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

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Integral Transforms and Special Functions
Vol. 00, No. 0, Month 2008, 1–15
Integral representations and integral transforms of some
families of Mathieu type series
Neven Elezović a , H.M. Srivastava b * and Živorad Tomovski c
a Department of Applied Mathematics, Faculty of Electrical Engineering and Computing, Zagreb, Croatia;
b Department of Mathematics and Statistics, University of Victoria, Victoria, British Columbia, Canada;
c Institute of Mathematics, Faculty of Natural Sciences and Mathematics, St. Cyril and Methodius
University, Skopje, Republic of Macedonia
By using some integral representations for several Mathieu type series (see P.L. Butzer, T.K. Pogány, and
H.M. Srivastava, A linear ODE for the Omega function associated with the Euler function E α (z) and
the Bernoulli function B α (z), Appl. Math. Lett. 19 (2006), pp. 1073–1077, P. Cerone and C.T. Lenard,
On integral forms of generalised Mathieu series, J. Inequal. Pure Appl. Math. 4 (5) (2003), Article 100,
pp. 1–11 (electronic), T.K. Pogány, H.M. Srivastava and Ž. Tomovski, Some families of Mathieu a-series
and alternating Mathieu a-series, Appl. Math. Comput. 173 (2006), pp. 69–108, H.M. Srivastava and
Ž. Tomovski, Some problems and solutions involving Mathieu’s series and its generalizations, J. Inequal.
Pure Appl. Math. 5 (2) (2004), Article 45, pp. 1–13 (electronic), Ž. Tomovski, Integral representations of
generalized Mathieu series via Mittag-Leffler type functions, Fract. Calc. Appl. Anal. 10 (2007), pp. 127–
138.) via the Bessel function J ν of the first kind, the Gauss hypergeometric function 2 F 1 , the generalized
hypergeometric function p F q and the Fox–Wright generalization p q of the hypergeometric function p F q ,
a number of integral representations of the Laplace, Fourier, and Mellin types are derived here for certain
general families of Mathieu type series. Some interesting corollaries and consequences of these integral
representations are also considered.
Keywords: integral representations; Mathieu series; alternating Mathieu series; Laplace transforms;
Fourier sine transforms; Fourier cosine transforms; Mellin transforms; Bessel function; Gauss hypergeometric
function; generalized hypergeometric function; Fox–Wright function; Riemann Zeta function;
Dirichlet Eta function; Sonine-Schafheitlin formula; Weber-Sonine integral; Legendre’s duplication
formula; Gauss–Legendre multiplication formula.
2000 Mathematics Subject Classifications: Primary: 33E20, 44A10; Secondary: 33C10, 33C20, 44A20
1. Introduction
The following familiar infinite series:
S(r) =
∞∑
n=1
*Corresponding author. Email: tomovski@iunona.pmf.ukim.edu.mk
ISSN 1065-2469 print/ISSN 1476-8291 online
© 2008 Taylor & Francis
DOI: 10.1080/10652460801965456
http://www.informaworld.com
2n
(n 2 + r 2 ) 2 (r ∈ R + ) (1)
Techset Composition Ltd, Salisbury GITR296711.TeX Page#: 15 Printed: 5/3/2008

