# Fractional Integral and Derivative Formulas by Using Marichev-Saigo-Maeda Operators Involving the S-Function

Authors: D. L. Suthar; Hafte Amsalu
Journal: Abstract and Applied Analysis
 (2019)

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## Abstract

We establish fractional integral and derivative formulas by using Marichev-Saigo-Maeda operators involving the S-function. The results are expressed in terms of the generalized Gauss hypergeometric functions. Corresponding assertions in terms of Saigo, Erdélyi-Kober, Riemann-Liouville, and Weyl type of fractional integrals and derivatives are presented. Also we develop their composition formula by applying the Beta and Laplace transforms. Further, we point out also their relevance.


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## Body

## 1. Introduction and Preliminaries

In recent times, the fractional calculus is the most fast growing subject of mathematical analysis. It is concern with applied mathematics that deals with integrals and derivatives of arbitrary orders. The fractional calculus operator linking diverse special functions has found substantial significance and applications in a variety of subfields of applicable mathematical analysis. Numerous applications of fractional calculus can be found in astrophysics, turbulence, nonlinear biological systems, fluid dynamics, stochastic dynamical system, plasma physics and nonlinear control theory, image processing, and quantum mechanics. Since last four decades, a number of workers like [1–9] so forth have studied, in depth, the properties, applications, and diverse extensions of a range of operators of fractional calculus. A comprehensive account of generalized fractional calculus operators along with their properties and applications can be found in [10–13] and also the research monographs [14, 15] and so forth.On relation of success of the Saigo operators [16, 17], in their study on various function spaces and their application in the integral equation and differential equations, Saigo