# Estimates of Classes of Generalized Special Functions and Their Application in the Fractionalk,s-Calculus Theory

Authors: S. Chandak; D. L. Suthar; S. AL-Omari; S. Gulyaz-Ozyurt
Journal: Journal of Function Spaces
 (2021)

---

## Abstract

In this article, we aim to develop newk,s-fractional integral and differential operators containing S-functions as kernels in a form of generalized k-Mittag-Leffer functions. We also set up various properties of such operators. Furthermore, we consider a variety of implications of the major outcomes that will be very useful in the implementation of scientific, engineering, and technical problems.


---

## Body

## 1. Introduction and Preliminaries

More focus has been given in recent years to the development of fractional calculus applications. The fractional calculus is very important in the development of integration and differentiation with the fractional calculus powers of real numbers or complex numbers (for example, integral and differential operators). The properties and application of the fractional calculus operator are described by [1, 2]. For more modern fractional calculus developments, the reader can refer to [3–7]. Some new results for ψ-Hilfer fractional pantograph-type differential equation depending on ψ-Riemann–Liouville integral are studied by Foukrach et al. [8]. In the frame of fractional derivatives, Alqahtani et al. [9] studied nonlinear F-contractions on b-metric spaces and differential equations with Mittag-Leffler kernel. Many scholars have computed numerous fractional integral inequalities containing the various fractional integration and differentiation operators over the past few years (see [10, 11]). The k symbols are well known from a number of sources related to the measurement of finite differences (see [12, 13]). In the literature, k-fractional integral operators have recently been considered by different scholars. For this function, we begin the literature with the following properties. The Pochhammer k-symbols and k-gamma function were introduced by Diaz and Pariguan (see [14])
(1)ζn,k=1,n=0,ζ∈ℂ,ζζ+k⋯ζ+n−1k,n∈ℕ,ζ∈ℂ,k>0,Γkρ=∫0∞zρ−1e−zk/kdz,ρ∈ℂ,Rρ>0.In the same paper, they spell out the relations(2)Γkρ+k=ρΓkρ,(3)Γkρ=kρ/k−1Γρk.Thek-fractional integral is develop by [15] as
(4)Ikςfx=1kΓkς∫0xx−zς/k−1fzdz.When we choosek=1, then Ikςfx shows the result of the Riemann-Liouville (R-L) fractional integration formula.We have(5)Iςfx=1Γς∫0xx−zς−1fzdz.Formulas fork-fractional integral are developed by [15] as
(6)Ikςxν/k−1=ΓkνΓkρ+νxρ/k+ν/k−1,Ikςx−uν/k−1=ΓkνΓkρ+νx−uρ/k+ν/k−1.The R-Lk,s-fractional integral of order ς>0 was elucidated by [16].
(7)ksIaςfx=s+11−ς/kkΓkς∫axxs+1−zs+1ς/k−1zsfzdz,where x∈a,b,k>0 and s∈R\−1. In the same paper, they defined the following result:
(8)ksIaςzs+1−as+1θ/k−1=Γkθs+1ς/k−1Γkθ+ςxs+1−as+1θ+ς/k−1.In recent years, the applications of the fractional calculus are given by the researchers (see [17, 18]). By using generalized k-fractional integrals, Gruss-type integral inequalities for generalized R-L k-fractional integrals, and k,s-R-L fractional integral inequalities for continuous random variables, analytical properties of k,s-Riemann-Liouville fractional integral several researchers have also provided such results including Hermite-Hadamard-type inequalities by using the definition of k,s-fractional integrals [19–23].TheS-function is given by [24]
(9)Sl,mς,ν,σ,ξ,kp1,⋯,pl;q1,⋯,qm;x=∑n=0∞p1n⋯plnσnξ,kxnq1n⋯qmnΓknς+νn!,k∈R;ς,ν,σ,ξ∈ℂ;Rς>0,pii=1,2,⋯,l,qjj=1,2,⋯,m,Rς>kRξ, and l<m+1.The Pochhammer symbolζn,ζ∈ℂ,n∈ℕ defined in terms of gamma function as (also see [7])
(10)ζn=1,n=0,ζ∈ℂ,ζζ+1⋯ζ+n−1,n∈ℕ,ζ∈ℂ.For some more result onS-function, the reader can refer to [25–29].