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is named after Émile Leonard Mathieu (1835–1890), who investigated it in his work [8] on
elasticity of solid bodies. An alternating version of the Mathieu series (1) in the form:
˜S(r) =
∞∑
(−1) n−1 2n
(r ∈ R + ) (2)
(n 2 + r 2 ) 2
n=1
was introduced by Pogány et al. [10].
Integral representations of the Mathieu series (1) and the alternating Mathieu series (2) are
given by (see [10])
S(r) = 1 ∫ ∞
t sin(rt)
dt (3)
r 0 e t − 1
and
˜S(r) = 1 ∫ ∞
t sin(rt)
dt. (4)
r 0 e t + 1
Recently, Srivastava and Tomovski [15] defined the following five-parameter family of
generalized Mathieu series:
S μ
(α,β) (r; a) = S μ (α,β) (r;{a k }) =
∞∑
n=1
2a β n
(a α n + r2 ) μ (r,α,β,μ∈ R + ), (5)
where (and throughout this paper)itistacitly assumed that the positive sequence:
(
)
a := {a k } ∞ k=1 ={a 1,a 2 ,a 3 ,...} lim a k =∞
(6)
k→∞
is so chosen (and then the positive parameters α, β, and μ) are so constrained that the infinite
series in the definition (5) converges, that is, that the following auxiliary series:
∞∑ 1
is convergent.
The following special cases:
n=1
a μα−β
n
S (2,1)
2 (r;{a k }), S μ (r) = S μ (2,1) (r;{k}), S μ (2,1) (r;{k γ }), and S μ
(α,α/2) (r;{k})
were investigated by Qi [11], Diananda [4], Tomovski [16], and Cerone and Lenard [3].
The main object of this sequel to the aforementioned investigations is to derive several formulas
involving the Laplace, Fourier, and Mellin transforms of various functions belonging to the family
of generalized Mathieu series considered here.
2. The Laplace transforms
In this section, we shall compute the Laplace transforms of various functions from the family of
generalized Mathieu series.
I. Using the table of Laplace transforms [6, Vol. I, p. 152, Entry 4.7 (16)], it follows that
∫ ∞
( ∫ 1 ∞
)
L(S(r))(x) = e −rx t sin(rt)
r e t − 1 dt dr
=
0
∫ ∞
0
t
e t − 1
0
(∫ ∞
−rx
sin(rt)
e
r
0
)
dr dt (7)

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Integral Transforms and Special Functions 3
∫ ∞
( )
t
t
=
0 e t − 1 arctan dt.
x
(8)
In the same way, one can obtain the Laplace transform given below:
∫ ∞
( )
t t
L( ˜S(r))(x) =
e t + 1 arctan dt. (9)
x
0
II. The function S μ+1 has the following integral representation via the Bessel function J ν of
the first kind (see Cerone and Lenard [3]):
S μ+1 (r) =
√ ∫ π
∞
t μ+1/2
(2r) μ−1/2 Ɣ(μ + 1) 0 e t − 1 J μ−1/2(rt)dt (r,μ ∈ R + ). (10)
Similarly, it can be shown that (see Srivastava and Tomovski [15])
S μ (α,0) (r;{k 2/α 2 √ ∫
π ∞
t μ−1/2
}) =
(2r) μ−1/2 Ɣ(μ) 0 e t − 1 J μ−1/2(rt)dt
(
r ∈ R + ; μ> 1 )
. (11)
2
We shall also need the following known result involving the Laplace–Mellin transform [6, Vol.
I, p. 182, Entry 4.14 (9)]:
∫ ∞
0
( ρ
) [ ν
e −st t λ−1 J ν (ρt)dt = s
−λ
Ɣ(ν + λ) 1
2s Ɣ(ν + 1) 2 F 1
2 (ν+λ), 1 ]
(ν+λ+1); ν+1;−ρ2
2 s 2
( )
R(s) > |I(ρ)|; R(ν + λ) > 0 .
Making use of the known result (12) and the Legendre duplication formula for the gamma function,
it follows that
√ ∫ π ∞
t μ+1/2 (∫ ∞
e −xr
)
L(S μ+1 (r))(x) =
Ɣ(μ + 1) 0 e t − 1 0 (2r) J μ−1/2(rt)dr dt
μ−1/2
√ ∫ π
∞
t μ+1/2 (∫ ∞
)
=
e −xr r 1/2−μ J
2 μ−1/2 Ɣ(μ + 1) 0 e t μ1/2 (rt)dr dt
− 1 0
√ ∫ π
∞
t 2μ ( 1
=
2 2μ−1 xƔ(μ + 1)Ɣ (μ + 1/2) 0 e t − 1 2 F 1
2 , 1; μ + 1 2 )
2 ;−t dt
x 2
∫
2 ∞
t 2μ (
=
xƔ(2μ + 1) e t − 1 2 F 1 1, 1 2 ; μ + 1 2 )
2 ;−t dt, (13)
x 2
provided that each member of (13) exists.
Similarly, we have
L( ˜S μ+1 (r))(x) =
where ˜S μ+1 (r) is the alternating version of S μ+1 (r).
0
(12)
∫
2 ∞
t 2μ (
xƔ(2μ + 1) 0 e t + 1 2 F 1 1, 1 2 ; μ + 1 2 )
2 ;−t dt, (14)
x 2