and Maeda [18] introduced the following generalized fractional integral and differential operators of any complex order with Appell function F3(.) in the kernel which is extension of Marichev [19], as follows.Letμ,μ′,v,v′,δ∈C and x> 0, then the generalized fractional calculus formulas (the Marichev-Saigo-Maeda operators) involving the Appell function or Horn’s F3-function are defined by the following equations:(1)I0,xμ,μ′,v,v′,δfx=x-μΓδ∫0xx-tδ-1t-μ′F3μ,μ′,v,v′,δ;1-tx,1-xtftdt,(2)=ddxkI0,xμ,μ′,v+k,v′,δ+kfx,Rδ≤0;k=-Rδ+1;(3)Ix,∞μ,μ′,v,v′,δfx=x-μ′Γδ∫x∞t-xδ-1t-μF3μ,μ′,v,v′,δ;1-xt,1-txftdt,(4)=-ddxkIx,∞μ,μ′,v,v′+k,δ+kfx,Rδ≤0;k=-Rδ+1,and(5)D0,xμ,μ′,v,v′,δfx=I0,x-μ′,-μ,-v′+k,-v,-δ+kfx(6)=ddxkI0,x-μ′,-μ,-v′+k,-v,-δ+kfx,Rδ>0;k=Rδ+1;(7)Dx,∞μ,μ′,v,v′,δfx=Ix,∞-μ′,-μ,-v′,-v,-δfx(8)=-ddxkIx,∞-μ′,-μ,-v′,-v+k,-δ+kfx,Rδ>0;k=Rδ+1.In (1) and (3), F3. denotes Appell function [20] in two variables defined as(9)F3μ,μ′,v,v′,δ;1-xt,1-tx=∑m.n=0∞μmμ′nvmv′nδm+nxmm!xnn!,max⁡x,y<1.Remark. The Appell function defined in above equation reduces to Gauss hypergeometric function 2F1 as given in the following relations:(10)F3μ,δ-μ,v,δ-v,δ;x,y=2F1μ,v;δ;x+y-xy,and(11)F3μ,0,v,v′,δ;x,y=2F1μ,v;δ;x,and(12)F30,μ′,v,v′,δ;x,y=2F1μ′,v′;δ;y.In view of the above reduction formula as given in (10), the general fractional calculus operators reduce to the Saigo operators [16] defined as follows:(13)I0,xμ,v,δfx=x-μ-vΓδ∫0xx-tμ-12F1μ+v,-δ;μ;1-xtftdt,(14)=ddxkI0,xμ+k,v-k,δ-kfx,Rμ≤0;k=R-μ+1;(15)Ix,∞μ,v,δfx=1Γδ∫x∞t-xμ-1t-μ-v2F1μ+v,-δ;μ;1-xtftdt,(16)=-ddxkIx,∞μ-k,v-k,δfx,Rμ≤0;k=R-μ+1;and(17)D0,xμ,v,δfx=I0,x-μ,-v,μ+δfx=ddxkI0,x-μ+k,-v-k,μ+δ-kfx(18)Dx,∞μ,v,δfx=Ix,∞-μ,-v,δ+μfx=-ddxkIx,∞-μ+k,-v-k,δ+μfx.where 2F1(.), a special case of the generalized hypergeometric function, is the Gauss hypergeometric function and the function f(t) is so constrained that the integrals in (13) and (15) converge.If we takev=0 in (13), (15), (17), and (18), we obtain the Erdélyi-Kober fractional integral and derivative operators [11, 21], defined as follows:(19)I0,xμ,δfx=x-μ-δΓμ∫0xx-tμ-1tμ-1ftdt,Rμ>0,(20)Ix,∞μ,δfx=xδΓμ∫x∞t-xμ-1t-μ-δftdt,Rμ>0,and(21)D0,xμ,δfx=ddxkI0,x-μ+k,-μ,μ+δ-kfx,Rμ>0;k=Rμ+1.