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

4 N. Elezović et al.
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Using the same procedure as described above, each of the following two Laplace transforms
can be derived:
L(S μ
(α,0) (r;{k 2/α }))(x) = 2√ ∫
π ∞
t μ−1/2 (∫ ∞
e −xr
)
Ɣ(μ) 0 e t − 1 0 (2r) J μ−1/2(rt)dr dt
μ−1/2
2 √ ∫
π ∞
t μ−1/2 (∫ ∞
)
=
e −xr r 1/2−μ J
2 μ−1/2 Ɣ(μ) 0 e t μ−1/2 (rt)dr dt
− 1 0
∫
2 ∞
t 2μ−1 (
=
xƔ(2μ) 0 e t − 1 2 F 1 1, 1 2 ; μ + 1 2 )
2 ;−t dt (15)
x 2
and
∫
L( ˜S μ (α,0) (r;{k 2/α 2 ∞
t 2μ−1 (
}))(x) =
xƔ(2μ) 0 e t + 1 2 F 1 1, 1 2 ; μ + 1 2 )
2 ;−t dt, (16)
x 2
where ˜S μ
(α,0) (r;{k 2/α }) is the alternating version of S μ (α,0) (r;{k 2/α }).
III. We next recall the following integral representation from the work of Srivastava and
Tomovski [15]:
S (α,β)
μ (r;{k q/α }) =
2
Ɣ (q [μ − β/α])
· 1F q
[μ;
∫ ∞
x q(μ−β/α)−1
0 e x − 1
(
q; q
[
μ − β α
])
;−r 2 ( x
q
where, for convenience, (q; λ) abbreviates the array of q parameters:
λ
q , λ + 1 ,..., λ + q − 1
q
q
For q = 2, we find from (17) that
S (α,β)
μ (r;{k 2/α }) =
(q ∈ N).
∫
2
∞
x 2(μ−β/α)−1
Ɣ (2 [μ − β/α]) 0 e x − 1
(
· 1F 2 μ; μ − β α ,μ− β α + 1 x 2
2 ;−r2 4
(
r, α, β ∈ R + ; μ − β α > 1 )
.
2
) q ]
dx, (17)
)
dx (18)
The following hypergeometric integral formula involving the Laplace–Mellin transform is
well-known (see, for example, [7, p. 60]):
(
p+2F q σ, σ + 1 ) ∫ ∞
2 ,α p; β q ;− 4ω2 = zσ
e −zt t σ −1 (
pF
z 2 q αp ; β q ;−ω 2 t 2) dt (19)
Ɣ(σ)
(
R(σ ) > 0; z ̸= 0; p ≦ q − 2; |arg(z)| < π 2
which, for σ = 1, yields the special case given below:
(
p+2F q 1, 3 ) ∫ ∞
2 ,α p; β q ;− 4ω2 = z e −zt (
pF
z 2 q αp ; β q ;−ω 2 t 2) dt (20)
0
(
z ̸= 0; p ≦ q − 2; |arg(z)| < π )
,
2
0
)
,