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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
)
,

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)

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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

Integral Transforms and Special Functions 7
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new integral representation for the Omega function:
(x) = 2 ( x
) ∫ ∞
( ) xt dt
π sin h cos
2
2π e t + 1
Since
∫ ∞
0
( x
) ( xt
e −sx sin h cos
2 2π
= 1 2
=
∫ ∞
0
e −(2s−1)x/2 cos
0
)
dx
( xt
2π
)
dx − 1 2
∫ ∞
0
(x ∈ R). (31)
( xt
e −(2s+1)x/2 cos
2π
2s − 1
(2s − 1) 2 + t 2 /π 2 − 2s + 1
(2s + 1) 2 + t 2 /π 2 (R(s) > 1 2
)
dx
)
, (32)
the Laplace transform for the Omega function (x) in (30) is given by
∫ ∞
∫ ∞
[ 2
( x
) ∫
L((x))(s) = e −sx (x)dx = e −sx ∞
( ) ]
xt dt
0
0 π sinh cos
dx
2 0 2π e t + 1
= 2 ∫ ∞
[∫ ∞
( ) xt
( x
) ] dt
e −sx cos sinh dx
π 0 0
2π 2 e t + 1
∫
2(2s − 1) ∞
dt
=
π [(2s − 1) 2 + t 2 /π 2 ](e t + 1)
−
2(2s + 1)
π
0
∫ ∞
0
dt
[(2s + 1) 2 + t 2 /π 2 ](e t + 1) . (33)
Several other integral representations of the Laplace type can be derived for the various Mathieu
type series in a similar manner. The details involved in all such derivations are being left as an
exercise for the interested reader.
3. The Fourier transforms
I. In this section, we shall make use of the available tables of Fourier sine and cosine transforms
(see, for example, [6, Vol. I, p. 78, Entry 2.6 (1)] and [6, Vol. I, p. 18, Entry 1.6 (1)]). First of all,
the Fourier sine transform of the Mathieu series S(r) is given by
∫ ∞
F s (S(r))(x) = sin(xr)S(r)dr
0
∫ ∞
( ∫ 1 ∞
)
t sin(rt)
= sin(xr)
0 r 0 e t − 1 dt dr
∫ ∞
(∫
t ∞
)
sin(rt) sin(xr)
=
dr dt
0 e t − 1 0 r
= 1 (∫ ∞
∣ )
2 · PV t ∣∣∣
e t − 1 ln t + x
t − x ∣ dt (x > 0), (34)
where the Cauchy Principal Value (PV) of the last integral is assumed to exist.
0

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For the Fourier cosine transform, we similarly find that
F c (S(r))(x) =
=
=
∫ ∞
0
∫ ∞
0
∫ ∞
= π 4
= π 2
0
∫ ∞
cos(xr)S(r)dr
( ∫ 1 ∞
cos(xr)
r 0
(∫ ∞
t
e t − 1
0
∫ ∞
x
0
)
t sin(rt)
e t − 1 dt dr
sin(rt) cos(xr)
dr
r
t
[1 + sgn(t − x)]dt
e t − 1
)
dt
t
dt
e t − 1
(x > 0). (35)
II. In order to obtain the Fourier sine and the Fourier cosine transforms of S μ+1 (r), we first
set ν = 1/2 and ν =−1/2 in the Sonine–Schafheitlin formula (see, for example, [18, p. 401,
Equation 13.4 (2)]):
∫ ∞
0
t −λ a μ Ɣ (μ + ν − λ + 1/2)
J μ (at)J ν (bt)dt =
2 λ b μ−λ+1 Ɣ(μ + 1)Ɣ (λ − μ + ν + 1/2)
( μ + ν − λ + 1
· 2F 1 , μ − ν − λ + 1 )
; μ + 1; a2
2
2
b 2
( )
R(μ + ν + 1) >R(λ) > −1; 0

