MATEMATIQKI VESNIK
originalni nauqni rad
research paper
66, 3 (2014), 323–332
September 2014
COMPOSITIONS OF SAIGO FRACTIONAL INTEGRAL
OPERATORS WITH GENERALIZED VOIGT FUNCTION
Deepa H. Nair and M. A. Pathan
Abstract. The principal object of this paper is to provide the composition of Saigo fractional integral operators with different forms of Voigt functions. An alternative explicit representation
of the generalized Voigt function in terms of Laplace integral transform is shown and the relations
between the left-sided and the right-sided Saigo fractional integral operators are established with
the 1 F1 -transform and the Whittaker transform, respectively. Many interesting results are deduced in terms of some relatively more familiar hypergeometric functions in one and two variables.
1. Introduction
This paper deals with the generalized Voigt function defined for µ, y ∈ R+ ,
x ∈ R, and <(µ + ν) > −1 by the integral
r Z ∞
µ
¶
2
x2 t2
x
µ −yt− t4
Ωµ,α,β,ν (x, y) =
t e
dt.
(1)
1 F2 α; β, 1 + ν; −
2 0
4
Here 1 F2 (α; β, γ; x) is the the function defined as
1 F2 (α; β, γ; x) =
∞
X
r=0
(α)r
xr
,
(β)r (γ)r r!
where α, β, γ ∈ C, γ 6= −p (p = 0, 1, 2, . . . ) and (α)r is the Pochhammer symbol
[1]. The function in (1) was introduced by Pathan and Shahwan [16]. The special
case of the function Ωµ,α,β,ν (x, y) in the form
r Z ∞
x
t2
Vµ,ν (x, y) =
tµ exp(−yt − )Jν (xt)dt, x, y ∈ R, <(µ + ν) > −1, (2)
2 0
4
with x, y ∈ R+ and <(µ + ν) > −1, known as generalized (unified) Voigt function,
was introduced and studied by Srivastava and Miller [22] and Srivastava and Chen
2010 Mathematics Subject Classification: 26A33, 44A15, 33C15, 33C55
Keywords and phrases: Fractional operators, Voigt function, Whittaker transform,
transform, Bessel function, Kampé de Fériet’s function.
323
1 F1 -
324
D. H. Nair, M. A. Pathan
[20]. Jν (x) appeared in (2) is the Bessel function [17] of the first kind of order ν
and is defined as
¡ ¢ν+2n
∞
X
(−1)n z2
, |z| < ∞.
Jν (z) =
Γ(n + 1)Γ(ν + n + 1)
n=0
The functions defined in (1) and (2) are connected through the relation [16, p. 77]
µ ¶ν
2
Ωµ,α,α,ν (x, y) = Γ(ν + 1)
Vµ−ν,ν (x, y).
x
The function Vµ,ν (x, y) is connected to the Voigt functions K(x, y) and L(x, y)
introduced by Voigt in 1899 through the following relations.
V 21 ,− 12 (x, y) = K(x, y) and
V 12 , 21 (x, y) = L(x, y),
where the functions K(x, y) and L(x, y) are respectively defined as follows:
Z ∞
1
t2
√
K(x, y) =
exp(−yt − ) cos(xt)dt
4
π 0
and
1
L(x, y) = √
π
Z
∞
exp(−yt −
0
t2
) sin(xt)dt,
4
with x ∈ R, y ∈ R+ . The function K(x, y) + i L(x, y) is identical to the so called
plasma dispersion function, except for a numerical factor, which is tabulated by
Fried and Conte [5] and by Fettis et al. [4]. Further, (1) can be connected to the
Struve function Hν (x) [2] through the relations
√ ³ ´1−β
Z ∞
t2
1
π x
Ωµ,1, 23 ,β−1 (x, y) =
Γ(β)
(3)
tµ−β+ 2 e−yt− 4 Hβ− 32 (xt)dt.
