# Some Fractional Operators with the Generalized Bessel–Maitland Function

Authors: R. S. Ali; S. Mubeen; I. Nayab; Serkan Araci; G. Rahman; K. S. Nisar
Journal: Discrete Dynamics in Nature and Society
 (2020)

---

## Abstract

In this paper, we aim to determine some results of the generalized Bessel–Maitland function in the field of fractional calculus. Here, some relations of the generalized Bessel–Maitland functions and the Mittag-Leffler functions are considered. We develop Saigo and Riemann–Liouville fractional integral operators by using the generalized Bessel–Maitland function, and results can be seen in the form of Fox–Wright functions. We establish a new operatorZν,η,ρ,γ,w,a+μ,ξ,m,σϕ and its inverse operator Dν,η,ρ,γ,w,a+μ,ξ,m,σϕ, involving the generalized Bessel–Maitland function as its kernel, and also discuss its convergence and boundedness. Moreover, the Riemann–Liouville operator and the integral transform (Laplace) of the new operator have been developed.


---

## Body

## 1. Introduction

During the last few years, many types of research studies developed the class of generalized fractional integrals containing a variety of special functions [1–5]. Watson [6] discussed applications of the Bessel function with some fields of applied sciences, biology, chemistry, physical sciences, and engineering. The generalization and extensions of the Bessel–Maitland function [7–14] dealt with special cases that gave useful results in different areas of mathematics. The recent work in the field of fractional calculus theory, differential equations of the Mittag-Leffler function, Sturm–Liouville problems in theoretical sense, Gronwall’s inequality, and exponential kernels of the differential operator [15–18] have found many applications in various subfields of mathematical analysis.The series representation of the Bessel–Maitland function [19] is defined as(1)Jβαs=∑n=0∞−snn!Γαn+β+1=ϕα,β+1;−s.The generalization of the Bessel–Maitland function introduced by Singh et al. [7] is(2)Jβ,qα,γs=∑n=0∞γqn−snΓαn+β+1n!,where α,γ,β∈ℂ, ℜα≥0,ℜβ≥−1,ℜγ≥0, and q∈0,1∪ℕ.The extended Bessel–Maitland function investigated in [20] is(3)Jβ,γ,δα,q,ps=∑n=0∞γqn−snΓαn+β+1δpn,where α,γ,β,δ∈ℂ, ℜα>0,ℜβ>−1,ℜγ>0,ℜδ≥0, p,q>0, and q<ℜα+p.The Saigo fractional integral operators are defined [21] for s>0, a,c,d∈ℂ, and ℜa>0:(4)ℱ0∗+a,c,dgs=s−a−cΓa∫0ss−τa−1R21a+c,−d;a;1−τsgτdτ,(5)ℱ0−a,c,dgs=1Γa∫s∞τ−sa−1τ−a−cR21a+c,−d;a;1−sτgτdτ.Samko et al. [22] defined the Riemann–Liouville fractional operators for ℜa>0 and n=Rea+1 as(6)ℱ0+ags=1Γa∫0ss−τa−1gτdτ,(7)D0+aϕs=ddsnℱ0+n−aϕs.The Gauss hypergeometric function defined by Saigo [21] for all a,c,d∈ℂ, a≠0, and s<1 is(8)R21b,−d;a;s=∑n=0∞bn−dnsnann!,where bn,−dn,andan are Pochhammer’s symbols.Pochhammer’s symbols defined by Petojevic [23] are(9)sn=ss+1s+2⋯s+n−1,forn≥1,1,forn=0,s≠0,where s∈ℂ and n∈ℕ, and in gamma form, they can be written as(10)sn=Γs+nΓs.The beta function is defined as given in [23], for ℜy>0andℜz>0, and also expressed in the gamma form, respectively:(11)βy,z=∫01gy−11−gz−1dg,(12)βy,z=ΓyΓzΓy+z.The gamma function is defined [23] for ℜu>0 as(13)Γu=∫0∞gu−1e−gdg.The generalized hypergeometric function is defined by Rainville [24]:(14)Rkrc1,…,ck,q1,…,qr;s=∑n=0∞c1n,…,cknq1n,…,qrnsnn!,where ci,qj∈ℂ, qj≠0,−1,…,i=−1,−2,…,k;j=−1,−2,…,r.The generalized Fox–Wright function is defined as [25](15)ψrsg=ψrsyj,pj1,rzi,qi1,sg≡∑n=0∞∏j=1rΓyj+pjn∏i=1sΓzi+qingnn!,where g∈ℂ, yj,zi∈ℂ, and pj,qi∈ℝj=1,2,…,r;i=1,2,…,s.The Gauss hypergeometric function in the gamma form can be written as(16)R21a,s;d;1=ΓdΓd−a−sΓd−aΓd−s,ℜd−a−s>0.The Laplace transform of functionfz is defined as(17)ℒft=fs=∫0∞e−stftdt.Dirichlet formula (Fubini’s theorem) [22] is given by(18)∫dcdy∫ycgy,τdτ=∫dcdτ∫dτgy,τdy.Definition 1.
The generalization of the generalized Bessel–Maitland function is defined and investigated as(19)Jν,η,ρ,γμ,ξ,m,σs=∑n=0∞ηξnγσn−snΓμn+ν+1ρmn,where μ,ν,η,ρ,γ∈ℂ, ℜμ>0, ℜν≥−1, ℜη>0, ℜρ>0, ℜγ>0, ξ,m,σ≥0, and m,ξ>ℜμ+σ.
The following notation is used in our results:(20)Qμ,νγ,σρ,m;nη,ξ=∑n=0∞ηξnγσn−1nΓμn+ν+1ρmn.Definition 2.
The extension of generalized Bessel–Maitland function (19) in multivariable function can be defined for μj,ν,ηj,ρj,γj∈ℂ, ℜμj>0, ℜν>−1, ℜηj>0, ℜρj>0, ℜγj>0, ξj,mj,σj≥0, and mj,ξj>ℜμj+σj, j=1,…,r, as(21)Jν;ηj;ρj;γjμj;ξj;mj;σjs1…sr=Jν;η1,…,ηr;ρj;γ1,…,γrμ1,…,μr;ξ1,…,ξr;m1,…,mr;σ1,…,σrs1…sr=∑n1,…,nr=0∞η1ξ1n1,…,ηrξrnrγ1σ1n1,…,γrσrnrΓν+1+∑j=1rnjμjρ1m1n1,…,ρrmrnr−s1n1…−srnr.Remark 1.
On settingj=1 in equation (21), we get generalized Bessel–Maitland function (19).Definition 3.
An integral operator which involves generalized Bessel–Maitland function (19) as its kernel is defined for μ,ν,η,w,γ,ρ∈ℂ, ℜμ>0, ℜν≥−1, ℜη>0, ℜw>0, ℜγ>0, ℜρ>0, ξ,m,σ≥0, and m,ξ>ℜμ+σ as follows:(22)Zν,η,ρ,γ,w,a+μ,ξ,m,σϕs=∫ass−τνJν,η,ρ,γμ,ξ,m,σws−τμϕτdτ.Remark 2.
If we putw=0 and replace ν by ν−1, then it will become a left-sided Riemann–Liouville fractional integral operator.
The new fractional operator (22) can be discussed to improve the results of some inequalities such as Polya–Szego inequality, Chebyshev inequality, and Hadamard inequality in the field of analysis.Definition 4.
The left inverse operator of integral operator (22), for μ,ν,η,w,γ,ρ∈ℂ, ℜμ>0, ℜν≥−1, ℜη>0, ℜw>0, ℜγ>0, ℜρ>0, ξ,m,σ≥0, m,ξ>ℜμ+σ, and n=ν as n−ν>0 is defined as follows:(23)Dν,η,ρ,γ,w,a+μ,ξ,m,σϕs=dndsnZν−n,η,ρ,γ,w,a+μ,ξ,m,σϕs=dndsn∫ass−τn−νJn−ν,η,ρ,γμ,ξ,m,σws−τμϕτdτ.Remark 3.
If we putw=0 and replace ν by ν−1, then equation (23) becomes the Riemann–Liouville fractional differential operator.Remark 4.
If we replaceσ=0, η=−η, ξ=ρ=m=1, and ν=ν−1 in equation (23), we get(24)Dν−1,−η,1,γ,w,a+μ,1,1,0ϕs=Dμ,ν,w,a+ηϕs,where the inverse operator Dμ,ν,w,a+ηϕs is described and discussed by Polito and Tomovski in [26].


