Research Article
Fractional q-IntegralOperatorsfortheProductofa q-Polynomial
and q-Analogueofthe I-Functions and Their Applications
V.K.Vyas ,
1
AliA.Al-Jarrah ,
1
D.L.Suthar ,
2
andNigussieAbeye
2
1
Sur University College, P. O. Box: 440, P. C.: 411, Sur, Oman
2
Department of Mathematics, Wollo University, P. O. Box: 1145, Dessie, Ethiopia
CorrespondenceshouldbeaddressedtoD.L.Suthar;dlsuthar@gmail.com
Received 17 April 2021; Revised 1 August 2021; Accepted 11 October 2021; Published 31 October 2021
AcademicEditor:AkbarZada
Copyright©2021V.K.Vyasetal.isisanopenaccessarticledistributedundertheCreativeCommonsAttributionLicense,
which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Inthisarticle,wederivefourtheoremsconcerningthefractionalintegralimagefortheproductofthe q-analogueofgeneralclass
of polynomials with the q-analogue of the I-functions. To illustrate our main results, we use q-fractional integrals of
Erd´ elyi–Kober type and generalized Weyl type fractional operators. e study concludes with a variety of results that can be
obtainedbyusingtherelationshipbetweentheErd´ elyi–KobertypeandtheRiemann–Liouville q-fractionalintegrals,aswellasthe
relationship between the generalized Weyl type and the Weyl type q-fractional integrals.
1.IntroductionandPreliminaries
It is commonly accepted that q-calculus is the basic ex-
tension of classical fractional calculus. e topic deals with
investigationsof q-calculusofanyorder,anditissignificant
and widespread because of its various applications in areas
similar to existing fractional calculus, patterns of the
q-transform analysis, q-distinction (differential), q-integral
equations, and so on (see [1, 3]). As a result of these ap-
plications, many researchers use the q-fractional calculus
formula to evaluate various special capacities such as the
q-hypergeometric function [4–7], the q-analogue of Fox’s
H-function [8, 9], the general class of q-polynomials [10],
the q-integral inequalities [11–14], and so on.
e q-analogueofRiemann–Liouvillefractionalintegral
operatorofafunction f(y) duetoAl-Salam[15]isdefined
as
I
μ
q
f(y)�
1
Γ
q
(μ)
y
0
[y − tq]
μ− 1
f(t)d(t; q),
(1)
where R(μ) > 0and0 < |q| < 1.
e q-analogue of the Erd´ elyi–Kober type fractional
integral operator is given by Agarwal [16] as
I
η,μ
q
f(y)�
y
− η− μ
Γ
q
(μ)
y
0
[y − tq]
μ− 1
t
η
f(t)d(t; q), (2)
where R(μ) > 0,0 < |q| < 1, and η is real or complex.
Al-Salam [15] has also established the basic q-analogue
oftheWeylfractionalintegraloperatorforarbitraryorder μ,
which is as follows:
K
− μ
q
f(y)�
q
− μ(μ− 1)/2
Γ
q
(μ)
∞
y
(t − y)
μ− 1
f tq
1− μ
d(t; q), (3)
where R(μ) > 0 and K
0
q
f(y)� f(y).
Inthesamearticle,Al-Salamdefinedthe q-extension of
the generalized Weyl fractional integral operator, which is
given as
K
η,μ
q
f(y)�
q
− η
y
η
Γ
q
(μ)
∞
y
(t − y)
μ− 1
t
− η− μ
f tq
1− μ
d(t; q),
(4)
where R(μ) > 0 and η is an arbitrary complex quantity.
According to Gasper and Rahman [17], the following
q-integrals were introduced:
Hindawi
Mathematical Problems in Engineering
Volume 2021, Article ID 7858331, 9 pages
https://doi.org/10.1155/2021/7858331