2
2
0
More special cases, representations and multivariate analogues of (1) can be seen
in [14]. For a review of various mathematical properties and computational methods, see Haubold and John [6], Srivastava and Miller [22], Srivastava et al. [23],
Yang [24], Khan et al. [7], Pathan et al. [15] etc. These functions have found
applications in various fields like astrophysical spectroscopy, neutron physics, plasma physics, statistical communication theory and some problems of mathematical
physics and engineering associated with multi-dimensional analysis of spectral harmonics. A further generalization of the Voigt function has been carried out in
the literature starting from the fact that the original function is defined for physical reasons as the convolution between the Gaussian and Lorentz densities. Since
these densities are both Levy stable densities, a probabilistic generalization can
be done by convolution of two general Levy densities. Due to the strong relation
between Levy densities and fractional calculus this probabilistic generalization is
worth mentioning. The details may be seen from [12] and [13].
The present paper is devoted to the study of the composition of Saigo fractional integral operators with different forms of the generalized Voigt function. The
Saigo fractional integral operators
325
fractional operators given by Saigo [18] for ν, β, η ∈ C, x ∈ R+ and for <(ν) > 0,
ν,β,η
ν,β,η
denoted by I0,x
and Jx,∞
, are defined as
ν,β,η
I0,x
f (x)
x−ν−β
=
Γ(ν)
Z
x
ν−1
(x − t)
0
¶
µ
t
f (t) dt
2 F1 ν + β, −η; ν; 1 −
x
(4)
and
ν,β,η
Jx,∞
f (x)
1
=
Γ(ν)
Z
∞
x
³
x´
(t − x)ν−1 t−ν−β 2 F1 ν + β, −η; ν; 1 −
f (t) dt,
t
(5)
where 2 F1 (a, b; c; x) is the Gauss hypergeometric series. Formula (4) is the leftsided Saigo fractional integral operator and (5) is the right-sided Saigo fractional
integral operator. Moreover
2 F1 (a, b; c; 1)
=
Γ(c)Γ(c − a − b)
, <(c − a − b) > 0.
Γ(c − a)Γ(c − b)
When β = −ν in (4) and (5) we get respectively, the classical Riemann-Liouville
fractional integral and the Weyl integral [9, 10, 19] of order ν as
Z x
1
−ν
D
f
(x)
=
(x − t)ν−1 f (t)dt, <(ν) > 0
0 x
Γ(ν) 0
and
ν
x W∞ f (x) =
1
Γ(ν)
Z
∞
(t − x)ν−1 f (t)dt, <(ν) > 0
x
whereas if we set β = 0 in (4) and (5), we obtain the Erdélyi-Kober (E-K) fractional
integration operators [9, 10, 19]
Z
x−ν−η x
ν,η
E0,x f (x) =
(x − t)ν−1 tη f (t)dt, x ∈ R+
Γ(ν) 0
and
ν,η
Kx,∞
f (x)
xη
=
Γ(ν)
Z
∞
(t − x)ν−1 t−ν−η f (t)dt, x ∈ R+ .
x
The paper is systemized into four sections. In the next section, we establish
an alternative representation of the generalized Voigt function, which provide an
extension for the result given by Pathan and Daman in 2010 (see reference [14]).
Section 3 is devoted to the study of the composition of the left-sided Saigo fractional
image of the generalized Voigt function and the results are expressed in closed
forms via Kampé de Fériet’s function. It is also shown that the image under Saigo
operator is related to 1 F1 -transform. Many interesting results are deduced from so
established results. Finally, in the last section, the effects of the right-sided Saigo
operator on generalized Voigt functions and its relation with Whittaker transform
are described.
326
D. H. Nair, M. A. Pathan
2. An alternate representation of generalized Voigt function
The Laplace transform of the function f (t) [2] having the transform parameter
p is defined as
Z
∞
L {f (u); p} =
e−pu f (u)du, <(p) > 0.
(6)
0
Theorem 1. [14] If L {h(t); p} = g(p) then
Z ∞
n
o
√
a2
3
a
−a t
e− 4t t− 2 g(t + p)dt, <(p) > 0, <(a) > 0.
L e
h(t); p = √
2 π 0
Now we will discuss about an alternative representation of the generalized
Voigt function with the aid of Theorem 1. By using the definition (6), we have
½
µ
¶ ¾
xt
L e−bt tµ−1 1 F2 α; β, 1 + ν; −
;p
2
µ
¶
Γ(µ)
x
=
F
α,
µ;
β,
1
+
ν;
−
= G(p + b), <(p) > 0, b > 0.
2 2
(p + b)µ
2(p + b)
Replacing x2 by ( x2 )2 and using Theorem 1, we find that
½
µ
¶ ¾
Z ∞
√
a2
3
xt
a
e− 4t t− 2 G(t + b)dt.
