# Composition Formulae for thek-Fractional Calculus Operator with the S-Function

Authors: Hagos Tadesse; Haile Habenom; Anita Alaria; Biniyam Shimelis
Journal: Journal of Mathematics
 (2021)

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## Abstract

In this study, the S-function is applied to Saigo’sk-fractional order integral and derivative operators involving the k-hypergeometric function in the kernel; outcomes are described in terms of the k-Wright function, which is used to represent image formulas of integral transformations such as the beta transform. Several special cases, such as the fractional calculus operator and the S-function, are also listed.


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## Body

## 1. Introduction and Preliminaries

Fractional calculus was first introduced in 1695, but only in the last two decades have researchers been able to use it efficiently due to the availability of computing tools. Significant uses of fractional calculus have been discovered by scholars in engineering and science. In literature, many applications of fractional calculus are available in astrophysics, biosignal processing, fluid dynamics, nonlinear control theory, and stochastic dynamical system. Furthermore, research studies in the field of applied science [1, 2], and on the application of fractional calculus in real-world problems [3, 4], have recently been published. A number of researchers [5–15] have also investigated the structure, implementations, and various directions of extensions of the fractional integration and differentiation in detail. A detailed description of such fractional calculus operators, as well as their characterization and application, can be found in research monographs [16, 17].Recently, a series of research publications with respect to generalized classical fractional calculus operators was published. Mubeen and Habibullah [18] broughtout k-fractional order integral of the Riemann–Liouville version and its applications. Dorrego [19] introduced an alternative definition for the k-Riemann–Liouville fractional derivative.Gupta and Parihar [20] introduced the left and right sides of Saigo k-fractional integration and differentiation operators connected with the k-Gauss hypergeometric function which are as follows:(1)I0+,kϑ,ς,γfx=x−ϑ−ς/kkΓkϑ∫0xx−tϑ/k−1F21,kϑ+ς,k,−γ,k;ϑ,k;1−txftdt;ℜϑ>0,k>0,(2)I−,kϑ,ς,γfx=1kΓkϑ∫x∞t−xϑ/k−1t−ϑ−ς/kF21,kϑ+ς,k,−γ,k;ϑ,k;1−xtftdt;ℜϑ>0,k>0.Mubeen and Habibullah [18] defined F21,kϑ,k,ς,k;γ,k;x, i.e., the k-Gauss hypergeometric function for x∈ℂ,x<1,ℜγ>ℜς>0:(3)F21,kϑ,k,ς,k;γ,k;x=∑n=0∞ϑn,kςn,kxnγn,kn!.Equations (1) and (2) are the left and right sides of fractional differential operators involving k-Gauss hypergeometric function, respectively:(4)D0+,kϑ,ς,γfx=ddxnI0+,k−ϑ+n,−ς−n,ϑ+γ−nfx;ℜϑ>0,k>0;n=ℜϑ+1=ddxnxϑ+ς/kkΓk−ϑ+n∫0xx−t−ϑ/k+n−1×F21,k−ϑ−ς,k,−γ−ϑ+n,k;−ϑ+n,k;1−txftdt,(5)D−,kϑ,ς,γfx=−ddxnI−,k−ϑ+n,−ς−n,ϑ+γfx;ℜϑ>0,k>0;n=ℜϑ+1=−ddxn1kΓk−ϑ+n∫x∞t−x−ϑ+n/k−1tϑ+ς/k×F21,k−ϑ−ς,k,−γ−ϑ,k;−ϑ+n,k;1−xtftdt,where x>0,ϑ∈ℂ,ℜϑ>0,k>0 and ℜϑ is the integer part of ℜϑ.Remark 1.
