# Fractionalq-Integral Operators for the Product of a q-Polynomial and q-Analogue of the I-Functions and Their Applications

Authors: V.K. Vyas; Ali A. Al-Jarrah; D. L. Suthar; Nigussie Abeye
Journal: Mathematical Problems in Engineering
 (2021)

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## Abstract

In this article, we derive four theorems concerning the fractional integral image for the product of theq-analogue of general class of polynomials with the q-analogue of the I-functions. To illustrate our main results, we use q-fractional integrals of Erdélyi–Kober type and generalized Weyl type fractional operators. The study concludes with a variety of results that can be obtained by using the relationship between the Erdélyi–Kober type and the Riemann–Liouville q-fractional integrals, as well as the relationship between the generalized Weyl type and the Weyl type q-fractional integrals.


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## Body

## 1. Introduction and Preliminaries

It is commonly accepted thatq-calculus is the basic extension of classical fractional calculus. The topic deals with investigations of q-calculus of any order, and it is significant 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 applications, 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.Theq-analogue of Riemann–Liouville fractional integral operator of a function fy due to Al-Salam [15] is defined as(1)Iqμfy=1Γqμ∫0yy−tqμ−1ftdt;q,where ℜμ>0 and 0<q<1.Theq-analogue of the Erdélyi–Kober type fractional integral operator is given by Agarwal [16] as(2)Iqη,μfy=y−η−μΓqμ∫0yy−tqμ−1tηftdt;q,where ℜμ>0, 0<q<1, and η is real or complex.Al-Salam [15] has also established the basic q-analogue of the Weyl fractional integral operator for arbitrary order μ, which is as follows:(3)Kq−μfy=q−μμ−1/2Γqμ∫y∞t−yμ−1ftq1−μdt;q,where ℜμ>0 and Kq0fy=fy.In the same article, Al-Salam defined theq-extension of the generalized Weyl fractional integral operator, which is given as(4)Kqη,μfy=q−ηyηΓqμ∫y∞t−yμ−1t−η−μftq1−μdt;q,where ℜμ>0 and η is an arbitrary complex quantity.According to Gasper and Rahman [17], the following q-integrals were introduced:(5)∫0yftdt;q=y1−q∑k=0∞qkfyqk,(6)∫y∞ftdt;q=y1−q∑k=1∞q−kfyq−k,(7)∫0∞ftdt;q=1−q∑k=−∞∞qkfqk.Also,(8)x−yν=xν∏n=0∞1−y/xqn1−y/xqν+n.In connection to result (5), (2) can be stated as follows:(9)Iqη,μfy=1−qΓqμ∑k=0∞qk1+η1−qk+1μ−1fyqk,where ℜμ>0 and η is real or