## 2. Relation with the Bessel–Maitland and the Mittag-Leffler Functions

In this section, we discuss some special cases of the generalized Bessel–Maitland function and developed its relations with generalized Mittag-Leffler functions:(i)
On replacingσ=0 in equation (19), we obtain the relation(25)Jν,η,ρ,γμ,ξ,m,0s=Jν,η,ρμ,ξ,ms,whereJν,η,ρμ,ξ,mz is the generalized Bessel–Maitland function investigated in [20].(ii)
On replacingσ=0andm=ρ=1 in equation (19), we obtain the relation(26)Jν,η,1,1μ,ξ,1,0s=Jν,ημ,ξs,whereJν,ημ,ξs is the generalization of Bessel–Maitland function defined by Singh et al. [7].(iii)
On replacingσ=ξ=0 and m=ρ=1 in equation (19), we obtain the relation [19](27)Jν,η,1,γμ,0,1,0s=Jνμs.(iv)
On settingσ=0 and replacing ν by ν−1 in equations (19) and (22), then we have(28)Jν−1,η,ρ,γμ,ξ,m,0−s=ℰμ,ν,mη,ρ,ξs,(29)Zν−1,η,ρ,γ,w,a+μ,ξ,m,0ϕs=Fμ,ν,m,w,a+η,ρ,ξϕw,whereℰμ,ν,mη,ρ,ξs is the Mittag-Leffler function investigated by Salim and Faraj [27].(v)
On settingσ=0 and ρ=m=1 and replacing ν by ν−1 in equation (19), then we have(30)Jν−1,η,1,γμ,ξ,1,0−s=ℰμ,νη,ξs,Zν−1,η,1,γ,w,a+μ,ξ,1,0ϕs=ℰa+;μ,νw;η,ξs,whereℰμ,νη,ρs is the Mittag-Leffler function defined by Shukla and Prajapati [28] and ℰa+;μ,νw;η,ξs is described by Srivastava and Tomovski in [29].(vi)
On settingσ=0 and ξ=m=ρ=1 and replacing ν by ν−1 in equations (19) and (22), then we have(31)Jν−1,η,1,γμ,1,1,0−s=ℰμ,νηs,Zν−1,η,1,γ,w,a+μ,1,1,0ϕs=F∗μ,ν;η;wϕs,whereℰμ,νηs is defined by Prabhakar in [30] and described the integral(32)F∗μ,ν;η;wϕs.(vii)
On settingσ=0 and ξ=m=η=ρ=γ=1 and replacing ν by ν−1 in equation (19), then we have(33)Jν−1,1,1,γμ,1,1,0−s=ℰμ,νs,whereℰμ,νs is the Mittag-Leffler function discussed by Wiman [31].(viii)
On settingσ=0, ξ=m=η=ρ=γ=1, and ν=0 in equation (19), then we have(34)J0,1,1,γμ,1,1,0−s=ℰμs,whereℰμs is the Mittag-Leffler function introduced in [32].