When we setk=1 in equations, operators (1), (2), (4), and (5) reduce into Saigo’s fractional integral and derivative operators, as stated in [9], respectively.
We consider the following basic results for our study.Lemma 1.
(see p. 497, equation 4.2, in [20]). Let ϑ,ς,γ,ε∈ℂ,ℜε>max0,ℜς−γ; then,(6)I0+,kϑ,ς,γtε/k−1x=∑n=0∞knΓkεΓkε−ς+γΓkε−ςΓkε+ϑ+γxε−ς/k−1.Lemma 2.
(see p. 497, equation 4.3, in [20]). Let ϑ,ς,γ,ε∈ℂ,ℜϑ>0,k∈ℜ+0,∞ and ℜε>maxℜ−ς,ℜ−γ; then,(7)I−,kϑ,ς,γt−ε/kx=∑n=0∞knΓkε+ςΓkε+γΓkεΓkε+ϑ+ς+γx−ε−ς/k.Lemma 3.
(see p. 500, equation 6.2, in [20]). Let ϑ,ς,γ,ε∈ℂ,n=ℜϑ+1,k∈ℜ+0,∞ such that ℜε>max0,ℜ−ϑ−ς−γ; then,(8)D0+,kϑ,ς,γtε/k−1x=∑n=0∞knΓkεΓkε+ς+γ+ϑΓkε+γΓkε+ς+n−nkxε+ς+n/k−n−1.Lemma 4.
(see p. 500, equation 6.3, in [20]). Let ϑ,ς,γ,ε∈ℂ and n=ℜϑ+1,k∈ℜ+, ℜε>maxℜ−ϑ−γ,ℜς−nk+n; then,(9)D−,kϑ,ς,γt−ε/kx=∑n=0∞knΓkε−ς−n+nkΓkε+ϑ+γΓkεΓkε−ς+γx−ε+ς+n/k−n.Recent time, theS-function is defined and studied by Saxena and Daiya [21], which is generalization of k-Mittag-Leffler function, K-function, M-series, Mittag-Leffler function (see [22–25]), as well as its relationships with other special functions. These special functions have recently found essential applications in solving problems in physics, biology, engineering, and applied sciences.TheS-function is defined for ϑ′,δ′,γ′ε∈ℂ, ℜϑ′>0, k∈ℜ, ℜϑ′>kℜε, lii=1,2,…,p, mjj=1,2,…,q, and p<q+1 as(10)Sp,qϑ′,δ′,γ′,ε,kl1,l2,…,lp;m1,m2,…,mq;x=∑n=0∞l1n…lpnγ′nε,km1n…mqnΓknϑ′+δ′xnn!.Here, Di´az and Pariguan [26] introduced the k-Pochhammer symbol and k-gamma function as follows:(11)γ′n,k=Γkγ′+nkΓkγ′,k∈ℜ,γ′∈ℂ/0,γ′γ′+k…γ′+n−1k,n∈ℕ,γ′∈ℂ,as well as the relationship with the classic Euler’s gamma function:(12)Γkγ′=kγ′/k−1Γγ′k,where γ′∈ℂ,k∈ℜ, and n∈ℕ. Refer to Romero and Cerutti’s papers [27] for more information on the k-Pochammer symbol, k-special functions, and fractional Fourier transforms.The following are some significant special cases of theS-function:(i)
Forp=q=0, the generalized k-Mittag-Leffler function [28](13)Ek,ϑ′,δ′γ′,εx=S0,0ϑ′,δ′,γ′,ε,k−;−;x=∑n=0∞γ′nε,kΓknϑ′+δ′xnn!,ℜϑ′k−ε>p−q.(ii)
Again, fork=ε=1, the S-function is the generalized K-function [29]:(14)Kp,qϑ′,δ′,γ′l1,l2,…,lp;m1,m2,…,mq;x=Sp,qϑ′,δ′,γ′,1,1l1,l2,…,lp;m1,m2,…,mq;x=∑n=0∞l1n…lpnγ′nm1n…mqnΓnϑ′+δ′xnn!,ℜϑ′>p−q.(iii)
Forε=k=γ′=1, the S-function reduced to generalized M-series [30]:(15)Mp,qϑ′,δ′l1,l2,…,lp;m1,m2,…,mq;x=Sp,qϑ′,δ′,1,1,1l1,l2,…,lp;m1,m2,…,mq;x=∑n=0∞l1n…lpnm1n…mqnΓnϑ′+δ′xnn!,ℜϑ′>p−q−1.For our purpose, we recall the definition of generalizedk-Wright function Ψpqkx, defined by Gehlot and Prajapati [31], for k∈ℝ+;x,ai,bj∈ℂ, ϑi,ςj∈ℜϑi,ςj≠0;i=1,2,…,p;j=1,2,…,q and ai+ϑin,bj+ςjn∈ℂ\kz−, as(16)Ψpqkx=Ψpqkai,ϑi1,pbj,ςj1,q;x=∑n=0∞∏i=1pΓkai+ϑin∏j=1qΓkbj+ςjnxnn!,which satisfies the condition(17)∑j=1qςjk−∑i=1pϑik>−1.