### 1.1. Special Cases

(i)
Choosingl=0=m in equation (9), it becomes generalized k-Mittag-Leffler function, defined by Saxena et al. [30].(11)S0,0ς,ν,σ,ξ,k−;−;x=∑n=0∞σnξ,kxnΓknς+νn!=Ek,ς,νσ,ξx,whereRς/k−ξ>l−m.
(ii)
Forξ=m, equation (9) yields(12)Sl,mς,ν,σ,q,kp1,⋯,pl;q1,⋯,qm;x=∑n=0∞p1n⋯plnσnm,kxnq1n⋯qmnΓknς+νn!,whereRς>kl.
(iii)
Whenξ=1, equation (9) yields(13)Sl,mς,ν,σ,1,kp1,⋯,pl;q1,⋯,qm;x=∑n=0∞p1n⋯plnσn,kxnq1n⋯qmnΓknς+νn!(14)=Sl,mς,ν,σ,kp1,⋯,pl;q1,⋯,qm;x,where Rς>kl.
(iv)
Fork=1, equation (14) yields K-function defined by Sharma [31](15)Sl,mς,ν,σ,1,1p1,⋯,pl;q1,⋯,qm;x=∑n=0∞p1n⋯plnσnxnq1n⋯qmnσnς+νn!(16)=Kl,mς,ν,σp1,⋯,pl;q1,⋯,qm;x,where Rς>l−m.
(v)
Whenσ=1 (equation (16)), it reduces to generalized M-series defined by [32] (see also [33])(17)Sl,mς,ν,1,1,1p1,⋯,pl;q1,⋯,qm;x=∑n=0∞p1n⋯plnxnq1n⋯qmnΓnς+νn!(18)=Ml,mς,νp1,⋯,pl;q1,⋯,qm;x,where Rς>l−m−1.
(vi)
Whenp=q=0 (equation (18)), it reduces to generalized Mittag-Leffler function, defined by [34](19)S0,0ς,ν,1,1,1−;−;x=∑n=0∞xnΓnς+νn!=Eς,νx,where ς,ν∈ℂ,Rς>0,Rν>0.


## 1.1. Special Cases

(i)
Choosingl=0=m in equation (9), it becomes generalized k-Mittag-Leffler function, defined by Saxena et al. [30].(11)S0,0ς,ν,σ,ξ,k−;−;x=∑n=0∞σnξ,kxnΓknς+νn!=Ek,ς,νσ,ξx,whereRς/k−ξ>l−m.
(ii)
Forξ=m, equation (9) yields(12)Sl,mς,ν,σ,q,kp1,⋯,pl;q1,⋯,qm;x=∑n=0∞p1n⋯plnσnm,kxnq1n⋯qmnΓknς+νn!,whereRς>kl.
(iii)
Whenξ=1, equation (9) yields(13)Sl,mς,ν,σ,1,kp1,⋯,pl;q1,⋯,qm;x=∑n=0∞p1n⋯plnσn,kxnq1n⋯qmnΓknς+νn!(14)=Sl,mς,ν,σ,kp1,⋯,pl;q1,⋯,qm;x,where Rς>kl.
(iv)
Fork=1, equation (14) yields K-function defined by Sharma [31](15)Sl,mς,ν,σ,1,1p1,⋯,pl;q1,⋯,qm;x=∑n=0∞p1n⋯plnσnxnq1n⋯qmnσnς+νn!(16)=Kl,mς,ν,σp1,⋯,pl;q1,⋯,qm;x,where Rς>l−m.
(v)
Whenσ=1 (equation (16)), it reduces to generalized M-series defined by [32] (see also [33])(17)Sl,mς,ν,1,1,1p1,⋯,pl;q1,⋯,qm;x=∑n=0∞p1n⋯plnxnq1n⋯qmnΓnς+νn!(18)=Ml,mς,νp1,⋯,pl;q1,⋯,qm;x,where Rς>l−m−1.
(vi)
Whenp=q=0 (equation (18)), it reduces to generalized Mittag-Leffler function, defined by [34](19)S0,0ς,ν,1,1,1−;−;x=∑n=0∞xnΓnς+νn!=Eς,νx,where ς,ν∈ℂ,Rς>0,Rν>0.