complex.Forfy=yν−1, then(10)Iqη,μyα−1=Γqα+ηΓqα+η+μyα−1.Now, by using (6) in (4), the q-generalized Weyl fractional integral operator can be written as(11)Kqη,μfy=q−ηyηΓqμ∫y∞t−yμ−1t−η−μftq1−μdt;q.Forfy=yα, then we obtain(12)Kqη,μyα=Γqη−αΓqη−α+μyαq−μα.We begin by returning to a form ofq-polynomials fn,Ny;q in terms of a bounded complex sequence Sn,qn=0∞, provided that(13)fn,Ny,q=∑j=0n/NnNjSj,qyj,n=0,1,2,….Also,q-gamma function is given (cf. Gasper and Rahman [17]) as(14)Γqβ=q;q∞1−q1−βqβ;q∞=1−qβ−11−qβ−1,where β≠0,−1,−2,… and(15)β;q∞=∏j=0∞1−βqj.Further, for real or complexβ and 0<q<1, the q-shifted factorial is defined as(16)β;qn=1,ifn=0,1−β1−βq⋯1−βqn−1,ifn∈ℕ,or(17)β;qn=β;q∞βqn;q∞.The definition forq-analogue of the I-function in expressions of the Mellin–Barnes type basic contour integral given by Saxena and Kumar [18] is as follows:(18)IPi,Qim,ny;q|aj,αj1,n,aji,αjin+1,Pibj,βj1,m,bj,βjm+1,Qi=12πω∫C∏j=1mGqbj−βjs∏j=1nGq1−aj+αjsπys∑i=1r∏j=m+1QiGq1−bji+βjis∏j=m+1PiGqaji−αjisGq1−ssinπsds,where 0≤m≤Qi;0≤n≤Pi; i=1,2,…,r;r is finite; ω=−1; and(19)Gqa=∏n=0∞1−qa+n−1=1qa;q∞,where αj,αji,βj,βji are real with positive and aj,aji,bj,bji are complex numbers.As of the contourC all of the poles of Gqbj−βjs, 1≤j≤m, are to its right side, and those of Gq1−aj+αjs, 1≤j≤n, are to its left side and at least some ε>0 distance away. The basic integral converges if ℜslogy−logsinπs<0, for huge values of s on the contour, that is, if argy<π. It reduces to the contour of integration C that can be replaced by other properly concave contours parallel to the imaginary axis.It is interesting to note that forr=1,P1=P;Q1=Q; definition (18) yields the q-analogue of Fox’s H-function due to Saxena et al. [19], namely,(20)HP,Qm,ny;q|a,αb,β=12πω∫C∏j=1mGqbj−βjs∏j=1nGq1−aj+αjsπys∏j=m+1QGq1−bj+βjs∏j=n+1PGqaj−αjsGq1−ssinπsds.If we putαi=βj=1,∀i and j in definition (20), it reduces to a basic analogue of Meijer’s G-function defined by Saxena et al. [19], that is,(21)HP,Qm,ny;q|a,1b,1≡GP,Qm,ny;q|a1,…,aPb1,…,bQ=12πω∫C∏j=1mGqbj−s∏j=1nq1−aj+sπys∏j=m+1QGq1−bj+s∏j=n+1PGqaj−sGq1−ssinπsds,where 0≤m1≤Q,0≤n1≤P, and ℜslogy−logsinπs<0.Additionally, if we taken1=0,m1=Q in definition (21), we attain the q-analogue of MacRobert’s E-function:(22)GP,QQ,0y;q|a1,…,aPb1,…,bQ=EqQ;bj:P;aj:y.A comprehensive description of many functions expressible in terms of Fox’sH-function can be found in research editions published by Mathai and Saxena [20, 21] and Mathai et al. [22].