## 2. Saigok-Fractional Integration in Terms of k-Wright Function

In this section, the results are displayed based on thek-fractional integrals associated with the S-function.Theorem 1.
Letϑ,ς,γ,ϑ′,ς′,γ′,ε,ε∈ℂ;k∈ℜ+,c∈ℜ and v>0, such that ℜϑ>0, ℜε>max0,ℜς−γ,ℜε+γ−ς>0, aii=1,2,…,p,bjj=1,2,…,q, ℜϑ′>kℜε; p<q+1. If condition (17) is satisfied and I0+,kϑ,ς,γ is the left-sided integral operator of the generalized k-fractional integration associated with S-function, then (18) holds true:(18)I0+,kϑ,ς,γtε/k−1Sp,qϑ′,ς′,γ′,ε,ka1,…,ap;b1,…,bq;ctv/kx=xε−ς/k−1Γkγ′∏j=1qΓbj∏i=1pΓaik∑j=1qbj−∑i=1pai×p+3Ψq+3ka1k,k…apk,k,γ′,εk,ε,v,ε+γ−ς,v,b1k,k…bqk,k,ς′,ϑ′,ε−ς,v,ε+ϑ+γ,v,kcxv/k.Proof.
We indicate the R.H.S. of equation (18) by I1; invoking equation (10), we have(19)I1=I0+,kϑ,ς,γtε/k−1∑n=0∞a1n…apnγ′nε,kb1n…bqnΓkς′+ϑ′nctv/knn!x=∑n=0∞a1n…apnγ′nε,kb1n…bqnΓkς′+ϑ′ncnn!I0+,kϑ,ς,γtε+vn/k−1x.
Now, applying equation (6) and (11), we obtain(20)I1=xε−ς/k−1Γkγ′∑n=0∞a1n…apnΓkγ′+nεkΓkε+vnΓkε+γ−ς+vnb1n…bqnΓkς′+ϑ′nΓkε−ς+vnΓkε+ϑ+γ+vnkcxv/knn!.
Using (12) and some important simplifications on the above equation, we obtain(21)I1=Γb1…ΓbqΓa1…Γapxε−ς/k−1Γkγ′kb1+⋯+bq−a1+⋯+ap×∑n=0∞Γka1k+nk…Γkapk+nkΓkγ′+nεkΓkε+vnΓkε+γ−ς+vnΓkb1k+nk…Γkbqk+nkΓkς′+ϑ′nΓkε−ς+vnΓkε+ϑ+γ+vnkcxv/knn!.
Interpreting the definition of Wright hypergeometric function (16) on the above equation, we arrive at the desired result (18).Theorem 2.
Letϑ,ς,γ,ϑ′,ς′,γ′,ε,ε∈ℂ;k∈ℜ+,c∈ℜ, and v>0, such that ℜϑ>0, ℜϑ′>0, and ℜε+ϑ>max−ℜς,−ℜγ, with ℜς≠ℜγ, aii=1,2,…,p,bjj=1,2,…,q, ℜϑ′>kℜε, and p<q+1. If condition (17) is satisfied and I−,kϑ,ς,γ is the right-sided integral operator of the generalized k-fractional integration associated with S-function, then (22) holds true:(22)I−,kϑ,ς,γt−ϑ−ε/kSp,qϑ′,ς′,γ′,ε,ka1,…,ap;b1,…,bq;ct−v/kx=k∑j=1qbj−∑i=1pai∏j=1qΓbj∏i=1pΓaix−ϑ−ε−ς/kΓkγ′×p+3Ψq+3ka1k,k…apk,k,γ′,εk,ϑ+ε+ς,v,ϑ+ε+γ,v,b1k,k…bqk,k,ς′,ϑ′,ϑ+ε,v,2ϑ+ε+ς+γ,v,kcx−v/k.Proof.