(22)Dx,∞μ,δfx=-1kddxkIx,∞-μ+k,-μ,μ+δfx,Rμ>0;k=Rμ+1.When v=-μ, then operators in (13), (15), (17), and (18) give the Riemann-Liouville and the Weyl fractional integral and derivative operators [11, 22] are defined as follows:(23)I0,xμfx=1Γμ∫0xx-tμ-1ftdt,Rμ>0,(24)Ix,∞μfx=1Γμ∫x∞t-xμ-1ftdt,Rμ>0,and(25)D0,xμfx=ddxkIo,x-μ+kfx,Rμ>0;k=Rμ+1.(26)Dx,∞μfx=-1kddxkIx,∞-μ+kfx,Rμ>0;k=Rμ+1.Power function formulas of the above discussed fractional operators are required for our present study as given in the following lemmas [16, 18, 23].Lemma 1.
Letμ,μ′,v,v′,δ,γ∈C, and x>0 be such that R(δ)>0, then the following formulas hold true:(27)I0,xμ,μ′,v,v′,δtγ-1x=ΓγΓγ+δ-μ-μ′-vΓγ+v′-μ′Γγ+v′Γγ+δ-μ-μ′Γγ+δ-μ′-vxγ+δ-μ-μ′-1,Rγ>max⁡0,Rμ+μ′+v-δ,Rμ′-v′.(28)Ix,∞μ,μ′,v,v′,δtγ-1x=Γ1-γ-vΓ1-γ-δ+μ+μ′Γ1-γ-δ+μ+v′Γ1-γΓ1-γ-δ+μ+μ′+v′Γ1-γ+μ-v×xγ+δ-μ-μ′-1,Rγ<1+min⁡R-v,Rμ+μ′-δ,Rμ+v′-δ,and(29)D0,xμ,μ′,v,v′,δtγ-1x=ΓγΓγ-δ+μ+μ′+v′Γγ-v+μΓγ-vΓγ-δ+μ+μ′Γγ-δ+μ+v′xγ-δ+μ+μ′-1,Rγ>max⁡0,Rδ-μ-μ′-v′,Rv-μ.(30)Dx,∞μ,μ′,v,v′,δtγ-1x=Γ1-γ+v′Γ1-γ+δ-μ-μ′Γ1-γ+δ-μ′-vΓ1-γΓ1-γ+δ-μ-μ′-vΓ1-γ-μ′+v′×xγ-δ+μ+μ′-1,Rγ<1+min⁡Rv′,Rδ-μ-μ′,Rδ-μ′-v.Lemma 2.
Letμ,v,δ,γ∈C, and x>0 be such that R(μ)>0, then the following formulas hold true:(31)I0,xμ,v,δtγ-1x=ΓγΓγ+δ-vΓγ-vΓγ+δ+μxγ-v-1,Rγ>max⁡0,Rv-δ.(32)Ix,∞μ,v,δtγ-1x=Γ1-γ+vΓ1-γ+δΓ1-γΓ1-γ+δ+μ+vxγ-v-1,Rγ<1+min⁡Rv,Rδ,and(33)D0,xμ,v,δtγ-1x=ΓγΓγ+δ+μ+vΓγ+δΓγ+vxγ+v-1,Rγ>-min⁡0,Rμ+v+δ.(34)Dx,∞μ,v,δtγ-1x=Γ1-γ-vΓ1-γ+μ+δΓ1-γΓ1-γ+δ-vxγ+v-1,Rγ<1+min⁡R-v-n,Rδ+μ,n=Rμ+1.Lemma 3.
Letμ,δ,γ∈C, and x>0 be such that R(μ)>0, then the following formulas hold true:(35)I0,xμ,δtγ-1x=Γγ+δΓγ+δ+μxγ-1,Rγ>-Rδ.(36)Ix,∞μ,δtγ-1x=Γ1-γ+δΓ1-γ+δ+μxγ-1,Rγ<1+Rδ,and(37)D0,xμ,δtγ-1x=Γγ+δ+μΓγ+δxγ-1,Rγ>-Rμ+δ.(38)Dx,∞μ,δtγ-1x=Γ1-γ+μ+δΓ1-γ+δxγ-1,Rγ<1+Rμ+δ-n.Lemma 4.
Letμ,γ∈C, and x>0 be such that R(μ)>0, then the following formulas hold true:(39)I0,xμtγ-1x=ΓγΓγ+μxγ+μ-1,Rγ>0.(40)Ix,∞μtγ-1x=Γ1-γ-μΓ1-γxγ+μ-1,0<Rγ<1-Rγ,and(41)D0,xμtγ-1x=ΓγΓγ-μxγ-μ-1,Rγ>Rμ>0.(42)Dx,∞μtγ-1x=Γ1-γ+μΓ1-γxγ-μ-1,Rγ<1+Rμ-n.