## 3. Convergence and Boundedness of the New Fractional Integral Operator

In this section, we discuss the convergence and boundedness of the fractional integral operator involving the generalized Bessel–Maitland function as its kernel in the form of a theorem.Theorem 1.
Let the operatorZν,η,ρ,γ,w,a+μ,ξ,m,σϕs be bounded on La,c with μ,ν,η,w,γ∈ℂ, ℜμ>0, ℜν≥−1, ℜη>0, ℜw>0, ℜγ>0, ξ,m,σ≥0, and m,ξ>ℜμ+σ; then, the following relation holds:(35)Zν,η,ρ,γ,w,a+μ,ξ,m,σϕc≤Bϕc,where(36)B=c−aReν∑n=0∞ηξnγσnρmn−wc−aReμnΓμn+ν+1Reμn+Reν+1.Proof.
LetKn denote the nth term of (36); then,(37)Kn+1Kn=ηξn+ξηξnγσn+σγσnρmnρmn+mΓμn+ν+1Γμn+μ+ν+1×Reμn+Reν+1Reμn+1+Reν+1−1n+1−1nwc−aReμ≈ξnξσnσwc−aReμρnρn+1μnReμ,asn⟶∞.
Hence,Kn+1/Kn⟶0 as n⟶∞, and ξ,σ<ρ+Reμ which means that the right-hand side of (36) is convergent and finite under the given condition. The condition of boundedness of the integral operator Zν,η,ρ,γ,w,a+μ,ξ,m,σϕx is discussed in the space of Lebesgue measure La,c of a continuous function on a,c, where c>a:(38)La,c=gx:gc=∫acgxdx<∞.
According to equations (19) and (22), we have(39)Zν,η,ρ,γ,w,a+μ,ξ,m,σϕc=∫ac∫ass−tνJν,η,ρ,γμ,ξ,m,σws−tμϕtdtds≤∫ac∫tcs−tνJν,η,ρ,γμ,ξ,m,σws−tμdsϕtdt.
By putting the valuess−t=y⇒ds=dy, s=c⇒y=c−t, and s=t⇒y=0 in equation (39), we have(40)Zν,η,ρ,γ,w,a+μ,ξ,m,σϕc=∫ac∫0c−tyReνJν,η,ρ,γμ,ξ,m,σwyμdyϕtdt≤∫ac∫0c−ayReνJν,η,ρ,γμ,ξ,m,σwyμdyϕtdt,B=∫0c−ayReνJν,η,ρ,γμ,ξ,m,σwyμdy,B=∑n=0∞ηξnγσn−wnρmnΓμn+ν+1∫0c−ayReμn+Reνdy=∑n=0∞c−aReνηξnγσn−wc−aReμnρmnΓμn+ν+1Reμn+Reν+1.
Therefore,(41)Zν,η,ρ,γ,w,a+μ,ξ,m,σϕc≤∫acBϕtdt≤Bϕc⇒Zν,η,ρ,γ,w,a+μ,ξ,m,σϕc≤Bϕc.