L e−a t tµ−1 1 F2 α; β, 1 + ν; −
;b = √
2
2 π 0
This can be simplified into the following form.
µ
¶
Z ∞
2
x2 t2
e−at−bt t2µ−1 1 F2 α; β, 1 + ν; −
dt
4
0
µ
¶
Z ∞
2
x2
a
−µ
− a4t − 32
t (t + b) 2 F2 α, µ; β, 1 + ν; −
e
dt,
= √
4(t + b)
2 π 0
<(µ) > 0, b > 0, x > 0, <(a) > 0.
On taking b = 14 , µ = µ+1
2 , a = y, the expression on the L.H.S. of the above
equation represents the generalized Voigt function Ωµ,α,β,ν introduced by Pathan
and Shahwan [16]. If we assume that α = β, this alternate representation will
coincides with the result established by Pathan and Daman [14].
3. Composition of the left-sided Saigo fractional operator
with generalized Voigt function
This section deals with the composition formulae of left-sided Saigo operator
and Voigt functions.
Theorem 2. If α∗ , β ∗ , η, λ ∈ C, µ ∈ R, p > 0, x > 0, <(α∗ ) > 0, <(µ + ν) >
−1, <(1 − λ) > 0, then the left-sided Saigo fractional image of the generalized Voigt
327
Saigo fractional integral operators
function in (1) is given by
½ µ+1
∞
X
(α∗ + β ∗ )r (−η)r Γ( 2 )
√
(1 − λ + α∗ )r r!
2
r=0
¸ √
· µ+1
1−λ 2−λ
2pΓ( µ2 + 1)(1 − λ)
1:2;1
2
2
2 : 2 , 2 ;α;
−
p
,
−x
+
× F0:3;2
∗
∗
+r 2−λ+α +r
−: 12 , 1−λ+α
,
;β, 1+ν;
(1 − λ + α∗ + r)
2
2
·µ
¸¾
2−λ 3−λ
1:2;1 2 +1: 2 , 2 ;α;
2
2
−
p
,
−x
. (7)
× F0:3;2
∗
∗
3 2−λ+α +r 3−λ+α +r
∗
α ,β
I0,p
∗
,η
[t−λ Ωµ,α,β,ν (x, t)] = C
−: 2 ,
√
µ −β ∗ −λ
Here C = x 2 p Γ(α∗B(1−λ,α
)
Fériet’s function [21].
∗
)
2
,
2
;β,1+ν;
p:q;k
and Fl:m;n
denotes the generalized Kampé de
Proof. Employing the definitions given in (1), (4) and on interchanging the
order of integrations which is permissible due to the uniform convergence, we have
r
µ
¶
∗
∗ Z ∞
2
x p−β −α
x2 u2
α∗ ,β ∗ ,η −λ
µ − u4
I0,p
[t Ωµ,α,β,ν (x, t)] =
u
e
F
α;
β,
1
+
ν;
−
1 2
2 Γ(α∗ ) 0
4
¶ ¸
·Z p
µ
∗
t
dt du.
×
(p − t)α −1 e−ut t−λ 2 F1 α∗ + β ∗ , −η; α∗ ; 1 −
p
0
With the help of series representation of Gauss hypergeometric function and on
taking t = ps, further simplification yields
r
µ
¶
Z
∗
x2 u2
x p−β −λ ∞ µ − u2
α∗ ,β ∗ ,η −λ
4
I0,p
[t Ωµ,α,β,ν (x, t)] =
u e
1 F2 α; β, 1 + ν; −
2 Γ(α∗ ) 0
4
·Z
¸
∞
1
X (α∗ + β ∗ )r (−η)r
∗
×
(1 − s)α +r−1 e−ups s−λ ds du. (8)
∗
(α )r r!
0
r=0
By using the definition of type-1 beta function and series expansion for the exponential function, we get
r
∗
∞ ∞
x p−β −λ B(1 − λ, α∗ ) X X (α∗ + β ∗ )r (−η)r
α∗ ,β ∗ ,η −λ
[t Ωµ,α,β,ν (x, t)] =
I0,p
2
Γ(α∗ )
(1 − λ + α∗ )r r!
r=0 k=0
µ
¶
Z
x2 u2
(1 − λ)k
(−p)k ∞ µ+k − u2
4
×
u
e
du.