Integral Transforms and Special Functions 9
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and [6, Vol. II, p. 37, Entry 8.7 (32)]
∫ ∞
√
πb
t −λ−1/2 J μ (at) cos(bt)dt =
0
2 · a μ Ɣ [1/2 (μ − λ + 1/2)]
2 λ b μ−λ+1 Ɣ(μ + 1)Ɣ [1/2 (λ − μ + 1/2)]
[ ( 1
· 2F 1 μ − λ + 1 )
, 1 (
μ − λ + 3 ) ]
; μ + 1; a2
2 2 2 2 b 2
( (
R μ + 1 )
)
> R(λ) > −1; 0

10 N. Elezović et al.
451
452
453
454
455
456
457
458
459
460
461
462
463
464
465
466
467
468
469
470
471
472
473
474
475
476
477
478
479
480
481
482
483
484
485
486
487
488
489
490
491
492
493
494
495
496
497
498
499
500
and
F c (S μ+1 (r))(x) =
√ π · 2
1/2−μ
∫ ∞
t μ+1/2
cos(xr)S μ+1 (r)dr =
0
Ɣ(μ + 1) x e t − 1 c(μ; x,t)dt
(x > 0; μ>0), (47)
∫ ∞
where (1)
s (μ; x,t), (2)
s (μ; x,t) and c (μ; x,t) are defined by (44), (45) and (46), respectively.
In a similar manner, we find from the integral representation (11) that
and
F s
(
S
(α,0)
μ (r;{k 2/α }) ) (x) =
∫ ∞
0
sin(xr)S μ
(α,0) (r;{k 2/α })dr
∫ ∞
t μ−1/2 (∫ ∞
)
r 1/2−μ J
Ɣ(μ) 0 e t μ−1/2 (rt) sin(xr)dr dt
− 1 0
( ∫ x
t μ−1/2
Ɣ(μ) 0 e t − 1 (1) s (μ; x,t)dt
t μ−1/2
)
(μ; x,t)dt
√ π · 2
3/2−μ
=
√ π · 2
3/2−μ
=
+
∫ ∞
x
e t − 1 (2) s
)
(
x>0; μ> 1 2
F c
(
S
(α,0)
μ (r;{k 2/α }) ) (x) =
∫ ∞
0
cos(xr)S μ
(α,0) (r;{k 2/α })dr
√ π · 2
3/2−μ
=
Ɣ(μ)
∫ ∞
t μ−1/2
x
(
x>0; μ> 1 2
(48)
e t − 1 c(μ; x,t)dt
)
, (49)
where (1)
s (μ; x,t), (2)
s (μ; x,t) and c (μ; x,t) are defined, as before, by (44), (45) and (46),
respectively.
III. In the evaluation of the Fourier sine and the Fourier cosine transforms of the general
Mathieu Series S μ
(α,β) (r;{k 2/α }), we shall make use of the following integral formulas proven
earlier by Miller and Srivastava [9, pp. 225 and 226]:
⎧
∫ ∞
⎨ (1)
sin(2ax) 1 F 2 (α; β,γ;−b 2 x 2 s (α, β, γ ; a,b) (0

Integral Transforms and Special Functions 11
501
502
503
504
505
506
507
508
509
510
511
512
513
514
515
516
517
518
519
520
521
522
523
524
525
526
527
528
529
530
531
532
533
534
535
536
537
538
539
540
541
542
543
544
545
546
547
548
549
550
where, and in what follows, (1)
s (α, β, γ ; a,b), (2)
s (α, β, γ ; a,b) and c (α, β, γ ; a,b) are
given by
(1)
s (α, β, γ ; a,b) := 1 ( )
1 b2
2a 3 F 2 , 1,α; β,γ;
2 a 2
(
0