## 4. The Generalized Bessel–Maitland Function with Some Fractional Integral Operators

In this section, we derive some results of Saigo fractional integral operators with the generalized Bessel-Maitland function, and these results are established in terms of the Fox–Wright function. Also, we develop the composition of Riemann–Liouville operators with the generalized Bessel–Maitland function.Theorem 2.
Leta,c,d,μ,ν,η,ρ,γ∈ℂ with ℜa>0, ρ>max0,ℜc−d, ℜμ>0, ℜν≥−1, ℜη>0, ℜρ>0, ℜγ>0, ξ,m,σ≥0, and m,ξ>ℜμ+σ; then, the following relation holds:(42)ℱ0+a,c,dτρJν,η,ρ,γμ,ξ,m,στδs=sρ−cΓρΓηΓγ×ψ54η,ξγ,σρ+1,δρ+d−c+1,δ1,1ν+1,μρ,mρ−c+1,δa+ρ+d+1,δ−sδ.Proof.
Consider the left-sided Saigo fractional integral operator (4), in which using the power function with generalized Bessel–Maitland function (19), we get(43)ℱ0+a,c,dτρJν,η,ρ,γμ,ξ,m,στδs=s−a−cΓa∫0ss−τa−1R21a+c,−d;a;1−τsτρ∑n=0∞ηξnγσn−τδnΓμn+ν+1ρmndτ.
By using equation (8) in equation (43), we have(44)ℱ0+a,c,dτρJν,η,ρ,γμ,ξ,m,στδs=s−c−1Qμ,νγ,σρ,m;nη,ξΓa∑p=0∞a+cp−dpapp!∫0s1−τsa+p−1τρ+δndτ.
By putting the valuesτ/s=u⇒dτ=sdu, τ=s⇒u=1, and τ=0⇒u=0 in equation (45), we obtain(45)ℱ0+a,c,dτρJν,η,ρ,γμ,ξ,m,στδs=sρ−cQμ,νγ,σρ,m;nη,ξsδn∑p=0∞a+cp−dpΓaapp!∫011−ua+p−1uρ+δndu.
By using equations (11) and (12) in equation (46), we get(46)ℱ0+a,c,dτρJν,η,ρ,γμ,ξ,m,στδs=sρ−cQμ,νγ,σρ,m;nη,ξ∑p=0∞a+cp−dpapp!sδnΓρ+δn+1Γa+pΓaΓρ+δn+a+p+1.
By using equations (10) and (16) in equation (47), we have(47)ℱ0+a,c,dτρJν,η,ρ,γμ,ξ,m,στδs=sρ−cQμ,νγ,σρ,m;nη,ξsδnΓρ+δn+d−c+1Γρ+δn+1Γρ+δn−c+1Γρ+δn+a+d+1.
By using equations (10) and (20) in equation (51), we get(48)ℱ0+a,c,dτρJν,η,ρ,γμ,ξ,m,στδs=∑n=0∞sρ−cΓρΓη+ξnΓηΓμn+ν+1Γρ+δn+d−c+1Γρ+δn+1Γγ+σn−sδnΓγΓρ+mnΓρ+δn−c+1Γρ+δn+a+d+1.
Hence, we attain the required result:(49)ℱ0+a,c,dτρJν,η,ρ,γμ,ξ,m,στδs=sρ−cΓρΓηΓγ×ψ54η,ξγ,σρ+1,δρ+d−c+1,δ1,1ν+1,μρ,mρ−c+1,δa+ρ+d+1,δ−sδ.Theorem 3.
Leta,c,d,μ,ν,η,ρ,γ∈ℂ with ℜa>0, ρ>max0,ℜc−d, ℜμ>0, ℜν≥−1, ℜη>0, ℜρ>0, ℜγ>0, ξ,m,σ≥0, and m,ξ>ℜμ+σ; then, the following relation holds:(50)ℱ0−a,c,dτ−ρJν,η,ρ,γμ,ξ,m,στ−δs=s−c−ρΓρΓηΓγ×ψ54η,ξγ,σρ+d,δρ+c,δ1,1ν+1,μρ,mρ,δa+ρ+d+c,δ−s−δ.Proof.
Consider the right-sided Sagio fractional integral operator (5), in which using the power function with generalized Bessel–Maitland function (19), we get(51)ℱ0−a,c,dτ−ρJν,η,ρ,γμ,ξ,m,στ−δs=1Γa∫s∞τ−sa−1τa+cR21a+c,−d;a;1−sτ∑n=0∞τ−ρηξnγσn−τ−δnΓμn+ν+1ρmndτ.
By using equation (8) in equation (52), we have(52)ℱ0−a,c,dτ−ρJν,η,ρ,γμ,ξ,m,στ−δs=Qμ,νγ,σρ,m;nη,ξ∫s∞1−sτa−1∑p=0∞a+cp−dp1−s/τpτc+1Γaapp!τρτδndτ=Qμ,νγ,σρ,m;nη,ξ∑p=0∞a+cp−dpΓaapp!∫s∞1−s/τa+p−1τρ+δnτ−c−1dτ.
By putting the valuess/τ=u⇒dτ=−s/u2du, τ=s⇒u=1, and τ=∞⇒u=0 in equation (53), we obtain(53)ℱ0−a,c,dτ−ρJν,η,ρ,γμ,ξ,m,στ−δs=Qρ,m;nη,ξμ,νγ,σΓa∑p=0∞a+cp−dpapp!∫101−ua+p−1s/uρ+δn+c+1−su2du=Qμ,νγ,σρ,m;nη,ξs−c∑p=0∞a+cp−dpsρΓaapp!sδn∫011−ua+p−1u−ρ−δn−c+1du.
By using equations (11) and (12) in equation (54), we have(54)ℱ0−a,c,dτ−ρJν,η,ρ,γμ,ξ,m,στ−δs=Qμ,νγ,σρ,m;nη,ξs−δns−ρ−c∑p=0∞a+cp−dpΓρ+δn+cΓa+pΓaapp!Γρ+δn+a+p+c.
Now, by using equations (10) and (16) in equation (56), we get(55)ℱ0−a,c,dτ−ρJν,η,ρ,γμ,ξ,m,στ−δs=Qμ,νγ,σρ,m;nη,ξs−δnΓρ+δn+dΓρ+δn+csρ+cΓρ+δnΓρ+δn+a+d+c.
By using equations (10) and (20) in (57), we have the result:(56)ℱ0−a,c,dτ−ρJν,η,ρ,γμ,ξ,m,στ−δs=∑n=0∞s−ρ−cΓη+ξnΓμn+ν+1ΓρΓγ+σnΓρ+δn+dΓρ+δn+c−s−δnΓηΓγΓρ+mnΓρ+δnΓρ+δn+a+d+c=s−c−ρΓρΓηΓγψ54η,ξγ,σρ+d,δρ+c,δ1,1ν+1,μρ,mρ,δa+ρ+d+c,δ−s−δ.Theorem 4.
Letℜλ>0, μ,ν,η,ρ,γ,w∈ℂ, ℜμ>0, ℜν≥−1, ℜη>0, ℜρ>0, ℜγ>0, ξ,m,σ≥0, and m,ξ>ℜμ+σ; then, the following relation holds:(57)ℱa+λJν,η,ρ,γμ,ξ,m,σwτ−aμτ−a−νs−a=Jν+λ,η,ρ,γμ,ξ,m,σws−aμs−a−λ−ν.Proof.
Consider the left-sided Riemann–Liouville fractional integral operator (6), in which using the power function with generalized Bessel–Maitland function (19), we get(58)ℱa+λτ−aνJν,η,ρ,γμ,ξ,m,σwτ−aμs−a=Qμ,νγ,σρ,m;nη,ξ∫asτ−aνwτ−aμnΓλs−τ−λ+1dτ=Qμ,νγ,σρ,m;nη,ξΓλw−n∫ass−τλ−1τ−aν+μndτ.
By putting the valuesτ−a/s−a=u⇒dτ=s−adu, τ=s⇒u=1, and τ=a⇒u=0 in equation (59), we obtain(59)ℱa+λτ−aνJν,η,ρ,γμ,ξ,m,σwτ−aμs−a=Qμ,νγ,σρ,m;nη,ξwnΓλ∫01s−s−au−aλ−1s−au−μn−νs−a−1du=s−aλ+νw−nΓλQμ,νγ,σρ,m;nη,ξs−aμn∫011−uλ−1uμn+νdu.
By using equations (11) and (12) in equation (60), we have(60)ℱa+λτ−aνJν,η,ρ,γμ,ξ,m,σwτ−aμs−a=s−aλ+νQμ,νγ,σρ,m;nη,ξwns−aμnΓμn+ν+1Γλ+μn+ν+1.
Now, by using equation (20) in equation (61), then the required result is obtained:(61)ℱa+λτ−aνJν,η,ρ,γμ,ξ,m,σwτ−aμs−a=s−aλ+ν∑n=0∞ηξnγσn−ws−aμnΓλ+μn+ν+1ρmn=s−aλ+νJν+λ,η,ρ,γμ,ξ,m,σws−aμ.