## 2. Main Results

In this section, we will prove theorems involving fractional orderq-integrals of Erdélyi–Kober type and generalized Weyl type for the product of q-analogue of general class of polynomial and q-analogue of I-function.Theorem 1.
Letℜμ>0, η,λ∈ℜ, q<1, and Iqη,μ⋅ be the Kober fractional q-integral operator for multiplication of q-general class of polynomial and q-analogue of I-function as(23)Iqη,μyλ−1fn,Ny,qIPi,Qim,nρyk;q|aj,αj1,n,aji,αjin+1,Pibj,βj1,m,bj,βjm+1,Qi=yλ−11−qμ∑j=0n/NnNjSj,qyj×IPi,Qi+1m,+1,nρyk;q|1−j−λ−η,kaj,αj1,n,aji,αjin+1,Pibj,βj1,m,bj,βjm+1,Qi1−j−μ−λ−η,k,provided ℜslogy−logsinπs<0.Proof.
By using definitions (13) and (18) on the left-hand side of equation (23), we have(24)L=Iqη,μyλ−1∑j=0n/NnNjSj,qyj×12πω∫C∏j=1mGqbj−βjs∏j=1nGq1−aj+αjsπρyks∑i=1r∏j=m+1QiGq1−bji+βjis∏j=n+1PiGqaji−αjisGq1−ssinπsds=∑j=0n/NnNjSj,q×12πω∫C∏j=1mGqbj−βjs∏j=1nGq1−aj+αjsπρs∑i=1r∏j=m+1QiGq1−bji+βjis∏j=n+1PiGqaji−αjisGq1−ssinπs×Iqη,μyj+λ+ks−1ds.
Using known result (10), we get(25)L=∑j=0n/NnNjSj,q×12πω∫C∏j=1mGqbj−βjs∏j=1nGq1−aj+αjsπρs∑i=1r∏j=m+1QiGq1−bji+βjis∏j=n+1PiGqaji−αjisGq1−ssinπsds×1−qμqj+λ+ks+μ+η;q∞qj+λ+ks+η;q∞yj+λ+ks−1,
We arrived at the right-hand side of result (23) by further simplification of above term.Theorem 2.
Letℜμ>0q<1 and the generalized Weyl fractional q-integral of the multiplication of basic analogue of general class of polynomials and q-analogue of I-function exist, and we have(26)Kqη,μyλfn,Ny,qIPi,Qim,nρyk;q|aj,αj1,n,aji,αjin+1,Pibj,βj1,m,bj,βjm+1,Qi=yλq−μλ1−qμ∑j=0n/NnNjSj,qyj×IPi+1,Qi+1m+1,nρyq−μk;q|aj,αj1,n;aji,αjin+1,Pi,η−j−λ,kη−λ−j+μ,kbj,βj1,m;bj,βjm+1,Qi,where 0≤m≤Qi;0≤n≤Pi; i=1,2,…,r;r is finite, q<1, and λ,ρ are arbitrary.Proof.
By using definition (13) and (18) on the left-hand side of equation (26), we have(27)L=∑j=0n/NnNjSj,q×12πω∫C∏j=1mGqbj−βjs∏j=1nGq1−aj+αjsπρs∑i=1r∏j=m+1QiGq1−bji+βjis∏j=n+1PiGqaji−αjisGq1−ssinπs×Kqη,μyj+λ+ksds.
If we use known result (12), the expression reduces to(28)L=yλq−μλ+ks2πω∑j=0n/NnNjSj,q×∫C∏j=1mGqbj−βjs∏j=1nGq1−aj+αjsπρyks∑i=1r∏j=m+1QiGq1−bji+βjis∏j=n+1PiGqaji−αjisGq1−ssinπs×Γqη−j−λ−ksΓqη−j−λ−ks+μ.
By simplifying further, we got to the right-hand side of result (26).Theorem 3.
Letℜμ>0, η∈ℜ, q<1, and Iqη,μ⋅ be the Kober fractional q-integral operator; then, the following result holds:(29)Iqη,μyλ−1fn,Ny,qHP,Qm1,n1yk;q|a1,α1,…,aP,αPb1,β1,…,bQ,βQ=yλ−11−qμ∑j=0n/NnNjSj,qyj×HP+1,Qm1,n1+1yk;q|1−j−λ−η,k,a1,α1,…,aP,αPb1,β1,…,bQ,βQ,1−j−μ−λ−η,k,provided ℜslogy−logsinπs<0,k>0.Proof.
By using definitions (13) and (20) on the left-hand side of equation (29), we have(30)L=Iqη,μyλ−1∑j=0n/NnNjSj,qyj×12πi∫C∏j=1mGqbj−βjs∏j=1nGq1−aj+αjsπyks∏j=m+1QGq1−bj+βjs∏j=n+1PGqaj−αjsGq1−ssinπsdsL=∑j=0n/NnNjSj,q×12πi∫C∏j=1mGqbj−βjs∏j=1nGq1−aj+αjsπ∏j=m+1QGq1−bj+βjs∏j=n+1PGqaj−αjsGq1−ssinπs×Iqη,μyj+λ+ks−1ds.
On using known result (10),(31)Iqη,μyν−1=Γqν+ηΓqν+η+μyν−1,and the above expression reduces to(32)L=∑j=0n/NnNjSj,q×12πi∫C∏j=1mGqbj−βjs∏j=1nGq1−aj+αjsπ1−qj+λ+ks∏j=m+1QGq1−bj+βjs∏j=n+1PGqaj−αjsGq1−ssinπs×1−qμqj+λ+ks+μ+η;q∞qj+λ+ks+η;q∞yj+λ+ks−1.
The desired right-hand side of (29) may be obtained by further simplification.Theorem 4.
Ifℜμ>0, η∈ℜ, q<1, then the generalized Weyl fractional q-integral operator for the basic analogue of product of general class of polynomial and basic analogue of H-function is given by(33)Kqη,μyλfn,Nx,qHP,Qm1,n1ρyk;q|a1,α1,…,aP,αPb1,β1,…,bQ,βQ;=yλq−μλ1−qμ∑j=0n/NnNjSj,qyj×HP+1,Qm1,n1+1ρyq−μk;q|a1,α1,…,aP,αP,η−j−λ,kη−λ−j+μ,k,b1,β1,…,bQ,βQ.Proof.
We can prove above theorem similar to the proof of Theorems2 and 3.


