Integral representations and integral transforms of some families of ...

Integral Transforms and Special Functions 5
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where α p and β q denote the p- and q-parameter arrays:
α 1 ,...,α p and β 1 ,...,β q ,
respectively. Indeed, under some additional parametric constraints, the hypergeometric integral
formulas (19) and (20) hold true also when p = q − 1 (see, for details, [7, p. 60]; see also Equation
(23) below).
In light of the hypergeometric integral formula (20), we find that
and
L(S μ (α,β) (r;{k q/α 2
}))(x) =
Ɣ (q [μ − β/α]) 0
[∫ ∞
· e −rx 1F q
[μ;
=
L( ˜S (α,β)
μ (r;{k q/α }))(x) =
0
2
xƔ (q [μ − β/α])
· 3F q
[
1, 3 2 ,μ; (q; q
∫
2
∞
xƔ (q [μ − β/α])
· 3F q
[
1, 3 2 ,μ; (q; q
∫ ∞
t q(μ−β/α)−1
e t − 1
(
q; q
[
μ − β α
∫ ∞
t q(μ−β/α)−1
0 e t − 1
0
[
μ − β α
t q(μ−β/α)−1
e t + 1
[
μ − β α
]) ( ) t q ] ]
;−r 2 dr dt
q
])
;− 4 x 2 ( t
q
])
;− 4 x 2 ( t
q
) q ]
dt (q ≧ 3)
) q ]
dt (q ≧ 3).
Next, by using the following special case a known Laplace transform of the class (20) when
p − 1 = q = 2 (see [7, p. 60]):
we obtain
(
3F 2 1, 3 )
2 ,α 1; β 1 ,β 2 ;− 4ω2 = z
z 2
L(S (α,β)
μ (r;{k 2/α }))(x) =
=
∫ ∞
(z ̸= 0; R(z) > 2|R(ω)| ≧ 0 ) ,
2
Ɣ (2 [μ − β/α])
[∫ ∞
·
0
0
(21)
(22)
e −zt 1F 2
(
α1 ; β 1 ,β 2 ;−ω 2 t 2) dt (23)
∫ ∞
t 2(μ−β/α)−1
0 e t − 1
e −rx 2F 1
(
μ; μ − β/α, μ − β α + 1 2 ;−r2 t 2
2
xƔ (2 [μ − β/α])
∫ ∞
t 2(μ−β/α)−1
0 e t − 1
4
) ]
dr dt
· 3F 2
(
1, 3 2 ,μ; μ − β α ,μ− β α + 1 2 ;−t 2
x 2 )
dt (24)