## 5. Riemann–Liouville Fractional Operators and Laplace Transform of the New Operator

In this section, we discuss the Riemann–Liouville fractional integral and differential operators with the fractional integral operator. Also, we developed a result which deals with the Laplace transform of the new fractional integral operator.Theorem 5.
Letw,λ,μ,ν,η,w,γ∈ℂ, ℜα>0, ℜβ>0, ℜλ>0, ℜμ>0, ℜν≥−1, ℜη>0, ℜw>0, ℜγ>0, ξ,m,σ≥0, ℜρ>0, and m,ξ>ℜμ+σ; then, the following relation holds:(62)ℱ0+λZν,η,ρ,γ,w,0+μ,ξ,m,σϕs=Zν+λ,η,ρ,γ,w,0+μ,ξ,m,σϕs.Proof.
Consider the left-sided Riemann–Liouville integral operator (6) involving new fractional integral operator (22) as(63)ℱ0+λZν,η,ρ,γ,w,0+μ,ξ,m,σϕs=1Γλ∫0ss−uλ−1∫0uu−τνJν,η,ρ,γμ,ξ,m,σwu−τμϕτdτdu.
By using equation (18) in equation (63), we have(64)ℱ0+λZν,η,ρ,γ,w,0+μ,ξ,m,σϕs=1Γλ∫0s∫τss−uλ−1u−τνJν,η,ρ,γμ,ξ,m,σwu−τμduϕτdτ.
By putting the valuest=u−τ⇒dt=du, u=s⇒t=s−τ, and u=τ⇒t=0 in equation (64), we get(65)ℱ0+λZν,η,ρ,γ,w,0+μ,ξ,m,σϕs=1Γλ∫0s∫0s−τtνJν,η,ρ,γμ,ξ,m,σwtμs−τ−t1−λdtϕτdτ=∫0s1Γλ∫0s−τtνJν,η,ρ,γμ,ξ,m,σwtμs−τ−t1−λdtϕτdτ.
By using equation (6) in equation (65), we have(66)ℱ0+λZν,η,ρ,γ,w,0+μ,ξ,m,σϕs=∫0sℱ0+λtνJν,η,ρ,γμ,ξ,m,σwtμs−τϕτdτ.
By using equation (58) in equation (66), we obtain(67)ℱ0+λZν,η,ρ,γ,w,0+μ,ξ,m,σϕs=∫0sJν+λ,η,ρ,γμ,ξ,m,σws−τμs−τ−ν−λϕτdτ=Zν+λ,η,ρ,γ,w,0+μ,ξ,m,σϕs.Theorem 6.
Letλ,μ,ν,η,w,γ∈ℂ, ℜα>0,ℜβ>0,ℜλ>0, ℜμ>0, ℜν≥−1, ℜη>0, ℜw>0, ℜγ>0, ξ,m,σ≥0, ℜρ>0, and m,ξ>ℜμ+σ; then, the following relation holds:(68)D0+λZν,η,ρ,γ,w,0+μ,ξ,m,σϕs=Zν−λ,η,ρ,γ,w,0+μ,ξ,m,σϕs.Proof.
Consider the left-sided Riemann–Liouville differential operator (7) involving new fractional integral operator (22); then,(69)D0+λZν,η,ρ,γ,w,0+μ,ξ,m,σϕs=1Γm−λddsm∫0ss−um−λ−1∫0uJν,η,ρ,γμ,ξ,m,σwu−τμu−τ−νϕτdτdu.
By using equation (18) in equation (69), we have(70)D0+λZν,η,ρ,γ,w,0+μ,ξ,m,σϕs=1Γm−λddsm∫0s∫τsτ−uνJν,η,ρ,γμ,ξ,m,σwu−τμs−uλ−m+1duϕτdτ.
By putting the valuest=u−τ⇒dt=du, u=s⇒t=s−τ, and u=τ⇒t=0 in equation (70), we get(71)D0+λZν,η,ρ,γ,w,0+μ,ξ,m,σϕs=1Γm−λddsm∫0s∫0s−τtνJν,η,ρ,γμ,ξ,m,σwtμs−τ−t−m+λ+1dtϕτdτ=∫0sddxm1Γm−λ∫0s−τtνJν,η,ρ,γμ,ξ,m,σwtμs−τ−t−m+λ+1dtϕτdτ.
Now, by using equation (6) in equation (71), we have(72)D0+λZν,η,ρ,γ,w,0+μ,ξ,m,σϕs=∫0sddsmℱ0+m−λtνJν,η,ρ,γμ,ξ,m,σwtμs−τϕτdτ.
By using equation (58) in equation (72), we obtain(73)D0+λZν,η,ρ,γ,w,0+μ,ξ,m,σϕs=∫0sddsmJm+ν−λ,η,ρ,γμ,ξ,m,σws−τμs−τλ−m−νϕτdτ.
By using equation (19) in equation (73) and then taking one-time derivative, we have(74)D0+λZν,η,ρ,γ,w,0+μ,ξ,m,σϕs=∑n=0∞−wnμn+m+ν−λΓμn+m+ν−λ+1×ηξnγσnwmnddsm−1∫0ss−τ−λ+μn+m+ν−1ϕτdτ=ddsm−1∫0sJν+m−λ−1,η,ρ,γμ,ξ,m,σws−τμs−τ−m−ν+λ+1ϕτdτ.