## 2.k,s-Fractional Integrals and Differentials of S-Function

In this section, we developk,s-fractional integration and differentiation operators containing S-function as its kernel. Also, we study k,s-fractional calculus; we define integral operators in terms of k,s as follows.Definition 1.
Ifk>0 and ς,ν,σ,ξ,ζ,τ∈ℂ,Rς>0,Rν>0,Rσ>0, then
(20)ksεa+;ς,ντ;σ,ξfx=1k∫axxs+1−zs+1ν/k−1Sl,mς,ν,σ,ξ,kτxs+1−zs+1ς/kzsfzdz,where x>ρ. Substituting s=0, then (20) reduces to the operator
(21)kεa+;ς,ντ;σfx=1k∫axx−zν/k−1Sl,mς,ν,σ,ξ,kτx−zς/kfzdz.
In particular, the integral operator in (21) decreases to the well-known R-L fractional integral operator defined as τ=0 and k=1.
(22)Ia+ςfx=1ς∫axx−zν−1fzdz.
The integral operator is described asksIa+ς and ksIa−ς, and also, k,s-fractional order left side and right side fractional differential operators are described as k,s- Dρ+,kς and Dρ−,kς. Also, we used left- and right-sided R-L k,s-fractional integral operators ksIa+ς and ksIa−ς. Similarly, the left- and right-sided R-L k,s-fractional differential operators are ksDa+,kς and ksDa−ς, respectively. By using the Lebesgue measurable integral of a real or complex valued function, we can describe both R-L k,s-fractional integral operators. The Lebesgue measurable integral of a valued function that is denoted and defined as real or complex form
(23)Lρ,ν=f:∥ψ∥1=∫ρν∣ψξ∣dξ<∞;ψ∈Lρ,ν.Definition 2.
Forψx∈Lρ,ν,ς∈ℂ,Rς>0,k>0, then we define the R-L left-sided k,s-fractional integral operator of order ς as
(24)a,ksDx−ςfx=a,ksIxςfx=ksIa+ςfx=ksIa+ςfx=s+11−ς/kkΓkς∫axxs+1−zs+1ς/k−1zsfzdzx>a.
The R-L right-sidedk,s-fractional integral operator of order ς is defined as
(25)ρ,ksDν−ςfx=ρ,ksIνςfx=ksIa−ςfx=ksIa+ςfx=s+11−ς/kkΓkς∫xνxs+1−zs+1ς/k−1zsfzdzx<ν.Definition 3.
Fork>0;s∈R−1;ς∈ℂ,Rς>0 and n=Rς+1, then the R-L left- and right-sided k,s-fractional differential operators are defined as
(26)ksDa+ςfx=1xsddxnknksIa+nk−ςfx,ksDa−ςfx=−1xsddxnknksIa−nk−ςfx,
respectively. Substitutingk=1 and s=0, then the R-L left- and right-sided k,s-fractional integrals and derivatives will reduce to the well-known R-L left-side and right-side fractional integrals and derivatives; see [35, 36].Definition 4.
The generalizedk,s-fractional derivative operator is denoted by ksDa+ς,ν where ς is the order such that 0<ς<1 and 0<ν<1 is defined as
(27)ksDa+ς,νfx=ksIρ+ςk−ν1xsddxkksIa+1−νk−ςfx.