1 F2 α; β, 1 + ν; −
(1 − λ + α∗ + r)k
k!
4
0
∗
α ,β
I0,p
∗
,η
[t−λ Ωµ,α,β,ν (x, t)]
r
∗
∞
∞
x 2µ p−β −λ B(1 − λ, α∗ ) X X (α∗ + β ∗ )r (−η)r
=
2
Γ(α∗ )
(1 − λ + α∗ )r r!
r=0 k=0
µ
¶
Γ( µ+k+1
)(1 − λ)k (−2p)k
µ+k+1
2
2
×
F
α,
;
β,
1
+
ν;
−x
. (9)
2 2
(1 − λ + α∗ + r)k
k!
2
328
D. H. Nair, M. A. Pathan
Now breaking the k-series into odd and even terms and using the formula (a)2n =
22n ( a2 )n ( a+1
2 )n and the definition of generalized Kampé de Fériet’s function
Qq
Qk
∞ Qp
h
i
r
s
X
j=1 (aj )r+s
j=1 (bj )r
j=1 (cj )s x y
p:q;k (ap ): (bq ); (ck );
Fl:m;n (αl ): (βm );(γn ) x, y =
,
Ql
Qm
Qn
r! s!
r,s=0
j=1 (αj )r+s
j=1 (βj )r
j=1 (γj )s
the result in (7) follows.
Corollary 2.1. A relation also can be established among left-sided Saigo
operator, Voigt function and 1 F1 -transform by evaluating (4) by using the series
expansion of exponential function and definition of type-1 beta function as
r
∞
x −β ∗ −λ−µ X (α∗ + β ∗ )r (−η)r
α∗ ,β ∗ ,η −λ
p
I0,p
[t Ωµ,α,β,ν (x, t)] =
2
r!
r=0
·
½
µ
¶ ¾¸
2
x2 u2
µ
− u4
× 1 F1
e
;p ,
1 F2 α; β, 1 + ν; −
4
where 1 F1 k (.) denotes an integral transform known as generalized 1 F1 -transform
[8], firstly considered by Erdélyi [3] and is defined as
Z
Γ(a) ∞
k
(1 F1 )(x) =
(xt)k 1 F1 (a; c; −xt)f (t)dt,
Γ(c) 0
with a, c ∈ C, <(a) > 0 and k ∈ R containing the confluent hypergeometric or
Kummer function 1 F1 (a; c; x) in the kernel.
Some special cases of the above theorem are explained in following corollaries.
Corollary 2.2. When α = β, (9) reduces to left-sided Saigo image of the
function Vµ,ν given as
∗
α ,β
I0,p
∗
,η
[t−λ Vµ,ν (x, t)] = C1
∞
X
(α∗ + β ∗ )r (−η)r
r=0
(1 − λ + α∗ )r r!
½ µ+ν+1
¸
·
2−λ
Γ( 2 ) 1:2;0 µ+ν+1
: 1−λ
2
2
2
2 , 2 ;−;
√
×
F0:3;1
−
p
,
−x
∗
∗
+r 2−λ+α +r
−: 12 , 1−λ+α
,
; 1+ν;
2
2
2
√
¸¾
·
µ+ν
2−λ 3−λ
2pΓ( µ+ν
2
2
2 + 1)(1 − λ) 1:2;0
2 +1: 2 , 2 ;−;
+
F0:3;1
− p , −x
,
∗ +r 3−λ+α∗ +r
−: 32 , 2−λ+α
,
;1+ν;
(1 − λ + α∗ + r)
2
2
1
∗
xν+ 2 2µ p−β −λ B(1 − λ, α∗ )
.
Γ(ν + 1)Γ(α∗ )
Corollary 2.3. Let α = β, µ = 12 . For ν = − 12 , we have the left-sided Saigo
image of the function K(x, t) and for ν = 12 , the result for L(x, t) can be obtained.
C1 =
Corollary 2.4. By using the relation (3), we can deduce a relation involving
Struve function and Saigo operator by taking α = 1, β = 32 , ν = β − 1. Similarly
for β = α + 1, ν = β − 1, result involving Lommel function [2] can be obtained.