12 N. Elezović et al.
551
552
553
554
555
556
557
558
559
560
561
562
563
564
565
566
567
568
569
570
571
572
573
574
575
576
577
578
579
580
581
582
583
584
585
586
587
588
589
590
591
592
593
594
595
596
597
598
599
600
and
F c
(
S
(α,β)
μ (r;{k 2/α }) ) (x) =
∫ ∞
0
cos(xr)S μ
(α,β) (r;{k 2/α })dr
√ π · 2
3/2−μ
=
Ɣ(μ)
(
x>0; μ>max
∫ ∞
t μ−1/2
x
where the -functions are given by (53), (54) and (55) with, of course,
a = x 2 ,b= t 2 ,α= μ, β = μ − β α
(
μ, μ − β α ,μ− β α + 1 2 ; x 2 , t )
dt
2
e t − 1 c
{ 2β
α , β α + 1 })
, (57)
2
and γ = μ − β α + 1 2 . (58)
In their special case when β = 0, our last results (55) and (56) would reduce immediately to
(49) and (50), respectively.
4. The Mellin transforms
I. In order to evaluate the Mellin transforms of S(r) and ˜S(r), we apply the familiar integral
formula [6, Vol. I, p. 68, Entry 2.3 (1)]:
We thus obtain
and
∫ ∞
0
sin t
t ν
M(S(r))(s) =
dt = Ɣ(1 − ν)cos
∫ ∞
0
∫ ∞
r s−1 S(r)dr =
( πν
) ( )
0 < R(ν) < 2 . (59)
2
∫ ∞
(∫
t ∞
sin(rt)
e t − 1 0 r 2−s
0
( 1
r s−1 r
)
dr dt
∫ ∞
=
0
( π
) ∫ ∞
= Ɣ(s − 1) cos
2 (2 − s) t
0 e t − 1 dt
( πs
)
=−Ɣ(s − 1)Ɣ(2)ζ(2) cos
2
=
π 3 ( πs
)
12Ɣ(2 − s) csc 2
M( ˜S(r))(s) =−Ɣ(s − 1) cos
0
t sin(rt)
e t − 1 dt )
dr
(0 < R(s) < 2) (60)
( πs
) ∫ ∞
2
=−Ɣ(2)η(2)Ɣ(s − 1) cos
=
π 3
24Ɣ(2 − s)
( πs
)
2
0
t
e t + 1 dt
)
( πs
2
(0 < R(s) < 2), (61)

Integral Transforms and Special Functions 13
601
602
603
604
605
606
607
608
609
610
611
612
613
614
615
616
617
618
619
620
621
622
623
624
625
626
627
628
629
630
631
632
633
634
635
636
637
638
639
640
641
642
643
644
645
646
647
648
649
650
where ζ(s)and η(s) denote the Riemann Zeta and the Dirichlet Eta functions, respectively, defined
by (see, for example, [12, pp. 96 and 139])
∞∑ 1 ( )
ζ(s) := R(s) > 1 and η(s) := ( 1 − 2 1−s) ∞∑ (−1) n−1
ζ(s) =
(R(s) > 0).
n s n s n=1
n=1
(62)
II. Using the Weber–Sonine integral (see, for example, [18, p. 391, Equation 13.24 (1)]; see also
[6, Vol. I, p. 326, Entry 6.8 (1)]):
∫ ∞
0
we find that
M(S μ+1 (r))(s) =
t ρ−1 J μ (at)dt =
∫ ∞
0
2ρ−1 Ɣ(μ + ρ/2)
a ρ Ɣ(1 + μ − ρ/2)
r s−1 S μ+1 (r)dr
(
a>0; −R(μ) < R(ρ) < 3 )
, (63)
2
√ ∫ π
∞
t μ+1/2 (∫ ∞
)
=
r s−μ−1/2 J
2 μ−1/2 Ɣ(μ + 1) 0 e t μ−1/2 (rt)dr dt
− 1 0
√ ∫ π
=
2 μ−1/2 Ɣ(μ + 1) · 2s−μ−1/2 Ɣ(s/2) ∞
t 2μ−s
Ɣ(μ + 1 − s/2) 0 e t − 1 dt
√ πƔ(s/2)
=
· Ɣ(2μ − s + 1) ζ(2μ − s + 1)
2 2μ−s Ɣ(μ + 1)Ɣ(μ + 1 − s/2)
( s
= B
2 ,μ− s )
2 + 1 ( )
ζ(2μ − s + 1) μ>0; 0 < R(s) 0
)
, (65)
:= {0, −1, −2,...}). (66)
III. Lastly, we recall that the Mellin transform of the Fox–Wright -function is given by (see,
for example, [13, p. 15, Equation (2.4.1); p. 19, Equation (2.6.11)])
∫ [ ]
∞
(a1 ,A
t ρ−1 1 ),...,(a p ,A p )
∏ p p q 0
(b 1 ,B 1 ),...,(b q ,B q ) ∣ ztσ j=1
dt =
Ɣ(a j − A j ρ/σ)
∏
σz ρ q
σ
j=1 Ɣ(b (67)
j − B j ρσ)
(
{ ( )}
aj
0 < R(ρ)