Obviously, whenν=0, then (27) reduces to the R-L k,s-fractional derivative operators ksDa+ς (24).Lemma 5.
Fork>0, the following result for k,s-fractional derivative operator Da+,kς,ν defined in (27) holds true:
(28)ksDa+,kς,νzs+1−as+1θ/k−1x=Γkθs+1−ς/kΓkθ−ςxs+1−as+1θ−ς/k−1,
withx>ρ,0<ς<1,0<ν<1, and Rθ>0.Proof.
We obtain from equation (8)
(29)ksIa+1−νk−ςzs+1−as+1θ/k−1x=Γkθs+11−νk−ς/k−1Γk1−νk−ς+θxs+1−as+11−νk−ς+θ/k−1,(30)1xsddxksIa+1−νk−ςzs+1−as+1θ/k−1x=1−νk−ς+θ−kΓkθks+11−νk−ς/k−1Γk1−νk−ς+θxs+1−as+11−νk−ς+θ/k−2,
which by applying the relation given by (2) yields
(31)ksDa+,kς,νzs+1−as+1θ/k−1x=ΓkθΓk1−νk−ς+θ−kksIa+νk−ςzs+1−as+11−νk−ς+θ/k−2x=Γkθs+1νk−ς+1−νk−ς/k−1Γk1−νk−ς+θ−k×Γk1−νk−ς+θ−kΓkνk−ς+1−νk−ς+θ−kxs+1−as+1θ−ς/k−1=Γkθs+1−ν/kΓkθ−νxs+1−as+1θ−ς/k−1,which is the desired proof.Theorem 6.
Fork>0, the following result always holds true:
(32)1xsddxαxs+1−as+1ν/k−1Sl,mς,ν,σ,ξ,kτxs+1−as+1ς/k=s+1αxs+1−as+1ν/k−α−1kαSl,mς,ν−αk,σ,ξ,kτxs+1−as+1ς/k,
wheres∈R\−1,ς,ν,σ,ξ∈ℂ,Rν>0,Rς>0,Rσ>0.Proof.
The proof is obvious by applying1/xsd/dxα where α=1,2,.Corollary 7.
Choosingl=0=m in equation (32), it becomes generalized k-Mittag-Leffler function and we get a result earlier given by Nisar et al. [37].Corollary 8.
Choosingξ=m in equation (32) and using equation (12), we get
(33)1xsddxαxs+1−as+1ν/k−1Sl,mς,ν,σ,m,kτxs+1−as+1ς/k=s+1αxs+1−as+1ν/k−α−1kαSl,mς,ν−αk,σ,m,kτxs+1−as+1ς/k.Corollary 9.
Choosingξ=1 in equation (32) and using (14) yield
(34)1xsddxαxs+1−as+1ν/k−1Sl,mς,ν,σ,1,kτxs+1−as+1ς/k=s+1αxs+1−as+1ν/k−α−1kαSl,mς,ν−αk,σ,kτxs+1−as+1ς/k.Corollary 10.
Choosingk=1 in equation (34) then S-function converted in to K-function and using (16) yield
(35)1xsddxαxs+1−as+1ν−1Sl,mς,ν,σ,1,1τxs+1−as+1ς=s+1αxs+1−as+1ν−α−1Kl,mς,ν−α,στxs+1−as+1ς.Corollary 11.
Choosingσ=1 in equation (35) and using equation (16), it becomes generalized M-series defined by Sharma and Jain [32]
(36)1xsddxmαxs+1−as+1ν−1Sl,mς,ν,1,1,1τxs+1−as+1ς=s+1αxs+1−as+1ν−α−1Ml,mς,ν−ατxs+1−as+1ς.Corollary 12.
Choosingl=0=m in equation (36), it becomes generalized Mittag-Leffler function [34]
(37)1xsddxαxs+1−as+1ν−1S0,0ς,ν,1,1,1τxs+1−as+1ς=s+1αxs+1−as+1ν−α−1Eς,ν−ατxs+1−as+1ς.Theorem 13.
Supposek>0,x>aa∈R+=0,∞ and ς,ν,σ,τ∈ℂ,Rν,Rς>0,Rσ>0,Rς>0, then
(38)ksIa+ςzs+1−as+1ν/k−1Sl,mς,ν,σ,ξ,kτzs+1−as+1ς/kx=xs+1−as+1ν+ς/k−1s+1ς/kSl,mς,ν+ς,σ,ξ,kτxs+1−as+1ς/k,(39)ksDa+ςzs+1−as+1ν/k−1Sl,mς,ν,σ,ξ,kτzs+1−as+1ς/kx=xs+1−as+1ν+ς/k−1s+1ς/kSl,mς,ν−ς,σ,ξ,kτxs+1−as+1ς/k,(40)ksDa+ς,νzs+1−as+1ν/k−1Sl,mς,ν,σ,ξ,kτzs+1−as+1ς/kx=xs+1−as+1ν+ς/k−1s+1−ς/kSl,mς,ν−ς,σ,ξ,kτxs+1−as+1ς/k.Proof.