The proof is parallel to that of Theorem1. Therefore, we omit the details.The results given in (18) and (22), being very general, can yield a large number of special cases by assigning some suitable values to the involved parameters. Now, we demonstrate some corollaries as follows.Corollary 1.
If we putp=q=0, then (18) leads to the subsequent result of S-function:(23)I0+,kϑ,ς,γtε/k−1Ek,ϑ′,δ′γ′,εctv/kx=xε−ς/k−1Γkγ′×3Ψ3kγ′,εk,ε,v,ε+γ−ς,v,ς′,ϑ′,ε−ς,v,ε+ϑ+γ,v,kcxv/k.Corollary 2.
Ifε=k=1, in (18), we obtain the subsequent result in term of S-function as(24)I0+ϑ,ς,γtε−1Kp,qϑ′,ς′,γ′a1,…,ap;b1,…,bq;ctvx=∏j=1qΓbj∏i=1pΓaixε−ς−1Γγ′×p+3Ψq+3a1,1…ap,1,γ′,1,ε,v,ε+γ−ς,v,b1,1…bq,1,ς′,ϑ′,ε−ς,v,ε+ϑ+γ,v,cxv.Corollary 3.
If we setε=1,γ′=1,andk=1, in equation (18), we obtain the following formula:(25)I0+ϑ,ς,γtε−1Mp,qϑ′,ς′a1,…,ap;b1,…,bq;ctvx=∏j=1qΓbj∏i=1pΓaixε−ς−1×p+3Ψq+3a1,1…ap,1,ε,v,ε+γ−ς,v,1,1,b1,1…bq,1,ς′,ϑ′,ε−ς,v,ε+ϑ+γ,v,cxv.Corollary 4.
Lettingp=q=0 in equation (22), then(26)I−,kϑ,ς,γt−ϑ−ε/kEk,ϑ′,δ′γ′,εct−v/kx=x−ϑ−ε−ς/kΓkγ′×Ψ33kγ′,εk,ϑ+ε+ς,v,ϑ+ε+γ,v,ς′,ϑ′,ϑ+ε,v,2ϑ+ε+ς+γ,v,kcx−v/k.Corollary 5.
Settingε=1, k=1, then equation (22) becomes(27)I−ϑ,ς,γt−ϑ−εKp,qϑ′,ς′,γ′a1,…,ap;b1,…,bq;ct−vx=∏j=1qΓbj∏i=1pΓaix−ϑ−ε−ςΓkγ′×p+3Ψq+3a1,1…ap,1,γ′,1,ϑ+ε+ς,v,ϑ+ε+γ,v,b1,1…bq,1,ς′,ϑ′,ϑ+ε,v,2ϑ+ε+ς+γ,v,cx−v.Corollary 6.
If we putε=1,γ′=1,andk=1 in equation (22), then equation becomes(28)I−ϑ,ς,γt−ϑ−εMp,qϑ′,ς′a1,…,ap;b1,…,bq;ct−vx=∏j=1qΓbj∏i=1pΓaix−ϑ−ε−δ×p+3Ψq+3a1,1…ap,1,ϑ+ε+ς,v,ϑ+ε+γ,v,1,1,b1,1…bq,1,ς′,ϑ′,ϑ+ε,v,2ϑ+ε+ς+γ,v,cx−v.


