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

6 N. Elezović et al.
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and
∫
L( ˜S μ (α,β) (r;{k 2/α 2
∞
t 2(μ−β/α)−1
}))(x) =
xƔ (2 [μ − β/α]) 0 e t + 1
(
· 3F 2 1, 3 2 ,μ; μ − β α ,μ− β α + 1 2 )
2 ;−t dt. (25)
x 2
IV. The following integral representation was derived by Srivastava and Tomovski [15]:
S (α,β)
μ (r;{k γ }) = 2
Ɣ(μ)
∫ ∞
x γ (μα−β)−1
0 e x − 1
1 1 [(μ, 1); (γ (μα − β),γα);−r 2 x γα ]dx (26)
(
r, α, β, γ ∈ R + ; γ (μα − β) > 1 ) ,
where p q denotes the Fox–Wright generalization of the hypergeometric function p F q with
p numerator and q denominator parameters (see, for example, [13, p. 19]). Moreover, the Laplace–
Mellin transform of the Fox–Wright p q -function can be found in the work of Srivastava et al. [14,
p. 944, Equation (6.1)]:
∫ ∞
[ ]
e −st t ρ−1 (a1 ,A 1 ),...,(a p ,A p )
p q 0
(b 1 ,B 1 ),...,(b q ,B q ) ∣ ztσ dt
⎡
⎤
(ρ, σ ), (a 1 ,A 1 ),...,(a p ,A p )
= s −ρ p+1 q
⎣
z
⎦
(b 1 ,B 1 ),...,(b q ,B q ) ∣ s σ
(
R(ρ) > 0; R(s) > 0; σ ∈ R
+ ) . (27)
Thus, by applying (26) as well as (27) with ρ = 1, we get
L(S μ (α,β) (r;{k γ }))(x) = 2
Ɣ(μ)
(∫ ∞
·
0
= 1
xƔ(μ)
∫ ∞
t γ (μα−β)−1
0 e t − 1
e −rx 1 1 [(μ, 1); (γ (μα − β),γα);−r 2 t γα ]dr
∫ ∞
t γ (μα−β)−1
0 e t − 1
)
dt
[
· 2 1 (1, 2); (μ, 1); ( γ (μα − β),γα ) ;− t γα ]
dt
x 2 (28)
and
L( ˜S μ (α,β) (r;{k γ }))(x) = 1 ∫ ∞
t γ (μα−β)−1
xƔ(μ) 0 e t + 1
[
· 2 1 (1, 2); (μ, 1); ( γ (μα − β),γα ) ;− t γα ]
dt. (29)
x 2
V. About a decade ago, in their investigation of the complex-index Euler function E α (z), Butzer
et al. [1] introduced the following special function:
(ω) = 2
∫ 1/2
0+
sin h(ωu) cot(πu)du (ω ∈ C), (30)
which they called the complete Omega function. Recently, Butzer et al. [2] made use of an
integral representation for the alternating Mathieu series ˜S μ (ω) in order to derive the following