Now, taking the (m−1) derivative of equation (74), we get(75)D0+λZν,η,ρ,γ,w,0+μ,ξ,m,σϕs=∫0sJν−λ,η,ρ,γμ,ξ,m,σws−τμs−τλ−νϕτdτ=Zν−λ,η,ρ,γ,w,0+μ,ξ,m,σϕs.Theorem 7.
Letw,ν´,η´,γ´,μ,ν,η,γ∈ℂ∈ℂ, ℜη´>0, ℜγ´>0, ℜμ>0, ℜν≥−1, ℜη>0, ℜw>0, and ℜγ>0; then, the following relation holds:(76)Zν,η,1,γ,w,0+μ,1,1,0Zν´,η´,1,γ´,w,0+μ,1,1,0ϕs=Zν+ν´+1,η+η´,1,γ,w,0+μ,1,1,0ϕs.Proof.
Consider the new fractional integral operator (22):(77)Zν,η,1,γ,w,0+μ,1,1,0Zν´,η´,1,γ´,w,0+μ,1,1,0ϕs=∫asJν,η,1,γμ,1,1,0ws−uμs−u−νdu∫auJν´,η´,1,γ´μ,1,1,0wu−tμu−t−ν´ϕtdt=∫asϕtdt∫tsJν,η,1,γμ,1,1,0ws−uμJν´,η´,1,γ´μ,1,1,0wu−tμs−u−νu−t−ν´du.
By reversing the order of integration and puttingτ=s−u/s−t, the inner integral is(78)Zν,η,1,γ,w,0+μ,1,1,0Zν´,η´,1,γ´,w,0+μ,1,1,0ϕs=∫01Jν,η,1,γμ,1,1,0ws−tτμJν,η´,1,γ´μ,1,1,σ´ws−s−tτ−tμs−tτ−νs−s−tτ−t−ν´s−t−1dτ=∑p=0∞ηp−wps−tμpΓμp+ν+1p!∑n=0∞η´n−wns−tμnΓμn+ν´+1n!×s−tν+ν´+1∫01τν+μp1−τν´+μndτ=s−tν+ν´+1∑n=0∞∑p=0∞ηp−wp+ns−tμp+nη´nΓμp+μn+ν+ν´+1+1p!n!=s−tν+ν´+1∑p=0∞−wps−tμpΓμp+ν+ν´+1+1∑n=0∞ηp−nη´np−n!n!.
This implies that(79)Zν,η,1,γ,w,0+μ,1,1,0Zν´,η´,1,γ´,w,0+μ,1,1,0ϕs=∫ass−tν+ν´+1Jν+ν´+1,η+η´,1,γμ,1,1,0ws−tμϕtdt=Zν+ν´+1,η+η´,1,γ,w,0+μ,1,1,0ϕs.Theorem 8.
Letμ,ν,η,ρ,γ,w∈ℂ, ℜμ>0, ℜν≥−1, ℜη>0, ℜρ>0, ℜγ>0, ξ,m,σ≥0, and m,ξ>ℜμ+σ; then, the following relation holds:(80)ℒZν,η,ρ,γ,w,0+μ,ξ,m,σϕ=s−v−1ΓρΓηΓγψ42η,ξγ,σν,μ1,1ν+1,μρ,m−wsμ.Proof.
Consider fractional integral operator (22):(81)ℒZν,η,ρ,γ,w,0+μ,ξ,m,σϕ=∫0∞e−st∫0tt−uν∑n=0∞ηξnγσn−wnt−uμnΓμn+ν+1ρmnϕududt.
Now, after changing the order of integration, we obtain(82)ℒZν,η,ρ,γ,w,0+μ,ξ,m,σϕ=∫0∞∫u∞t−uνest∑n=0∞ηξnγσn−wnt−uμnΓμn+ν+1ρmndtϕudu=∑n=0∞ηξnγσn−wnΓμn+ν+1ρmn∫0∞∫u∞t−uμn+νestdtϕudu.
By puttingt−u=τ, then(83)ℒZν,η,ρ,γ,w,0+μ,ξ,m,σϕ=∑n=0∞ηξnγσn−wnΓμn+ν+1ρmn∫0∞ϕuesu∫0∞e−sττμn+νdτdu=∑n=0∞ηξnγσn−wnΓμn+ν+1ρmnΓμn+νsμn+ν+1ϕs=s−v−1ΓρΓηΓγ∑n=0∞Γη+ξnΓγ+σn−ws−μnΓμn+νΓμn+ν+1Γρ+mn=s−v−1ΓρΓηΓγψ42η,ξγ,σν,μ1,1ν+1,μρ,m−wsμ.Theorem 9.
Letλ,δ,χ,μ,ν,η,ρ,γ,w∈ℂ, ℜμ>0, ℜν≥−1, ℜη>0, ℜρ>0, ℜγ>0, ξ,m,σ≥0, and m,ξ>ℜμ+σ; then, the following relation holds:(84)Zν,η,ρ,γ,w,0+μ,ξ,m,σtλ+δ/χ−1s=sν+λ+δ/χΓρΓλ+δ/χΓηΓγψ32η,ξγ,σ1,1ν+λ+δχ+1,μρ,m−wsμ.Proof.
Consider the new fractional integral operator (22):(85)Zν,η,ρ,γ,w,0+μ,ξ,m,σtλ+δ/χ−1s=∫0ss−τνJν,η,ρ,γμ,ξ,m,σws−τμτλ+δ/χ−1dτ=∑n=0∞ηξnγσn−wnsμn+νΓμn+ν+1ρmn∫0s1−τsμn+ντλ+δ/χ−1dτ.
By puttingτ/s=u, then we have(86)Zν,η,ρ,γ,w,0+μ,ξ,m,σtλ+δ/χ−1s=∑n=0∞ηξnγσn−wnsμn+νΓμn+ν+1ρmn∫0s1−uμn+νsuλ+δ/χ−1sdu=∑n=0∞ηξnγσn−wnsμn+ν+λ+δ/χΓμn+ν+1ρmnΓμn+ν+1Γλ+δ/χΓμn+ν+λ+δ/χ+1=sν+λ+δ/χΓρΓηΓγ∑n=0∞Γη+ξnΓγ+σn−wsμnΓλ+δ/χΓμn+ν+λ+δ/χ+1Γρ+mn=sν+λ+δ/χΓρΓλ+δ/χΓηΓγψ32η,ξγ,σ1,1ν+λ+δχ+1,μρ,m−wsμ.