## 3. Saigok-Fractional Differentiation in Terms of k-Wright Function

In this section, the results are displayed based on thek-fractional derivatives associated with the S-function.Theorem 3.
Letϑ,ς,γ,ϑ′,ς′,γ′,ε,ε∈ℂ;k∈ℜ+,c∈ℜ, and v>0, such that ℜϑ>0, ℜϑ′>0, ℜε>max0,ℜ−ϑ−ς−γ,ℜε+γ+ς>0, aii=1,2,…,p, bjj=1,2,…,q,ℜϑ′>kℜε, and p<q+1. If condition (17) is satisfied and D0+,kϑ,ς,γ is the left-sided differential operator of the generalized k-fractional integration associated with S-function, then (29) holds true:(29)D0+,kϑ,ς,γtε/k−1Sp,qϑ′,ς′,γ′,ε,ka1,…,ap;b1,…,bq;ctv/kx=xε+ς/k−1Γkγ′∏j=1qΓbj∏i=1pΓaik∑j=1qbj−∑i=1pai×p+3Ψq+3ka1k,k…apk,k,γ′,εk,ε,v,ε+ς+γ+ϑ,v,b1k,k…bqk,k,ς′,ϑ′,ε+γ,v,ε+δ,1−k+v,cxv+1/k−1.Proof.
For the sake of convenience, let the left-hand side of (29) be denoted by I2. Using definition (10), we arrive at(30)I2=∑n=0∞a1n…apnγ′nε,kb1n…bqnΓkς′+ϑ′vcnn!D0+,kϑ,ς,γtε+vn/k−1x.
Now, applying equation (8) and (11), we obtain(31)I2=xε+ς/k−1Γkγ′∑n=0∞a1n…apnΓkγ′+εnkb1n…bqnΓkς′+ϑ′n×Γkε+vnΓkε+ς+γ+ϑ+vnΓkε+γ+vnΓkε+ς+n−nk+vnn!cxv+1/k−1n.
Using (12) and simplifications on the above equation, we obtain(32)I2=kb1+⋯+bq−a1+⋯+apΓb1…ΓbqΓa1…Γapxε+ς/k−1Γkγ′∑n=0∞Γkγ′+nεkΓkς′+ϑ′n×Γka1k+nk…Γkapk+nkΓkε+vnΓkε+ς+γ+ϑ+vnΓkb1k+nk…Γkbqk+nkΓkε+γ+vnΓkε+ς+n−nk+vnn!cxv+1/k−1n.
In accordance with (16), we obtain the required result (29). This completed the proof of Theorem 3.Theorem 4.
Letϑ,ς,γ,ϑ′,ς′,γ′,ε,ε∈ℂ;k∈ℜ+,c∈ℜ, and v>0, such that ℜϑ>0, ℜϑ′>0, ℜε>maxℜϑ+ς+n−ℜγ,andℜϑ+ς−γ+n≠0, where n=ℜϑ+1, aii=1,2,…,p,bjj=1,2,…,q,ℜϑ′>kℜε, and p<q+1. If condition (17) is satisfied and D−,kϑ,ς,γ is the right-sided differential operator of the generalized k-fractional integration associated with S-function, then (33) holds true:(33)D−,kϑ,ς,γtϑ−ε/kSp,qϑ′,ς′,γ′,ε,ka1,…,ap;b1,…,bq;ct−v/kx=xϑ−ε+ς/kΓkγ′∏j=1qΓbj∏i=1pΓaik∑j=1qbj−∑i=1pai×p+3Ψq+3ka1k,k…apk,k,γ′,εk,ε−ϑ−δ,v+k−1,ε+γ,v,b1k,k…bqk,k,ς′,ϑ′,ε−ϑ,v,ε−ϑ−ς+γ,v,cx−v+1/k−1.Proof.