Saigo fractional integral operators
329
Corollary 2.5. Taking α = 0, due to the relation [14]
√
√
µ
t2
Ωµ,0,β,ν (x, t) = x 2 2 Γ(µ + 1)e 4 D−(µ+1) ( 2t),
where Dµ (.) is the parabolic cylindrical function [2], we can establish a relation
between the Saigo operator and the parabolic cylindrical function from (7) as follows:
∞
X
√
t2
(α∗ + β ∗ )r (−η)r
α∗ ,β ∗ ,η −λ
I0,p
[t
e 4 D−(µ+1) ( 2t)] = C2
(1 − λ + α∗ )r r!
r=0
¶
½ µ+1
µ
Γ( 2 )
µ + 1 1 − λ 2 − λ 1 1 − λ + α∗ + r 2 − λ + α∗ + r
2
√
,
,
; ,
,
; −p
×
3 F3
2
2
2
2
2
2
2
√
2pΓ( µ2 + 1)(1 − λ)
+
(1 − λ + α∗ + r)
µ
¶¾
2 − λ 3 − λ 3 2 − λ + α∗ + r 3 − λ + α∗ + r
µ
2
× 3 F3
+ 1,
,
; ,
,
; −p
,
2
2
2
2
2
2
µ
where C2 =
∗
2 2 p−β −λ B(1−λ,α∗ )
.
Γ(α∗ )Γ(µ+1)
Corollary 2.6. Further, on taking β ∗ = 0, the results for Erdélyi-Kober
operator can be deduced from above results.
Corollary 2.7. Instead of β ∗ = 0, if we take β ∗ = −α∗ the corresponding
results for Riemann-Liouville operator can be obtained.
4. Right-sided Saigo fractional operator and generalized Voigt function
The function Ωµ,α,β,ν (x, y) can also be written as
r
µ
½
¶ ¾
u2
x2 u2
x
Ωµ,α,β,ν (x, s) =
L uµ e− 4 1 F2 α; β, 1 + ν; −
;s ,
(10)
2
4
where L {f (u); s} represents the Laplace transform of the function f with the transform parameter s, see [2].
By using the above representation, we will next discuss about the compositions
of right-sided Saigo operator with Voigt functions and hence some of their special
cases are discussed.
Theorem 3. If α∗ , β ∗ , η, λ ∈ C, µ ∈ R, p > 0, x > 0, <(α∗ ) > 0, <(µ + ν) >
−1, then the Saigo fractional image of the the generalized Voigt function is given
by
∞
X
(α∗ + β ∗ )r (−η)r
α∗ ,β ∗ ,η −λ
Jp,∞ [t Ωµ,α,β,ν (x, t)] =
r!
r=0
½ 2m
∞
p Γ(−2m) X ( 12 + m − k)s ps
Ωµ+s,α,β,ν (x, p)
×
Γ( 12 − m − k) s=0 (2m + 1)s s!
¾
∞
X
( 12 − m − k)q pq
Γ(2m)
+ 1
Ωµ+q−2m,α,β,ν (x, p) , (11)
Γ( 2 + m − k) q=0 (1 − 2m)q q!
330
D. H. Nair, M. A. Pathan
where
k=
1 − 2α∗ − 2r − β ∗ − λ
(β ∗ + λ)
; m=−
.
2
2
(12)
Proof. Consider the right-sided Saigo fractional integral operator. By using
(5) and (10), we have
∗
α ,β
Jp,∞
∗
,η
[t−λ Ωµ,α,β,ν (x, t)]
Z ∞
³
∗
∗
∗
p ´ −λ
1
(t − p)α −1 t−α −β 2 F1 α∗ + β ∗ , −η; α∗ ; 1 −
t
=
∗
Γ(α ) p
t
½r
½
¶ ¾¾
µ
2
x
x2 u2
µ − u4
×
L u e
;t
dt.
1 F2 α; β, 1 + ν; −
2
4
On taking t − p = y and interchanging the order of integration, due to the uniform
convergence of the integral,
r
µ
¶
Z ∞
2
x 1
x2 u2
α∗ ,β ∗ ,η −λ
µ −pu− u4
Jp,∞ [t Ωµ,α,β,ν (x, t)] =
u e
1 F2 α; β, 1 + ν; −
2 Γ(α∗ ) 0
4
¶ ¾¸
· ½
µ
∗
∗
∗
y
× L y α −1 (p + y)−α −β −λ 2 F1 α∗ + β ∗ , −η; α∗ ;
;u
dt.
p+y
On simplification by using Gauss hypergeometric series, the Laplace integral on the
right-side can be written as
½
µ
¶ ¾
∗
∗
∗
y
L y α −1 (p + y)−α −β −λ 2 F1 α∗ + β ∗ , −η; α∗ ;
;u
p+y
∞
o
X
(α∗ + β ∗ )r (−η)r n α∗ +r−1
−α∗ −β ∗ −λ−r
=
L
y
(p
+
y)
;
u
.
