download - Ijsrp.org

International Journal of Scientific and Research Publications, Volume 2, Issue 10, October 2012 47
ISSN 2250-3153
K 2 , (a, a, a, a; b, b, b, c; d 1 , d 2 , d 3 , d 4 ; x,y,l,t)
Γ(
d
4)
Γ(
d
4
− a − b − q − 2m
− 2n)
Γ(
d
4
− a − m − n − q)
Γ(
d
4
− b − m − n)
F E (a,a,a,b,b,c,; d 1 ,d 2 ,d 3 ; x, y, t) (2.4.2)
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)
= F G ( a, a, a, b 1 ,b 3 , b 4 , c 1 ,c 2 , c 2 ; x,z,t) 2 F 1 (a+m+p+q,b 2 ;c 1 +m;y) (2.4.3)
K 12 ( a, a, a, a; b 1 , b 2 , b 3 , b 4 , c 1 , c 1 ,c 2 , c 2 ; x,l,z,t)
Γ(
c1
+ m)
Γ(
c1
− a − b2
− p − q)
Γ(
c1
+ m − b2
) Γ(
c1
− a − p − q)
F G ( a, a, a, b 1 ,b 3 , b 4 ,; c 1 , c 2 , c 2 ; x, y, z, t ) (2.4.4)
K 15 ( a, a, a, b 5 ;b 4 , b 1 ,b 2 ,b 3 ; c, c, c,c; x, y, z, t )
= F S ( b 5 , a, a, b 4 ,b 1 ,b 2 ;c,c,c; x,y,t) 2 F 1 (a+m+n,b 3 ;c+q+m+n; z) (2.4.5)
K 15 ( a, a, a, b 5 ; b 4 , b 1 , b 2 , b 3 ;c, c, c, c; x, y, z, t)
Γ(
c + q + m + n)
Γ(
c + q − a − b3
)
Γ(
c + q + m + n − b3
) Γ(
c + q − a)
F S (b 5 , a, a, b 4 , b 1 , b 2 ,; c,c,c; x,y,t) (2.4.6)
K 6 (a,a,a,a; b,b, c 1 ,c 2 ;e, d,d,d;x,y,z,t)
= F F ( a, a, a,b, c 1 , b, ;e,d,d; x,z,y) 2 F 1 ( a+m+p+n,c 2 ;d+p+n; t) (2.4.7)
K 6 ( a,a,a,a; b,b, c 1 ,c 2 ;e, d,d,d;x,y,z,t)
Γ(
d + p + n)
Γ(
d − m − a − c2)
Γ(
d − a − m)
Γ(
d + p + n − c )
=
2 F F ( a, a, a,b, c 1 , b, ;e,d,d; x,z,y) (2.4.8)
Where (F 4 ,F 14 , F 8 ,F 7 ) & (F E , F F , F G ,F S ) are Lauriceila's set & Saran Triple hypergeometric Series.
(1) Proof:-
Now Quadruple hypergeometric function can be reduced to Lauricella's set and Saran Triple hypergometric Series.
K 2 ( a,a,a,a; b,b,b,c; d 1 d 2 ,d 3 ,d 4 ;x,y,z,t)
=
∞
m n p q
( a)
m+
n+
p+
q(
b)
m+
n+
p(
c)
q x y z t
∑
, , , = 0 ( d ) ( d ) ( d ) ( d ) m!
n!
p!
q!
m n p q
=
⎡
⎢
⎢
= ⎣
1
( a)
m
2
m+
n+
q
n
( b)
3
p
4
q
( a + m + n + q)
( b + m + n)
, , p,
q=
0 ( d ) ( d ) ( d ) m!
n!
( d )
p!
m n
∞
∑
( a)
1
m
2
( b)
m+
n
n
( c)
x
4
q
q
( c)
x
m
y
y
n
t
t
q
3
p
p
⎤⎡ ∞
( a + m + n + q)(
b + m + n)
z
⎥⎢∑
⎥⎦
⎢⎣
p=
0 ( d3)
p
∞
m n q
p
m+
n+
q m+
n q
∑
q, m,
n= 0 ( d1)
m
( d2)
n
( d4)
q
m!
n!
q!
p!
= F E ( a, a, a, b, b,c,; d 1 , d 2 , d 3 , x,y, t) 2 F 1 ( a+m+n+q, b+m+n, d 4 ; z) (2.4.10)
This completes the derivation of (2.4.1)
(2) Proof:
If z = 1, in euqation (2.4.10)
K 2 (a, a, a, a; b, b, b, c; d 1 , d 2 , d 3 , d 4 ; x,y, l,t)
= F E ( a, a, a, b, b,c,; d 1 , d 2 , d 3, x,y, t) 2 F 1 ( a+m+n+q, b+m+n, d 4 ; 1) (2.4.11)
Now Apply Gauss's summation theorem in equation ( 2.4.11)
γ
γ
Γ(
) Γ(
− − )
F 1 (α,β,γ;1)= Γ(
γ − α ) Γ(
γ − β −)
K 2 ( a, a, a, a; b, b, b, c; d 1 , d 2 , d 3 , d 4 ; x,y, l,t)
α
β
p
z
p
⎤
⎥
⎥⎦
(2.4. 9)
www.ijsrp.org