The proof is parallel to that of Theorem3. Therefore, we omit the details.The results given in (29) and (33) are reduced as special cases by assigning some suitable values to the involved parameters. Now, we demonstrate some corollaries as follows.Corollary 7.
Ifp=q=0, then (29) holds the following formula:(34)D0+,kϑ,ς,γtε/k−1Ek,ϑ′,δ′γ′,εct−v/kx=xε+ς/k−1Γkγ′×Ψ33kγ′,εk,ε,v,ε+ς+γ+ϑ,v,ς′,ϑ′,ε+γ,v,ε+δ,1−k+v,cxv+1/k−1.Corollary 8.
If we putε=1 and k=1, then (29) gives the result in term of S-function as follows:(35)D0+ϑ,ς,γtε/k−1Kp,qϑ′,ς′,γ′a1,…,ap;b1,…,bq;ctvx=xε+ς−1Γγ′∏j=1qΓbj∏i=1pΓai×p+3Ψq+3a1,1…ap,1,γ′,1,ε,v,ε+ς+γ+ϑ,v,b1,1…bq,1,ς′,ϑ′,ε+γ,v,ε+δ,v,cxv.Corollary 9.
If we putε=1,γ′=1,andk=1, in equation (29), then(36)D0+ϑ,ς,γtε/k−1Mp,qϑ′,ς′a1,…,ap;b1,…,bq;ctvx=∏j=1qΓbj∏i=1pΓaixε+ς−1×p+3Ψq+3a1,1…ap,1,ε,v,ε+ς+γ+ϑ,v,1,1,b1,1…bq,1,ς′,ϑ′,ε+γ,v,ε+δ,v,cxv.Corollary 10.
If we setp=q=0, then (33) provides the result as(37)D−,kϑ,ς,γtϑ−ε/kEk,ϑ′,δ′γ′,εct−v/kx=xϑ−ε+ς/k−1Γkγ′=Ψ33kγ′,k,ε−ϑ−δ,v+k−1,ε+γ,v,ς′,ϑ′,ε−ϑ,v,ε−ϑ−ς+γ,v,cx−v+1/k−1.Corollary 11.
By lettingε=1 and k=1, in equation (33), then(38)D−ϑ,ς,γtϑ−εKp,qϑ′,ς′,γ′a1,…,ap;b1,…,bq;ct−vx=xϑ−ε+δΓγ′∏j=1qΓbj∏i=1pΓai×p+3Ψq+3a1,1…ap,1,γ′,1,ε−ϑ−ς,v,ε+γ,v,b1,1…bq,1,ς′,ϑ′,ε−ϑ,v,ε−ϑ−ς+γ,v,cx−v.Corollary 12.
Whenε=1,γ′=1,andk=1, in equation (33), then equation becomes(39)D−ϑ,ς,γtϑ−εMp,qϑ′,ς′a1,…,ap;b1,…,bq;ct−vx=∏j=1qΓbj∏i=1pΓaixϑ−ε+δ×p+3Ψq+3a1,1…ap,1,ε−ϑ−δ,v,ε+γ,v,1,1,b1,1…bq,1,ς′,ϑ′,ε−ϑ,v,ε−ϑ−ς+γ,v,cx−v.