14 N. Elezović et al.
651
652
653
654
655
656
657
658
659
660
661
662
663
664
665
666
667
668
669
670
671
672
673
674
675
676
677
678
679
680
681
682
683
684
685
686
687
688
689
690
691
692
693
694
695
696
697
698
699
700
By making use of the Mellin transform formula (66) in conjunction with the integral
representation (26), we obtain
M ( S (α,β)
μ (r;{k γ }) ) =
∫ ∞
= 2 ∫ ∞
x γ (μα−β)−1
Ɣ(μ) 0 e x − 1
0
r s−1 S μ
(α,β) (r;{k γ })dr
(∫ ∞
0
r s−1 [
1 1 (μ, 1) ; (γ (μα − β) ,γα) ;−r 2 x γα] )
dr dx
Ɣ (s/2) Ɣ (μ − s/2)
1
x γ (μα−β)−γαs/2−1
= ·
dx
Ɣ(μ) Ɣ (γ (μα − β) − γαs/2) 0 e x − 1
( s
= B
2 ,μ− s ) (
ζ γ (μα − β) − γαs ) (
0 < R (s) < 2μ; γ ∈ R
+ ) . (68)
2 2
Upon setting γ = q/α (q ∈ N) in (67), we readily get the following special case:
M(S (α,β)
μ (r;{k q/α })) =
( s
= B
2 ,μ− s )
ζ
2
∫ ∞
0
(
q
∫ ∞
r s−1 S μ
(α,β) ( {
r; k
q/α }) dr
([
μ − β α
which, in the further special case when q = 2, yields
M(S (α,β)
μ (r;{k 2/α })) =
( s
= B
2 ,μ− s )
ζ
2
∫ ∞
0
(
2
]
− s 2
)) (0
< R(s) < 2μ
)
, (69)
r s−1 S μ
(α,β) ( {
r; k
2/α }) dr
[
μ − β α
]
− s) (0
< R(s) < 2μ
)
. (70)
Alternatively, in view of the integral representation (68), the Mellin transform formula (68) can
be proven directly by applying the relatively more familiar analogue of (66) for the generalized
hypergeometric p F q -function and the Gauss–Legendre multiplication formula for the Gamma
function [12, p. 8, Equation 1.1 (51)]:
Acknowledgements
Ɣ(mz) =
mmz−1/2
(2π) 1/2(m−1)
m ∏
j=1
(
Ɣ z + j − 1 )
m
(z ̸= 0, − 1 m , − 2 m , ··· ; m ∈ N )
.
The present investigation was supported, in part, by the Ministry of Education and Sciences of Macedonia and the Ministry
of Education, Sciences and Sports of Croatia under Projects No. 05-437/1 and No. 13-272/1, and also, in part, by the
Natural Sciences and Engineering Research Council of Canada under Grant OGP0007353.
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Integral Transforms and Special Functions 15
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