Research Article
Marichev-Saigo-Maeda Fractional Integration Operators
Involving Generalized Bessel Functions
Saiful R. Mondal
1
and K. S. Nisar
2
1
Department of Mathematics & Statistics, College of Science, King Faisal University, P.O. Box 400, Hofuf, Al-Ahsa 31982, Saudi Arabia
2
Department of Mathematics, College of Arts and Science, Salman bin Abdulaziz University, P.O. Box 54, Wadi Al-Dawaser 11991,
Saudi Arabia
Correspondence should be addressed to Saiful R. Mondal; saiful786@gmail.com
Received 12 February 2014; Accepted 4 March 2014; Published 8 April 2014
Academic Editor: Santanu Saha Ray
Copyright © 2014 S. R. Mondal and K. S. Nisar. Tis is an open access article distributed under the Creative Commons Attribution
License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly
cited.
Two integral operators involving Appell’s functions, or Horn’s function in the kernel are considered. Composition of such
functions with generalized Bessel functions of the frst kind is expressed in terms of generalized Wright function and generalized
hypergeometric series. Many special cases, including cosine and sine function, are also discussed.
1. Introduction
Let ,
, ,
, ∈ C and >0; then the generalized
fractional integral operators involving Appell’s functions or
Horn’s function are defned as follows:
(
,
,,
,
0,+
)()
=
−
Γ()
∫
0
(−)
−1
−
3
×(,
,,
;;1−
,1−
)(),
(1)
(
,
,,
,
0,−
)()
=
−
Γ()
∫
∞
(−)
−1
−
3
×(,
,,
;;1−
,1−
)(),
(2)
with Re() > 0. Te generalized fractional integral operators
of the types (1) and (2) have been introduced by Marichev [1]
and later extended and studied by Saigo and Maeda [2]. Tese
operators together are known as the Marichev-Saigo-Maeda
operator.
Te fractional integral operator has many interesting
applications in various subfelds in applicable mathematical
analysis; for example, [3], it has applications related to a
certain class of complex analytic functions. Te results given
in [4–6] can be referred to for some basic results on fractional
calculus.
Te purpose of this work is to investigate compositions
of integral transforms (1) and (2) with the generalized Bessel
function of the frst kind W
,,
defned for complex ∈ C
and ,,∈ C by
W
,,
() =
∞
∑
=0
(−1)
Γ(+)!
(
2
)
2+
, (3)
where :=+(+1)/2. More details related to the function
W
,,
and its particular cases can be found in [7, 8] and
references therein. It is worth mentioning that W
,1,1
=
is Bessel function of order and W
,1,−1
=
is modifed
Bessel function of order . Also, W
,2,1
= 2
/√ is spherical
Bessel function of order and W
,2,−1
= 2
/√ is modifed
spherical Bessel function of order . Tus the study of the
integral transform of W
,,
will give far reaching results than
the result in [9, 10].
Te present paper is organized as follows. In Sections 2
and 3, composition of integral transforms (1) and (2) with
Hindawi Publishing Corporation
Mathematical Problems in Engineering
Volume 2014, Article ID 274093, 11 pages
http://dx.doi.org/10.1155/2014/274093