88 ON q-LAPLACE TRANSFORMS OF A GENERAL CLASS OF q ...

ON q-LAPLACE TRANSFORMS OF A GENERAL CLASS... 87
1
[ ]
a1 , · · · , a
4.6 sinh r ;
q(x) rΦ s q, tx
2s Φ 1 : 0 ; r
(
q : −; a1 , · · · , a r ;
q ; 1 0 : 0 ; s − : − ; b 1 , · · · , b s ; s , t )
s
b 1 , · · · , b s ;
−
2s 1 Φ 1 : 0 ; r
(
q : −; a1 , · · · , a r;
q ; −1
0 : 0 ; s − : − ; b 1 , · · · , b s; s , t )
s
[ ] 1
a1 , · · · , a
4.7 cosh r ;
2s
Φ 1 : 0 ; r
( )
q : −; a1 , · · · , a r;
q ;
0 : 0 ; s − : − ; b
q(x) rΦ s q, tx
1 , · · · , b s ;
1 s , s
t
b 1 , · · · , b s ;
+ 1 2s Φ 1 : 0 ; r
(
q : −; a1 , · · · , a r ;
q ; −1
0 : 0 ; s − : − ; b 1 , · · · , b s ; s , t )
[ ] s
(q; q) ν (q; q)ν −;
4.8 (x + a) ν
s 1+ν or
(−as; q) ∞ s 1+ν 0 Φ 0 q, −as
−;
[
]
4.9 x λ (q; q) ν+λ
q −ν ;
(x + a) ν ; λ > 0
s 1+ν+λ 1 Φ 1
q −ν−λ q , as/q
;
[
4.10
K Aff
n (x; a, N; q)
n > N
4.11 J v (x ; q)
4.12 J −v (x; q)
4.13 P (α,β)
n (x; γ, δ; q)
4.14 R m,v (x ; q)
(q; q) ∞
s
[
1
s 1+ν · 4Φ 4
Φ 1 : 0 ; 2
0 : 1 ; 2
e ivπ [
(q; q) −ν
(q; q) ν s 1−ν · 4Φ 4
1/s : − ; q −n , 0 ;
− : 1/s ; aq, q −N ;
q v+1
2 , −q v+1
2 , q v+2
2 , −q v+2
2 ;
q v+1 , 0, 0, 0;
q 1−ν
2 , −q 1−ν
2 , q 2−ν
2 , −q 2−ν
2 ;
q 1−ν , 0 , 0 , 0;
(αq, −δα q/γ, q) n (γ/α q) n (q; q) ∞
(q, ⎡−q; q) n s
qα
1+ρ
Φ 1 : 0; 2
⎢ sγ : − ; q−n , αβ q n+1 ;
0 : 1; 2 ⎣
− : qα
sγ
−δαq
; αq, ;
γ
(q; q) −m (q ν [
; q) m q
−n , q
ν+m−n ;
s 1−m 2Φ 1
[
q ν ;
q 1−m
2 , −q 1−m
2 , q 2−m
2 , −q 2−m
2 ;
4Φ 4
q 1−ν−m , 0 , 0 , 0 ;
]
q, q, q
q ,
q, q 1+ρ , q⎥
⎦
q, q n+1 ]
]
q
s 2
]
q, q−ν+1
s 2
⎤
]
q, q1−ν−m
s 2
To prove the result (4.2), we take f(x) = x ν e q (ax k ) in the equation (1.9) and
make use the definition (1.4), which yields
{
qL s x ν e q (ax k ) } = (q; q) ∑ ∞
∞ q j(1+ν) ∞
{
∑ a(s −1 q j ) k} r
s 1+ν
.
(q; q) j (q, q) r
j=0
On interchanging the order of summations and then summing the resulting 0 Φ 0 (.)
series with the help of equation (2.2), the right hand side of the above expression
(q; q) ∞ (
∞
∑
) a/s
k r
(q 1+ν , q 1+ν+1 , · · · , q 1+ν+k−1 ; q k ) r
s 1+ν
(4.15)
(q; q)
r=0
r
[ ]
a1 , · · · , a
For the proof of the result (4.3), we take f(x) = e q (x) r Φ r ;
s q, tx in
b 1 , · · · , b s ;
the equation (1.9) and make use of definition (1.4) and (1.16), this yields;
{ [ ]}
a1 , · · · , a
qL s e q (x) r Φ r ;
s q, tx =
b 1 , · · · , b s ;
(q; q) ∞
∞∑ q j ∑ ∞
(s −1 q j ) r ∑ ∞
(a 1 , · · · , a r ; q) k (ts −1 q j ) k
s (q; q) j (q; q) r (q, b 1 , · · · , b s ; q) k
j=0
r=0
On interchanging the order of summations and then summing the inner 0 Φ 0 (.)
series with the help of equation (2.2), the right hand side of the above expression
reduces
(q; q) ∞
s
∞∑
∞∑
k=0 r=0
k=0
r=0
(a 1 , · · · , a r ; q) k (t/s) k (1/s) r
(q 1+r+k ; q) ∞ (q; q) r (q, b 1 , · · · , b s ; q) k
(4.16)