(α∗ )r r!
r=0
Employing the formula [2]
n
o Γ( 1 − k + m)
1
1
1 ap
2
L tm−k− 2 (t + a)m+k− 2 ; p =
pm− 2 e 2 Wk,m (ap),
1
−m
a2
| arg a| < π, <( 12 − k + m) > 0, where Wk,m (.) denotes the Whittaker function [2,
12] defined as
µ
¶
1
+m
1 Γ(−2m)z 2
1
Wk,m (z) = e− 2 z 1
+ m − k; 2m + 1; z
1 F1
2
Γ( 2 − m − k)
µ
¶
1
2 −m
1
− 12 z Γ(2m)z
+e
− m − k; 1 − 2m; z , (13)
1 F1
2
Γ( 21 + m − k)
and on further simplification, we get
r
∞
x m− 1 X (α∗ + β ∗ )r (−η)r
α∗ ,β ∗ ,η −λ
Jp,∞
[t Ωµ,α,β,ν (x, t)] =
p 2
2
r!
r=0
¶
µ
Z ∞
pu
1
u2
x2 u2
Wk,m (pu)du. (14)
×
uµ−m− 2 e− 2 − 4 1 F2 α; β, 1 + ν; −
4
0
By using (14) and rewriting the result using (1), we get the Theorem.
Saigo fractional integral operators
331
Corollary 3.1. We also have a relation between (11) and generalized Whittaker transform [8] as follows:
r
∞
x 2m−µ X (α∗ + β ∗ )r (−η)r
α∗ ,β ∗ ,η −λ
Jp,∞ [t Ωµ,α,β,ν (x, t)] =
p
2
r!
r=0
µ
¶ ¾
½
2
x2 u2
µ−m
− u4
;p ,
× Wk,m
e
1 F2 α; β, 1 + ν; −
4
σ
where Wk,m
(.) denotes the generalized Whittaker transform given by
Z ∞
1
1
σ
Wk,m {f (t); p} =
e− 2 pt (pt)σ− 2 Wk,m (pt)f (t)dt,
0
Wk,m (.) is the function defined in (13). It is known that the Meijer transform and
Varma transform are particular case of generalized Whittaker transform.
It is interesting to note that if we take α∗ = −β ∗ , the relation (14) reduces to
the one established by Pathan and Daman [14] for the Weyl fractional operator.
Corollaty 3.2. When α = β, we have the following result:
∗
α ,β
Jp,∞
∗
,η
[t−λ Vµ−ν,ν (x, t)]
½
∞
∞
X
(α∗ + β ∗ )r (−η)r p2m Γ(−2m) X ( 12 + m − k)s ps
=
Vµ−ν+s,ν (x, p)
r!
Γ( 21 − m − k) s=0 (2m + 1)s s!
r=0
¾
∞
X
( 12 − m − k)q pq
Γ(2m)
+ 1
Vµ−ν+q−2m,ν (x, p) ,
Γ( 2 + m − k) q=0 (1 − 2m)q q!
where m and k are the same given in (12).
Corollary 3.3. Under the same restrictions of parameters given for Corollaries 2.3, 2.4 and 2.5, the corresponding results for the right-sided Saigo operator
and the functions K(x, y), L(x, y), Struve function, Parabolic cylindrical function
can be obtained from (11).
Corollary 3.4. Further, for β ∗ = 0, the images of Voigt functions can be
obtained from above results under the right-sided Erdélyi-Kober fractional operator.
Acknowledgement. The authors would like to thank the Department of Science and Technology, Government of India, New Delhi, for the financial assistance
for this work under project No. SR/S4/MS:287/05, and the Centre for Mathematical Sciences for providing all facilities.We are also thankful to the referee for giving
us various suggestions for the improvement of the paper.