By using equation (7), we get
(41)ksIa+ςzs+1−as+1ν/k−1Sl,mς,ν,σ,ξ,kτzs+1−as+1ς/kx=s+11−ς/kkΓkς∫axzs+1−as+1ν/k−1Sl,mς,ν,σ,ξ,kτzs+1−as+1ς/kzsxs+1−as+11−ς/kdz=s+11−ς/kkΓkς∑n=0∞p1n⋯plnσnξ,kτnq1n⋯qmnΓknς+νn!×∫axzs+1−as+1ν+ςn/k−1xs+1−as+1ς/k−1zsdz.
Substitutingzs+1=as+1+yxs+1−as+1, this implies zsdz=xs+1−as+1/s+1dy; we get
(42)ksIa+ςzs+1−as+1ν/k−1Sl,mς,ν,σ,ξ,kτzs+1−as+1ς/kx=s+11−ς/ks+1kΓkς∑n=0∞p1n⋯plnσnξ,kτnq1n⋯qmnΓknς+νn!×∫01xs+1−as+1ν+ςn+ς/k−11−yς/k−1yν+ρn/k−1dy=xs+1−as+1ν+ς/k−1s+1ς/kkΓkς∑n=0∞p1n⋯plnσnξ,kτxs+1−as+1ς/knΓknς+νΓkςq1n⋯qmnΓknς+νΓkς+ν+nςn!=xs+1−as+1ν+ς/k−1s+1ς/k∑n=0∞p1n⋯plnσnξ,kτxs+1−as+1ς/knq1n⋯qmnΓkς+ν+nςn!=xs+1−as+1ν+ς/k−1s+1ς/kSl,mς,ν+ς,σ,ξ,kτzs+1−as+1ς/k.
This completes the proof of (38).Now, to prove (39),
(43)ksDa+ςzs+1−as+1ν/k−1Sl,mς,ν,σ,ξ,kτzs+1−as+1ς/kx=1xsddxnknksIa+nk−ςzs+1−as+1ν/k−1Sl,mς,ν,σ,ξ,kτzs+1−as+1ς/kx,and using (38), this take the following form:
(44)ksDa+ςzs+1−as+1ν/k−1Sl,mς,ν,σ,ξ,kτzs+1−as+1ς/kx=kn1xsddxns+1−ς/k−nxs+1−as+1ν−ς/k+n−1Sl,mς,ν−ς+nk,σ,ξ,kτxs+1−as+1ς/k.Applying (32), we have
(45)ksDa+ςzs+1−as+1ν/k−1Sl,mς,ν,σ,ξ,kτzs+1−as+1ς/kx=s+1−ς/kxs+1−as+1ν−ς/k+n−1Sl,mς,ν−ς+nk,σ,ξ,kτxs+1−as+1ς/k.This completes the desired proof.Now, to prove (40), we have
(46)ksDa+ς,νzs+1−as+1ν/k−1Sl,mς,ν,σ,ξ,kτzs+1−as+1ς/kx=ksDa+ς,ν∑n=0∞p1n⋯plnσnξ,kτnq1n⋯qmnΓknς+νn!zs+1−as+1ςn+ν/k−1x=∑n=0∞p1n⋯plnσnξ,kτnq1n⋯qmnΓknς+νn!ksDa+ς,νzs+1−as+1ν+ςn/k−1x.Using (28), we get
(47)ksDa+ςzs+1−as+1ν/k−1Sl,mς,ν,σ,ξ,kτzs+1−as+1ς/kx=∑n=0∞p1n⋯plnσnξ,kτnq1n⋯qmnΓknς+νn!Γkν+ςns+1−ς/kΓkν+ςn−ςxs+1−as+1ν+ςn−ς/k−1=xs+1−as+1ν−ς/k−1s+1−ς/k∑n=0∞p1n⋯plnσnξ,kτxs+1−as+1ς/knq1n⋯qmnΓknς+ν−ςn!=xs+1−as+1ν−ς/k−1s+1−ς/kSl,mς,ν−ς,σ,ξ,kτxs+1−as+1ς/k,which completes the desired proof.Corollary 14.
Choosingp=q=0 in equations (38), (39), and (40), it reduces to generalized k-Mittag-Leffler function and we get a result earlier given by Nisar et al. [37].Corollary 15.
Choosingξ=q in equations (38), (39), and (40) and using equation (12), we get
(48)ksIa+ςzs+1−as+1ν/k−1Sl,mς,ν,σ,q,kτzs+1−as+1ς/kx=xs+1−as+1ν+ς/k−1s+1ς/kSl,mς,ν+ς,σ,q,kτxs+1−as+1ς/k,ksDa+ςzs+1−as+1ν/k−1Sl,mς,ν,σ,q,kτzs+1−as+1ς/kx=xs+1−as+1ν−ς/k−1s+1ς/kSl,mς,ν−ς,σ,q,kτxs+1−as+1ς/k,ksDa+ς,νzs+1−as+1ν/k−1Sl,mς,ν,σ,q,kτzs+1−as+1ς/kx=xs+1−as+1ν−ς/k−1s+1−ς/kSl,mς,ν−ς,σ,q,kτxs+1−as+1ς/k.Corollary 16.