## 6. Inverse Operator with Some Special Functions

In this section, we discuss some applications of the inverse fractional operator. We derive some results of the inverse fractional operator with the Mittag-Leffler function and Bessel–Maitland function, and results can be seen in the form of Wright functions.Theorem 10.
Considerδ,α,β,μ,ν,η,ρ,γ,λ,∈ℂ with minℜδ,ℜα,ℜβ>0, p,q>0, ℜμ>0, ℜν≥−1, ℜη>0, ℜρ>0, ℜγ>0, ξ,m,σ≥0, and m,ξ>ℜμ+σ; then, the following relation holds:(87)Dν,η,ρ,γ,w,a+μ,ξ,m,στ−aρ−1Eα,β,pγ,δ,qτ−aλs=ΓρΓδΓγΓηΓγ`∑m=0∞Γγ`+qmΓαm+β×s−aλm+ρ−νψ32η,ξγ,σ1,1ρ,mλm+ρ−ν+1,μ−ws−aμ.Proof.
Consider inverse fractional integral operator (23) with Mittag-Leffler function (28); then, the following results hold:(88)Dν,η,ρ,γ,w,a+μ,ξ,m,στ−aρ−1Eα,β,pγ,δ,qτ−aλs=ddsp∫ass−τp−ν∑n=0∞ηξnγσn−ws−τμnρmnΓμn+p−ν+1τ−aρ−1∑m=0∞γ`qmτ−aλmΓαm+βδpmdτ=dpdspQμ,p−νγ,σρ,m;nη,ξwn∑m=0∞γ`qmΓαm+βδpm∫ass−τμn+p−ντ−aρ+λm−1dτ.
Substitutingu=s−τ/s−a in equation (88), we obtain(89)Dν,η,ρ,γ,w,a+μ,ξ,m,στ−aρ−1Eα,β,pγ,δ,qτ−aλs=dpdspQμ,p−νγ,σρ,m;nη,ξwn∑m=0∞γ`qms−ap−ν+μn+λm+ρΓαm+βδpm∫011−uμn+p−νuρ+λm−1du.
By using equations (11) and (12), we get(90)Dν,η,ρ,γ,w,a+μ,ξ,m,στ−aρ−1Eα,β,pγ,δ,qτ−aλs=dpdspQμ,p−νγ,σρ,m;nη,ξwn∑m=0∞γ`qms−ap−ν+μn+λm+ρΓαm+βδpmΓμn+p−ν+1Γρ+λmΓμn+p−ν+ρ+λm+1.
Now, back-substitutingQμ,p−νγ,σρ,m;nη,ξ in equation (90), we have(91)Dν,η,ρ,γ,w,a+μ,ξ,m,στ−aρ−1Eα,β,pγ,δ,qτ−aλs=∑m,n=0∞ηξnγσndp/dsp−wns−ap−ν+μn+λm+ρρmnΓμn+ρ+λm+p−ν+1γ`qmΓρ+λmΓαm+βδpm=∑m,n=0∞ηξnγσns−a−ν+μn+λm+ρρmnΓμn+ρ+λm−ν+1γ`qmΓρ+λmΓαm+βδpm=ΓρΓδΓγΓηΓγ`∑m=0∞Γγ`+qms−aλm+ρ−νΓαm+β×ψ32η,ξγ,σ1,1ρ,mλm+ρ−ν+1,μ−ws−aμ.Corollary 1.
On settingν=−ν in Theorem 10, we obtain the result in the form of the differential operator:(92)Zν,η,ρ,γ,w,a+μ,ξ,m,στ−aρ−1Eα,β,pγ,δ,qτ−aλs=ΓρΓδΓγΓηΓγ`∑m=0∞Γγ`+qmΓαm+β×s−aλm+ρ+νψ32η,ξγ,σ1,1ρ,mλm+ρ+ν+1,μ−ws−aμ.Theorem 11.
Considerα,β,μ,ν,η,ρ,γ,γ´,λ,∈ℂ with ℜγ´>0,ℜα>0,ℜβ>−1, q>0, ℜμ>0, ℜν≥−1, ℜη>0, ℜρ>0, ℜγ>0, ξ,m,σ≥0, and m,ξ>ℜμ+σ; then, the following relation holds:(93)Dν,η,ρ,γ,w,0+μ,ξ,m,στ−1JβατJβ,qα,γ´τ−λs=ΓρΓη−1ΓγΓγ´∑m,n=0∞Γη+ξnΓγ+σnΓρ+mn×−wsμn−sms−νΓαm+β+1m!ψ22γ´,qm,−λβ+1,αμn+m−ν+1,−λ−s−λ.Proof.
Consider inverse fractional integral operator (23) with the product of two Bessel–Maitland functions (1) and (2); then, the following results hold:(94)Dν,η,ρ,γ,w,0+μ,ξ,m,στ−1JβατJβ,qα,γ´τ−λs=dpdsp∫0ss−τp−ν∑n=0∞ηξnγσn−ws−τμnρmnΓμn+p−ν+1×∑m=0∞τ−1−τmm!Γαm+β+1∑w=0∞γ´qw−1wτ−λww!Γαw+β+1dτ=dpdspQμ,p−νγ,σρ,m;nη,ξ∑m,w=0∞−1m+wsp−ν+μnm!Γαm+β+1×γ´qwwnw!Γαw+β+1∫0s1−τsp−ν+μnτ−1+m−λwdτ.
By substitutingu=τ/s in equation (94), we have(95)Dν,η,ρ,γ,w,0+μ,ξ,m,στ−1JβατJβ,qα,γ´τ−λs=dpdspQμ,p−νγ,σρ,m;nη,ξ∑m,w=0∞wn−1m+wsμnm!Γαm+β+1γ´qwsm−λw+p−νw!Γαw+β+1∫011−up−ν+μnu−1+m−λwdu.
By using beta-gamma relations (11) and (12), we obtain(96)Dν,η,ρ,γ,w,0+μ,ξ,m,στ−1JβατJβ,qα,γ´τ−λs=Qμ,p−νγ,σρ,m;nη,ξ∑m,w=0∞wn−1m+wsμnm!Γαm+β+1γ´qwsm−λw+p−νw!Γαw+β+1Γp−ν+μn+1Γm−λwΓp−ν+μn+m−λw+1.