REFERENCES
[1] A. Erdélyi, W. Magnus, F. Oberhettinger and F. G. Tricomi, Higher Transcendental Functions, Vol. 1, McGraw-Hill, New York (1953).
332
D. H. Nair, M. A. Pathan
[2] A. Erdélyi, W. Magnus, F. Oberhettinger and F. G. Tricomi, Tables of Integral Transforms,
Vol. 1, McGraw-Hill, New York (1954).
[3] A. Erdélyi, W. Magnus, F. Oberhettinger and F. G. Tricomi, Higher Transcendental Functions, Vol. 3, McGraw-Hill, New York (1955).
[4] H. E. Fettis, J. C. Caslin, K. R. Cramer, An Improved Tabulation of the Plasma Dispersion
Function, Parts I and II, ARL 72-0056 and 72-0057, Air Force Systems Command, WrightPatterson AFB, Ohio, 1972.
[5] B. D. Fried, S. F. Conte, The Plasma Dispersion Function, Academic Press, New York, 1961.
[6] H. J. Haubold, R. W. John, Spectral line profiles and neutron cross sections: New results
concerning the analysis of Voigt functions, Astrophys. Space Sci. 65 (1979), 477–491.
[7] S. Khan, B. Agrawal, M. A. Pathan, Some connections between generalized Voigt functions
with the different parameters, Appl. Math. Comput., 181 (2006), 57-=64.
[8] A. A. Kilbas, M. Saigo, H-transforms, Theory and Applications, Chapman & Hall/CRC,
Boca Raton, London, New York, 2004.
[9] V. Kiryakova, Generalized Fractional Calculus and Applications, Longman Harlow & Wiley,
New York, 1994.
[10] A. M. Mathai, H. J. Haubold, Special Functions for Applied Scientists, Springer, New York
(2008).
[11] A. M. Mathai, R. K. Saxena, H. J. Haubold, The H-function Theory and Applications,
Springer, New York (2010).
[12] C. E. G. Otiniano, T. R. Sousa, P. N. Rathie, Stable random variables: Convolution and
reliability, J. Comput. Appl. Math., 242 (2013), 1–11.
[13] G. Pagnini, F. Mainardi,Evolution equation for the probabilistic generalization of the Voigt
profile function, J. Comput. Appl. Math., 233 (2010), 1590–1595.
[14] M. A. Pathan, O. A. Daman, On Weyl fractional integral operators and generalized Voigt
functions, Ganita Sandesh, 24 (1) (2010), 1–10.
[15] M. A. Pathan, M. Kamarujjamma, M. K. Alam, On multiindices and multivariable presentations of the Voigt functions, J. Comput. Appl. Math. 160 (2003), 251–257.
[16] M. A. Pathan, M. J. S. Shahwan, New representations of the Voigt functions, Demonstratio
Math., 39 (2006), 75–80.
[17] E. D. Rainville, Special Functions, Macmillan, New York, 1960; Reprinted by Chelsea, Bronx,
New York, 1971.
[18] M. Saigo, A remark on integral operators involving the Gauss hypergeometric functions,
Math. Rep. Kyushu Univ., 11 (1978), 135–143.
[19] S. G. Samko, A. A. Kilbas, O. I. Marichev, Fractional Integrals and Derivatives, Theory and
Applications, Gordon and Breach, New York, 1993.
[20] H. M. Srivastava, M. P. Chen, Some unified presentations of the Voigt functions, Astrophys.
Space Sci., 192 (1992), 63–74.
[21] H. M. Srivastava, H. L. Manocha, A Treatise on Generating Functions, Ellis Horwood Limited, England, 1984.
[22] H. M. Srivastava, E. A. Miller, A unified presentation of the Voigt functions, Astrophys.
Space Sci. 135 (1987), 111–118.
[23] H. M. Srivastava, M. A. Pathan, M. Kamarujjama, Some unified presentations of the generalized Voight functions, Comm. Appl. Anal. 2(1), (1998), 49–64.
[24] S. Yang, A Unification of the Voigt functions, Internat. J. Math. Ed. Sci. Tech. 25 (6),
(1994), 845–851.
(received 19.01.2013; in revised form 06.09.2013; available online 15.12.2013)
Centre for Mathematical Sciences, Arunapuram P.O., Palai, Kerala-686 574, India
E-mail: deepanaircms@gmail.com, mapathan@gmail.com