Choosingξ=1 in equations (38), (39), and (40) and using (14) yield
(49)ksIa+ςzs+1−as+1ν/k−1Sl,mς,ν,σ,1,kτzs+1−as+1ς/kx=xs+1−as+1ν+ς/k−1s+1ς/kSl,mς,ν+ς,σ,kτxs+1−as+1ς/k,(50)ksDa+ςzs+1−as+1ν/k−1Sl,mς,ν,σ,1,kτzs+1−as+1ς/kx=xs+1−as+1ν−ς/k−1s+1ς/kSl,mς,ν−ς,σ,kτxs+1−as+1ς/k,(51)ksDa+ς,νzs+1−as+1ν/k−1Sl,mς,ν,σ,1,kτzs+1−as+1ς/kx=xs+1−as+1ν−ς/k−1s+1−ς/kSl,mς,ν−ς,σ,kτxs+1−as+1ς/k.Corollary 17.
Choosingk=1 in equations (49), (50), and (51) then S-function reduces to K-function, and using (16), it yields
(52)sIa+ςzs+1−as+1ν−1Sl,mς,ν,σ,1,1τzs+1−as+1ςx=xs+1−as+1ν+ς−1s+1ςKl,mς,ν+ς,στxs+1−as+1ς,(53)sDa+ςzs+1−as+1ν−1Sl,mς,ν,σ,1,1τzs+1−as+1ςx=xs+1−as+1ν−ς−1s+1ςKl,mς,ν−ς,στxs+1−as+1ς,(54)sDa+ς,νzs+1−as+1ν−1Sl,mς,ν,σ,1,1τzs+1−as+1ςx=xs+1−as+1ν−ς−1s+1−ςKl,mς,ν−ς,στxs+1−as+1ς.Corollary 18.
Choosingσ=1 in equations (52), (53), and (54) then S-function becomes M-series defined by [32], and using (16), it yields
(55)sIa+ςzs+1−as+1ν−1Sl,mς,ν,1,1,1τzs+1−as+1ςx=xs+1−as+1ν+ς−1s+1ςMl,mς,ν+ςτxs+1−as+1ς,(56)sDa+ςzs+1−as+1ν−1Sl,mς,ν,1,1,1τzs+1−as+1ςx=xs+1−as+1ν−ς−1s+1ςMl,mς,ν−ςτxs+1−as+1ς,(57)sDa+ς,νzs+1−as+1ν−1Sl,mς,ν,1,1,1τzs+1−as+1ςx=xs+1−as+1ν−ς−1s+1−ςMl,mς,ν−ςτxs+1−as+1ς.Corollary 19.
Choosingp=q=0 in equations (55), (56), and (57) then S-function turns into generalized Mittag-Leffler function defined by Mittag-Leffler [34], and using (16), it yields
(58)sIa+ςzs+1−as+1ν−1S0,0ς,ν,1,1,1τzs+1−as+1ςx=xs+1−as+1ν+ς−1s+1ςEς,ν+ςτxs+1−as+1ς,sDa+ςzs+1−as+1ν−1S0,0ς,ν,1,1,1τzs+1−as+1ςx=xs+1−as+1ν−ς−1s+1ςEς,ν−ςτxs+1−as+1ς,sDa+ς,νzs+1−as+1ν−1S0,0ς,ν,1,1,1τzs+1−as+1ςx=xs+1−as+1ν−ς−1s+1−ςEς,ν−ςτxs+1−as+1ς.
If we substitutes=0 in (38), (39), and (40), then we have some special results of the S-function.Corollary 20.
Supposek>0,x>aa∈R+=0,∞ and ς,ν,σ,τ∈ℂ, Rν,Rς>0,Rσ>0,Rς>0, then
(59)kIa+ςt−aν/k−1Sl,mς,ν,σ,ξ,kτt−aς/kx=x−aν+ς/k−1Sl,mς,ν+ς,σ,ξ,kτx−aς/k,(60)kDa+ςt−aν/k−1Sl,mς,ν,σ,ξ,kτt−aς/kx=x−aν−ς/k−1Sl,mς,ν−ς,σ,ξ,kτx−aς/k,(61)kDa+ς,νt−aν/k−1Sl,mς,ν,σ,ξ,kτt−aς/kx=x−aν−ς/k−1Sl,mς,ν−ς,σ,ξ,kτx−aς/k.
If we substitutek=1 in (59), (60), and (61), then we have some special results of the S-function.Corollary 21.
Supposex>aa∈R+=0,∞ and ς,ν,σ,τ∈ℂ,Rν,Rς>0,Rσ>0,Rς>0, then
(62)Ia+ςt−aν−1Sl,mς,ν,σ,ξ,1τt−aςx=x−aν+ς−1Sl,mς,ν+ς,σ,ξ,1τx−aς,Da+ςt−aν−1Sl,mς,ν,σ,ξ,1τt−aςx=x−aν−ς−1Sl,mς,ν−ς,σ,ξ,1τx−aς,Da+ς,νt−aν−1Sl,mς,ν,σ,ξ,kτt−aςx=x−aν−ς−1Sl,mς,ν−ς,σ,ξ,1τx−aς.