## 4. Image Formulas Associated with Integral Transforms

In this section, we establish some theorems involving the results obtained in previous sections pertaining with the integral transform. Here, we definedk-beta function as follows.Thek-beta function [32] is defined as(40)Bkg,h=1k∫01zg/k−11−zh/k−1dz,g>0,h>0.They have the following important identities:(41)Bkg,h=1kBgk,hk=ΓkgΓkhΓkg+h.Now, we definek-beta function in the form(42)Bkfz;g,h=1k∫01zg/k−11−zh/k−1fzdz,g>0,h>0.Theorem 5.
Letϑ,ς,γ,ϑ′,ς′,ε,γ′,ε∈ℂ;k∈ℜ+,c∈ℜ, and v>0, such that ℜϑ>0, ℜε>max0,ℜς−γ,andℜε+γ−ς>0; then, the leading fractional order integral holds true:(43)BkI0+,kϑ,ς,γtε/k−1Sp,qϑ′,ς′,γ′,ε,ka1,…,ap;b1,…,bq;cztv/kx;g,h=xε−ς/k−1ΓkhΓkγ′∏j=1qΓbj∏i=1pΓaik∑j=1qbj−∑i=1pai×p+4Ψq+4ka1k,k…apk,k,γ′,εk,ε,v,ε+γ−ς,v,g,v,b1k,k…bqk,k,ς′,ϑ′,ε−ς,v,ε+ϑ+γ,v,g+h,v,kcxv/k.Proof.
LetI3 be the left-hand side of (43), and using (42), we have(44)I3=1k∫01zg/k−11−zh/k−1I0+,kϑ,ς,γtε/k−1Sp,qϑ′,ς′,γ′,ε,ka1,…,ap;b1,…,bq;cztv/kxdz,which, using (10) and changing the order of integration and summation, is valid under the conditions of Theorem 1 and yields(45)I3=∑n=0∞a1n…apnγ′nε,kb1n…bqnΓkς′+ϑ′ncnn!I0+,kϑ,ς,γtε+vn/k−1x×1k∫01zg+vn/k−11−zh/k−1dz.
From Lemma1 and substituting (41) in (45), we obtain(46)I1=kb1+⋯+bq−a1+⋯+apxε−ς/k−1Γkγ′Γb1…ΓbqΓa1…Γap∑n=0∞Γka1k+nk…Γkapk+nkb1k+nk…Γkbqk+nk×Γkγ′+nεkΓkε+vnΓkε+γ−ς+vnΓkg+vnΓkhΓkΓkς′+ϑ′nΓkε−ς+vnΓkε+ϑ+γ+vnΓkg+h+vnkcxv/knn!.
Using the definition of (16) in the right-hand side of (46), we arrive at result (43).Theorem 6.
Letϑ,ς,γ,ϑ′,ς′,γ′,ε,ε∈ℂ;k∈ℜ+,c∈ℜ, and v>0, such that ℜϑ>0, ℜϑ′>0, and ℜε+ϑ>max−ℜς,−ℜγ, with ℜς≠ℜγ; then, the following fractional integral holds true:(47)BkI−,kϑ,ς,γt−ϑ−ε/kSp,qϑ′,ς′,γ′ε,ka1,…,ap;b1,…,bq;czt−v/kx;g,h=k∑j=1qbj−∑i=1pai∏j=1qΓbj∏i=1pΓaiΓkhx−ϑ−ε−ς/kΓkγ′×p+4Ψq+4ka1k,k…apk,k,g,−v,γ′,εk,ϑ+ε+ς,v,ϑ+ε+γ,v,b1k,k…bqk,k,g+h,−v,ς′,ϑ′,ϑ+ε,v,2ϑ+ε+ς+γ,v,kcx−v/k.Proof.
The proof is similar of Theorem5. Therefore, we omit the details.Theorem 7.
Letϑ,ς,γ,ϑ′,ς′,γ′,ε,ε∈ℂ;k∈ℜ+,c∈ℜ, and v>0, such that ℜϑ>0, ℜϑ′>0, ℜε>max0,ℜ−ϑ−ς−γ,andℜε+γ+ς>0; then, the following fractional derivative holds true:(48)BkD0+,kϑ,ς,γtε/k−1Sp,qϑ′,ς′,γ′,ε,ka1,…,ap;b1,…,bq;cztv/kx;g,h=ΓkhΓkγ′∏j=1qΓbj∏i=1pΓaixε+ς/k−1k∑j=1qbj−∑i=1pai×p+4Ψq+4ka1k,k…apk,k,γ′,εk,ε,v,ε+ς+γ+ϑ,v,g,v,b1k,k…bqk,k,ς′,ϑ′,ε+γ,v,g+h,v,ε+δ,1−k+v,cxv+1/k−1.Proof.