Putting the value ofQμ,p−νγ,σρ,m;nη,ξ in equation (96), we have(97)Dν,η,ρ,γ,w,0+μ,ξ,m,στ−1JβατJβ,qα,γ´τ−λs=∑m,n,w=0∞ηξnγσn−wndp/dspsp−ν+μn+λm+ρρmnΓμn+ρ+λm+p−ν+1wn−1m+wm!Γαm+β+1γ´qwΓm−λww!Γαw+β+1=∑m,n,w=0∞ηξnγσn−wns−ν+μn+λm+ρρmnΓμn+ρ+λm−ν+1wn−1m+wm!Γαm+β+1γ´qwΓm−λww!Γαw+β+1=ΓρΓη−1ΓγΓγ´∑m,n=0∞Γη+ξnΓγ+σnΓρ+mn−wsμn−sms−νΓαm+β+1m!×ψ22γ´,qm,−λβ+1,αμn+m−ν+1,−λ−s−λ.Corollary 2.
If we replaceν=−ν in Theorem 11, we have the result in the sense of the left inverse operator:(98)Zν,η,ρ,γ,w,0+μ,ξ,m,στ−1JβατJβ,qα,γ´τ−λs=ΓρΓη−1ΓγΓγ´∑m,n=0∞Γη+ξnΓγ+σnΓρ+mn×−wsμn−smsνΓαm+β+1m!ψ22γ´,qm,−λβ+1,αμn+m+ν+1,−λ−s−λ.Theorem 12.
Letα,β,μ,ν,η,ρ,γ∈ℂ, ℜμ>0, ℜν≥−1, ℜη>0, ℜρ>0, ℜγ>0, ξ,m,σ≥0, and m,ξ>ℜμ+σ; then, there exists the following relation:(99)Dν,η,ρ,γ,w,0+μ,ξ,m,στα/β−1R21αβ+μ,−η:αβ:τs=∑m,n=0∞α/β+μm−ηmm!−ν+μn+α/β+1m×Γη+ξnΓγ+σns−ν+m+α/βΓ−ν+μn+α/β+1Γρ+mn−wsμnΓρΓηΓγ.Proof.
Consider fractional integral operator (22) with Gauss hypergeometric function (8); then, the following results hold:(100)Dν,η,ρ,γ,w,0+μ,ξ,m,στα/βτR21αβ+μ,−η;αβ;τs=dpdsp∫0ss−τp−ν∑n=0∞ηξnγσn−ws−τμnρmnΓμn+p−ν+1τα/β−1∑m=0∞α/β+μm−ηmα/βmm!τmdτ=dpdspQμ,p−νγ,σρ,m;nη,ξwn∑m=0∞α/β+μm−ηmsp−ν+μnα/βmm!∫0s1−τsp−ν+μnτm+α/β−1dτ.
Puttingu=τ/s in equation (100), we get(101)Dν,η,ρ,γ,w,0+μ,ξ,m,στα/βτR21αβ+μ,−η;αβ;τs=dpdspQμ,p−νγ,σρ,m;nη,ξwn∑m=0∞α/β+μm−ηmsp−ν+μnα/βmm!s−m−α/β∫011−up−ν+μnum+α/β−1du.
Using equations (11) and (12), we have(102)Dν,η,ρ,γ,w,0+μ,ξ,m,στα/βτR21αβ+μ,−η;αβ;τs=dpdspQμ,p−νγ,σρ,m;nη,ξwn∑m=0∞α/β+μm−ηmsp−ν+μnα/βmm!s−m−α/βΓp−ν+μn+1Γm+α/βΓp−ν+μn+m+α/β+1.
By substitutingQμ,p−νγ,σρ,m;nη,ξ in equation (102), we get the required result:(103)Dν,η,ρ,γ,w,0+μ,ξ,m,στα/β−1R21αβ+μ,−η;αβ;τs=∑n,m=0∞ηξnγσn−wnΓm+α/β−ηmρmnΓp−ν+μn+m+α/β+1α/β+μmα/βmm!dpdspsp−ν+μn+m+α/β=∑m,n=0∞α/β+μm−ηmm!−ν+μn+α/β+1m−wsμnΓρΓηΓγΓγ+σnΓρ+mnΓη+ξns−ν+m+α/βΓ−ν+μn+α/β+1.Corollary 3.
By puttingν=−ν in Theorem 12, we get the following form:(104)Zν,η,ρ,γ,w,0+μ,ξ,m,στα/β−1R21αβ+μ,−η;αβ;τs=∑m,n=0∞α/β+μm−ηmm!ν+μn+α/β+1m×Γη+ξnΓγ+σnsν+m+α/βΓν+μn+α/β+1Γρ+mn−wsμnΓρΓηΓγ.


## 7. Conclusion

In this paper, we discussed some relations of generalized Bessel–Maitland functions and the Mittag-Leffler functions and also developed Saigo and Riemann–Liouville fractional integral operators with the generalized Bessel–Maitland function, and results can be seen in the form of Fox–Wright functions. Also, we established a new operatorZν,η,ρ,γ,w,a+μ,ξ,m,σϕ and also discussed its convergence and boundedness. Moreover, we discussed the Riemann–Liouville fractional operator and the integral transform (Laplace) of the new operator and also developed some important applications of the left inverse operator.



---

*Source: 1378457-2020-07-24.xml*