## 3. Applications of the Main Results

In this section, we shall discuss some special cases of the main result, by assigning appropriate values to the parameters involved in the main results.If we use the relation between basic analogue of Weyl fractionalq-integral operator and generalized Weyl fractional q-integral operator for particular fx=xλ as(34)Kqμfy=yμq−μμ+1/2Kq−μ,μfy,then for the basic analogue of the product of the general class of polynomials and the q-analogue of the I-function, we include Weyl fractional q-integral(35)Kqμyλfn,Nx,qIPi,Qim,nρyk;q|aj,αj1,n,aji,αjin+1,Pibj,βj1,m,bj,βjm+1,Qi=yλ+μq−μλ−μμ+1/21−qμ∑j=0n/NnNjSj,q×IPi+1,Qi+1m+1,nρyq−μk;q|aj,αj1,n;aji,αjin+1,Pi,−μ−j−λ,k−λ−j,kbj,βj1,m;bj,βjm+1,Qi,and also Weyl fractional q-integral image of the q-analogue for product of general class of polynomial and q-analogue of H-function(36)Kqμyλfn,Nx,qHP,Qm1,n1ρyk;q|a1,α1,…,aA,αPb1,β1,…,bB,βQ=yλ+μq−μλ+μμ+1/21−qμ∑j=0n/NnNjSj,q×HP+1,Qm1,n1+1ρyq−μk;q|a1,α1,…,aP,αP,−μ−j−λ,k−λ−j,k,b1,β1,…,bQ,βQ.Similarly, if we use the relation between Kober fractionalq-integral and Riemann–Liouville fractional q-integral operator (cf. Yadav et al. [6]) as(37)Iqμfy=yμIq0,μfy,then we have Riemann–Liouville fractional q-integral for the basic analogue of product of general class of polynomial and basic analogue of I-function(38)Iqμyλ−1fn,Ny,qIPi,Qim,nρyk;q|aj,αj1,n,aji,αjin+1,Pibj,βj1,m,bj,βjm+1,Qi=yλ+μ−11−qμ∑j=0n/NnNjSj,q×IPi,Qi+1m,+1,nρyk;q|1−j−λ,kaj,αj1,n,aji,αjin+1,Pibj,βj1,m,bj,βjm+1,Qi1−j−μ−λ,k.Furthermore, Riemann–Liouville fractionalq-integral for the basic analogue of product of general class of polynomial and basic analogue of H-function is given by(39)Iqμyλ−1fn,Ny,qHP,Qm1,n1yk;q|a1,α1,…,aP,αPb1,β1,…,bQ,βQ=yλ+μ−11−qμ∑j=0n/NnNjSj,q×HP+1,Qm1,n1+1yk;q|1−j−λ,k,a1,α1,…,aP,αPb1,β1,…,bQ,βQ1−j−μ−λ,k.


## 4. Conclusion

We arrive at the conclusion that the results presented in this article are general and contribute to theq-calculus theory. It is also important to note that the results presented in this paper can be applied to the arrangements of specific q-differentials and q-integrals related to the q-analogues of Meijer’s G-function, Fox’s H-function, and I-function. Furthermore, by using the appropriate coefficient Sj,q the q-polynomial family fn,Ny,q yields a number of known q-polynomials as special cases. These include, among others, the q-Hermite polynomials, the q-Laguerre polynomials, the q-Jacobi polynomials, the Wall polynomials, the q-Konhauser polynomials, and several others.



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