q + 2) ⎤
+
(2)
2Γ(
q + 1) 2Γ(
q + 1)
⎥ Γ
⎦
Γ(
p + q + 2)
Γ(
q + 1) Γ(p
+ 1
= ) [Γ(q+1)+Γ(p+1)] (2.2.13)
This completes the derivation of (2.2.8)
2.3 In this Section Quadruple hypergeometric function reduced to the Appell hypergeometric function
(1) Theorem By specializing the parameters of K 12 ,K 10 ,K 15 we obtain the following
K 12 (a, a, a, a; b 1 , b 2 , b 3 , b 4 , c 1 , c 1 , c 2 , c 2 ; x,y,z,t)
= F1 (a, b 1 , b 2 ; c 1 ; x, y) F 1 (a+m+n, b 3 , b 4 ; c 2 ; z,t) (2.3.1)
K 12 (a, a, a, a; b 1 , b 2 ,b 3 ,b 4 , c 1 ,c 1 , c 2 ,c 2 ; 1,1,1,1)
=
Γ(
c1)
Γ(
c1
− a − b
Γ(
c − a)
Γ(
c − b
1
1
1
1
− b2
) Γ(
c2
) Γ(
c2
− a − m − n − b3
− b4
)
.
− b ) Γ(
c − a)
Γ(
c − b − b )
2
2
2
3
4
(2.3.2)
K 10 (a, a, a, a; b,b,c 1 ,c 2 ; d 1 , d 2 , d 3 , d 4 ; x, y, z, t)
= F 4 (a, b; d 1 , d 2 ; x, y) F 2 (a+m+n, c 1 ,c 2 ;d 3 , d 4 ; z, t) (2.3.3)
K 15 (a, a, a, b 5 ; b 1 , b 2 , b 3 , b 4 ; c, c, c, c; x, y, z, t)
= F 1 (a, b 1 , b 2 ; c; x, y) F 3 (a+m+n, b 5 , b 3 , b 4 ; c+m+n; z, t) (2.3.4)
where (F 1 , F 2 , F 3 , F 4 ,) are Appell hypergeometric function of two variables.
(1) Proof:
Now Quadruple hypergemotric function can be reduced to Appell function of two variable.
K 12 (a, a, a, a; b 1 , b 2 , b 3 , b 4 , c 1 , c 1 , c 2 , c 2 ; x, y, z, t)
=
∞
m n p q
( a)
m+
n+
p+
q
( b1
)
m(
b2
)
n(
b3
)
p(
b4
)
q x y z t
∑
, , , 0 ( c ) m!
n!
m!
n!
p!
q!
m n p q=
1 m+
n
(2.3.5)
=
∞
m n
p
( a)
m+
n+
p+
q
( b1
)
m
( b2
)
n
x y ( a + m + n)
p+
q
( b3
)
p
( b4
)
q
z t
∑
, n,
p,
q=
0 ( c ) m!
n!
( c )
p!
q!
m 1 m+
n
2 p+
q
⎡
⎤⎡ + +
⎢ ∑ ∞
m n
( a)
∞
m+
n+
p+
q
( b1
)
m
( b2
)
n
x y ( a m n)
p+
q
( b3
)
p
( b4
)
q
z
⎥⎢
∑
⎢
⎥
= ⎣ = ( )
+
! ! ⎦⎢⎣
!
m, n 0 c1
m n
m n
p,
q=
0 ( c2)
p+
q
p!
q
= F 1 (a, b 1 , b 2 ; c ; x, y) F 1 ( a+m+n,b 3 ,b 4 ; c 2 ; z, t) (2.3.6)
where F 1 is Appell function of two variable
This completes the derivation of (2.3.1)
(2) Proof :-
now putting x = y = z = t= 1 in equation (2.3.6)
K 12 ( a, a, a, a; b 1 , b 2 , b 3 , b 4 , c 1 , c 1 , c 2 , c 2 ; 1,1,1,1 )
= F 1 (a,b 1 ,b 2 ;c 1 ; 1,1) F 1 ( a+m+n,b 3 ,b 4 ; c 2 ; 1,1) (2.3.7)
q
p
t
q
⎤
⎥
⎥⎦
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