LetI4 be the left-hand side of (48), and using the definition of Beta transform, we have(49)I4=1k∫01zg/k−11−zh/k−1D0+,kϑ,ς,γtε/k−1Sp,qϑ′,ς′,γ′,ε,ka1,…,ap;b1,…,bq;cztv/kxdz,which, using (10) and changing the order of integration and summation, is reasonable under the conditions of Theorem 3 and yields(50)I4=∑n=0∞a1n…apnγ′nε,kb1n…bqnΓkς′+ϑ′vcnn!D0+,kϑ,ς,γtε+vn/k−1x×1k∫01zg+vn/k−11−zh/k−1dz.
From Lemma3 and substituting equation (41) in (50), we obtain(51)I4=kb1−a1+⋯+bq−apΓb1…ΓbqΓa1…Γapxε+ς/k−1Γkγ′∑n=0∞Γkγ′+nεkΓka1k+nk…Γkς′+ϑ′nΓkb1k+nk…×Γkapk+nkΓkε+vnΓkε+ς+γ+ϑ+vnΓkg+vnΓkhΓkbqk+nkΓkε+γ+vnΓkε+ς+n−nk+vnΓkg+h+vnn!cxv+1/k−1n.
Using the definition of (16) in the above equation, we obtain the required result (48). This completed the proof of Theorem 7.Theorem 8.
Letϑ,ς,γ,ϑ′,ς′,γ′,ε,ε∈ℂ;k∈ℜ+,c∈ℜ, and v>0, such that ℜϑ>0, ℜϑ′>0, ℜε>maxℜϑ+ς+n−ℜγ,andℜϑ+ς−γ+n≠0, where n=ℜϑ+1; then, the following fractional derivative holds true:(52)BkD−,kϑ,ς,γtϑ−ε/kSp,qϑ′,ς′,γ′,ε,ka1,…,ap;b1,…,bq;czt−v/kx=ΓkhΓkγ′∏j=1qΓbj∏i=1pΓaixϑ−ε+ς/k−1k∑j=1qbj−∑i=1pai×p+4Ψq+4ka1k,k…apk,k,γ′,εk,ε−ϑ−δ,v+k−1,g,−vε+γ,v,b1k,k…bqk,k,ς′,ϑ′,g+h,−v,ε−ϑ,v,ε−ϑ−ς+γ,v,cx−v+1/k−1.Proof.
The proof is identical to that of Theorem7. As a result, we exclude the specifics.


## 5. Conclusion

The strength of generalizedk-fractional calculus operators, also known as general operators by many scholars, is that they generalize classical Riemann-Liouville (R-L) operators and Saigo’s fractional calculus operators. For k⟶1, operators (1) to (5) reduce to Saigo’s [9] fractional integral and differentiation operators. If we set δ=−ϑ, operators (1) to (5) reduce to k-Riemann-Liouville operators as follows:(53)I0+,kϑ,−ϑ,γfx=I0+,kϑfx,I−,kϑ,−ϑ,γfx=I−,kϑfx,D0+,kϑ,−ϑ,γfx=D0+,kϑfx,D−,kϑ,−ϑ,γfx=D−,kϑfx.On the account of the most general character of the S-function, numerous other interesting special cases of results (18), (22), (29), 2and (33) can be obtained, but for lack of space, they are not represented here.



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*Source: 7379820-2021-07-12.xml*