Integral representations and integral transforms of some families of ...
Integral representations and integral transforms of some families of ...
Integral representations and integral transforms of some families of ...
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14 N. Elezović et al.<br />
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By making use <strong>of</strong> the Mellin transform formula (66) in conjunction with the <strong>integral</strong><br />
representation (26), we obtain<br />
M ( S (α,β)<br />
μ (r;{k γ }) ) =<br />
∫ ∞<br />
= 2 ∫ ∞<br />
x γ (μα−β)−1<br />
Ɣ(μ) 0 e x − 1<br />
0<br />
r s−1 S μ<br />
(α,β) (r;{k γ })dr<br />
(∫ ∞<br />
0<br />
r s−1 [<br />
1 1 (μ, 1) ; (γ (μα − β) ,γα) ;−r 2 x γα] )<br />
dr dx<br />
Ɣ (s/2) Ɣ (μ − s/2)<br />
1<br />
x γ (μα−β)−γαs/2−1<br />
= ·<br />
dx<br />
Ɣ(μ) Ɣ (γ (μα − β) − γαs/2) 0 e x − 1<br />
( s<br />
= B<br />
2 ,μ− s ) (<br />
ζ γ (μα − β) − γαs ) (<br />
0 < R (s) < 2μ; γ ∈ R<br />
+ ) . (68)<br />
2 2<br />
Upon setting γ = q/α (q ∈ N) in (67), we readily get the following special case:<br />
M(S (α,β)<br />
μ (r;{k q/α })) =<br />
( s<br />
= B<br />
2 ,μ− s )<br />
ζ<br />
2<br />
∫ ∞<br />
0<br />
(<br />
q<br />
∫ ∞<br />
r s−1 S μ<br />
(α,β) ( {<br />
r; k<br />
q/α }) dr<br />
([<br />
μ − β α<br />
which, in the further special case when q = 2, yields<br />
M(S (α,β)<br />
μ (r;{k 2/α })) =<br />
( s<br />
= B<br />
2 ,μ− s )<br />
ζ<br />
2<br />
∫ ∞<br />
0<br />
(<br />
2<br />
]<br />
− s 2<br />
)) (0<br />
< R(s) < 2μ<br />
)<br />
, (69)<br />
r s−1 S μ<br />
(α,β) ( {<br />
r; k<br />
2/α }) dr<br />
[<br />
μ − β α<br />
]<br />
− s) (0<br />
< R(s) < 2μ<br />
)<br />
. (70)<br />
Alternatively, in view <strong>of</strong> the <strong>integral</strong> representation (68), the Mellin transform formula (68) can<br />
be proven directly by applying the relatively more familiar analogue <strong>of</strong> (66) for the generalized<br />
hypergeometric p F q -function <strong>and</strong> the Gauss–Legendre multiplication formula for the Gamma<br />
function [12, p. 8, Equation 1.1 (51)]:<br />
Acknowledgements<br />
Ɣ(mz) =<br />
mmz−1/2<br />
(2π) 1/2(m−1)<br />
m ∏<br />
j=1<br />
(<br />
Ɣ z + j − 1 )<br />
m<br />
(z ̸= 0, − 1 m , − 2 m , ··· ; m ∈ N )<br />
.<br />
The present investigation was supported, in part, by the Ministry <strong>of</strong> Education <strong>and</strong> Sciences <strong>of</strong> Macedonia <strong>and</strong> the Ministry<br />
<strong>of</strong> Education, Sciences <strong>and</strong> Sports <strong>of</strong> Croatia under Projects No. 05-437/1 <strong>and</strong> No. 13-272/1, <strong>and</strong> also, in part, by the<br />
Natural Sciences <strong>and</strong> Engineering Research Council <strong>of</strong> Canada under Grant OGP0007353.<br />
References<br />
[1] P.L. Butzer, S. Flocke, <strong>and</strong> M. Hauss, Euler functions E α (z) with complex α <strong>and</strong> applications, inApproximation,<br />
Probability <strong>and</strong> Related Fields (G.A.Anastassiou <strong>and</strong> S.T. Rachev, eds), Plenum Press, NewYork, 1994, pp. 127–150.<br />
[2] P.L. Butzer, T.K. Pogány, <strong>and</strong> H.M. Srivastava, A linear ODE for the Omega function associated with the Euler<br />
function E α (z) <strong>and</strong> the Bernoulli function B α (z), Appl. Math. Lett. 19 (2006), pp. 1073–1077.<br />
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