## 3. Some Properties of the Operatorksεa+;ς,ντ;σfx

Theorem 22.
Fork>0 and ς,ν,σ,ξ,ζ,τ∈ℂ,Rς>0,Rν>0,Rσ>0 and Rς>0, we have
(63)ksεa+;ς,ντ;σ,ξzs+1−as+1ς/kx=xs+1−as+1ς+ν/k−1Γkςs+1Sl,mς,ν+ς,σ,ξ,kτxs+1−as+1ς/k.Proof.
From (20),
(64)ksεa+;ς,ντ;σ,ξfx=1k∫axxs+1−zs+1ν/k−1Sl,mς,ν,σ,ξ,kτxs+1−zs+1ς/kzsfzdz.
Therefore, we have(65)ksεa+;ς,ντ;σ,ξzs+1−as+1ς/kx=1k∫axxs+1−zs+1ν/k−1zs+1−as+1ς/k−1Sl,mς,ν,σ,ξ,kτxs+1−zs+1ς/kzsdz=∑n=0∞p1n⋯plnσnξ,kτnq1n⋯qmnΓknς+νn!1k∫axzs+1−as+1ς/k−1xs+1−zs+1ν+ςn/k−1zsdz=∑n=0∞a1n⋯apnσnξ,kτnq1n⋯qmns+1Γknς+νn!xs+1−as+1ν+ς+ςn/k−1Bkν+ςn,ς=∑n=0∞a1n⋯apnσnξ,kτnxs+1−as+1ν+ς+ςn/k−1q1n⋯qmns+1Γknς+νn!Γkν+ςnΓkςΓkν+ς+ςn=xs+1−as+1ν+ς/k−1Γkςs+1Sl,mς,ν+ς,σ,ξ,kτxs+1−as+1ς/k,which completes the desired proof.Theorem 23.
Suppose letf∈L1a,b,s∈R\−1,k>0 and ς,ν,σ,ξ,ζ,τ∈ℂ,Rς>0,Rν>0,Rσ>0 and Rς>0, then ksεa+;ς,ντ;σ,ξfx exit for any x∈a,b.Proof.
Assume that∇=a,b×a,b and U:∇⟶R such that Ux,t=xs+1−zs+1zs for all x∈a,b. It is obvious that U=U++U− where
(66)U+x,t=xs+1−zs+1ν/k−1zs,a≤t≤x≤b.0,a≤x≤t≤b.U−x,t=zs+1−xs+1ν/k−1xs,a≤t≤x≤b.0,a≤x≤t≤b.
AsU is measurable on ζ, we can write
(67)∫abUx,zdz=∫axUx,zdz=∫axxs+1−zs+1ν/k−1zsdz=kνxs+1−zs+1ν/k.So, we get
(68)=∑n=0∞p1n⋯plnσnξ,kτnq1n⋯qmnΓknς+νn!∫axxs+1−zs+1ν+ςn/k−1zsdz=∑n=0∞a1n⋯apnσnξ,kτxs+1−zs+1ς/knq1n⋯qmnΓknς+νn!kν+ςnxs+1−as+1ν/k.
Again, integral the above term, we have(69)∫ab∫abUx,tSl,mς,ν,σ,ξ,kτxs+1−zs+1ς/k∣fx∣dzdx=∫ab∣fx∣∫axUx,zSl,mς,ν,σ,ξ,kτxs+1−zs+1ς/kdzdx=∑n=0∞a1n⋯apnσnξ,kτnq1n⋯qmnΓknς+νn!kν+ςn∫abxs+1−as+1ν+ςn/k∣fx∣dx≤∑n=0∞a1n⋯apnσnξ,kτbs+1−as+1ς/knq1n⋯qmnΓknς+νn!k2ν+ςnν+ςn+kbs+1−as+1ν/k+1×∫ab∣fx∣dx≤bs+1−as+1ν/k+1∑n=0∞a1n⋯apnσnξ,kτbs+1−as+1ς/knq1n⋯qmnΓknς+νn!×k2ν+ςnν+ςn+k∥f∥1≤∞.
TheV:ζ⟶R function is then such that Vx,t=Ux,tfx can be integrated into ζ by Toneli’s theorem. Thus, by Fubini’s theorem ∫abUx,tSl,mς,ν,σ,ξ,kτxs+1−zs+1ς/kfxdx is an integrable function on a,b, as a function of t∈a,b. Thus, ksεa+;ς,ντ;σ,ξfx exists.Theorem 24.
Forς∈ℂ,ς,ν,σ,ξ,ζ,τ∈ℂ,Rς>0,Rν>0,Rσ>0 and Rς>0,k>0,s∈R\−1 and x>a, the following result holds:
(70)ksIa+ςksεa+;ς,ντ;σ,ξfx=1s+1ς/kksεa+;ς,ν+ςτ;σ,ξfx=ksεa+;ς,ν+ςτ;σ,ξksIa+ςfx,for any f∈Lς,ν.Proof.
From (20) and (24), we observe
(71)ksIa+ςksεa+;ς,ντ;σ,ξfx=s+11−ς/kkΓkς∫axksεa+;ς,ντ;σ,ξfzxs+1−zs+11−ς/kzsdz=s+11−ς/kk2Γkς∫axxs+1−zs+1ς/k−1×∫azzs+1−us+1ν/k−1Sl,mς,ν,σ,ξ,kτxs+1−zs+1ς/kfuusduzsdz.
By changing the order of integration, we obtain(72)ksIa+ςksεa+;ς,ντ;σ,ξfx=1k∫axs+11−ς/kkΓkς∫uxxs+1−zs+1ς/kzs+1−us+1ν/k−1Sl,mς,ν,σ,ξ,kτxs+1−zs+1ς/kzsdzusfudu.
Using (38), we have
(73)ksIa+ςksεa+;ς,ντ;σ,ξfx=1ks+1ς/k∫axxs+1−zs+1ν+ς/k−1×Sl,mς,ν+ς,σ,ξ,kτxs+1−zs+1ς/kusfudu.
Thus, we get(74)ksIa+ςksεa+;ς,ντ;σ,ξfx=1s+1ς/kksεa+;ς,ν+ςτ;σ,ξfx.
To prove the second part of the theorem, consider the RHS of (70); then, by applying (20), we get
(75)ksIa+ςksεa+;ς,ντ;σ,ξfx=1k∫axxs+1−zs+1ν/k−1Sl,mς,ν,σ,ξ,kτxs+1−zs+1ς/kksIa+ςfzzsdz=1k∫axxs+1−zs+1ν/k−1Sl,mς,ν,σ,ξ,kτxs+1−zs+1ς/k×s+11−ς/kkΓkς∫azfuzs+1−us+11−ς/kusdudz.
By interchanging the order of integration, we have(76)ksIa+ςksεa+;ς,ντ;σ,ξfx=1k∫axs+11−ς/kkΓkς×∫uxxs+1−zs+1ν/k−1zs+1−us+1ς/k−1Sl,mς,ν,σ,ξ,kτxs+1−zs+1ς/kzsdz×fuusdu.
Again, by making use of (24) and using (38), we get
(77)ksIa+ςksεa+;ς,ντ;σ,ξfx=1s+1ς/kksεa+;ς,ν+ςτ;σ,ξfx.
Thus, (74) and (77) complete the proof of (70).


## 4. Concluding Remarks

In this article, we develop some new results ofk,s-fractional integral and differential operators involving S-function under the extension of left and right R-L k,s-fractional integral and differential operators. Also, we find some special cases of functions like k-Mittag-Leffler function, K-function, and M-series. If we set l=0=m, then we obtained results of the generalized k-Mittag-Leffler function given by [37], and if we set s=0, the results obtained are reduced to the well-known results given by [38]. If we select k=1 and s=0, then the results obtained are reduced to the Mittag-Leffler function’s well-known results.



---

*Source: 9582879-2021-10-06.xml*