## 2. S-Function

The S-function is defined by Saxena and Daiya [24] as(43)Sp,qα,β,ε,τ,ka1,a2,…,ap;b1,b2,…,bq;x=∑n=0∞a1n…apnεnτ,kb1n…bqnΓknα+βxnn!,k∈R,α,β,ε,τ∈C,R(α)>0,ai(i=1,2,3,…,p),bj(i=1,2,3,…,q), R(α)>kR(τ), and p<q+1. The k-Pochhammer symbol and k-gamma function introduced by Diaz and Pariguan [25] are as follows:(44)εn,k=Γkε+nkΓkε,k∈R,ε∈C/0εε+k⋯ε+n-1k,n∈C,ε∈Cand the relation with the classical Euler’s gamma function is as follows:(45)Γkε=kε/k-1Γεkwhere ε∈C,k∈R, and n∈N. For further details of k-Pochhammer symbol and k-special functions one can refer to the papers by Romero et al. [26].Special Cases (1)
Whenp=q=0 in (43), the S-function reduced to generalized k-Mittag-Leffler function, defined by Saxena et al. [27]:(46)S0,0α,β,ε,τ,k-;-;x=∑n=0∞εnτ,kxnΓknα+βn!(47)=Ek,α,βε,τx,Rαk-τ>p-q. (2)
Whenk=τ=1 the S-function reduced to generalized K-function, defined by Sharma [28]:(48)Sp,qα,β,ε,1,1a1,a2,…,ap;b1,b2,…,bq;x=∑n=0∞a1n…apnεnxnb1n…bqnΓnα+βn!(49)=Kp,qα,β,εa1,a2,…,ap;b1,b2,…,bq;x,Rα>p-q. (3)
Whenτ=k=ε=1, the S-function reduced to generalized M-series defined by Sharma and Jain [29]:(50)Sp,qα,β,1,1,1a1,a2,…,ap;b1,b2,…,bq;x=∑n=0∞a1n…apnb1n…bqnΓnα+βxnn!(51)=Mp,qα,βa1,a2,…,ap;b1,b2,…,bq;x,Rα>p-q-1.Here, our aim is to establish composition formula of Marichev-Saigo-Maeda fractional integral and derivative operators of the product of S-function. The main formulas obtained here are represented in terms of the generalized Wright function pψq(z) defined for Z∈C, ai,bj∈C, and Ai,Bj∈R(Ai,Bj≠0; i=1,2,…,p;j=1,2,…,q) which is given by the series(52)pψqz=pψqai,Ai1,pbj,Bj1,qz=∑k=0∞Γa1+A1k…Γap+ApkzkΓb1+B1k…Γbp+Bpkk!,where Γz is the Euler gamma function and the function was introduced by Wright [30] and is known as generalized Wright function. Several theorems are on the asymptotic expansion of pψq(z) for all values of the argument z, under the condition(53)1+∑j=1qBj-∑i=1pAi≥0.For detailed study of various properties, generalization and application of Wright function, and generalized Wright function, we refer to, for instance, [30, 31].ForAi=Bi=1, (52) reduces to the generalized hypergeometric function pFq (see [32]).(54)pFqa1,…,apb1,…,bqz=Γb1…ΓbqΓa1…Γappψqai,11,pbj,11,qz.


## 3. Approach to Fractional Calculus

Throughout this paper, we assume thatμ,μ′,v,v′,δ,γ,αi,βi,εi,τi∈C, ki∈R, pi,qi∈N, and x>0, such that Rξ>0, Rαi>kiR(τi), and pi<qi+1. Further, let the constants satisfy the condition ai,bj∈C, and Ai,Bj∈R(Ai,Bj≠0; i=1,2,…,p;j=1,2,…,q), such that condition (53) is also satisfied.


### 3.1. Left- and Right-Sided Generalized Fractional Integration of S-Function

In this section, we establish image formulas for the product of S-function involving left- and right-sided operators of Marichev-Saigo-Meada fractional integral operators (1) and (3), respectively, in terms of the generalized Wright function. These formulas are given by the following theorems.Theorem 5.
LetR(δ)>0, R(γ)>max⁡0,Rμ+μ′+v-δ,Rμ′-v′, then the generalized fractional integration I0,xμ,μ′,v,v′,δ of the product of S-function is given by(55)I0,xμ,μ′,v,v′,δtγ-1∏i=1rSpi,qiαi,βi,εi,τi,kia1i,…,api;b1i,…,bqi;tx=xγ+δ-μ-μ′-1∏i=1rki1-βi/kiΓb1i…ΓbqiΓεi/kiΓa1i…Γapip+1r+3ψq+1r+3·a11,1…a1r,1,…,ap1,1…apr,1,ε1k1,τ1,…,εrkr,τr,γ,r,γ+δ-μ-μ′-v,r,γ-μ′+v′,rb11,1…b1r,1,…,bq1,1…bqr,1,β1k1,α1k1,…,βrkr,αrkr,γ+v′,r,γ+δ-μ-μ′,r,γ+δ-μ′-v,rk1τ1-α1/k1…krτr-αr/krxr.Proof.
On using (43), writing the function in the series form, the left-hand side of (55) leads to(56)I0,xμ,μ′,v,v′,δtγ-1∏i=1rSpi,qiαi,βi,εi,τi,kia1i,…,api;b1i,…,bqi;tx(57)=I0,xμ,μ′,v,v′,δtγ-1∏i=1r∑n=0∞a1in…apinεinτi,kib1in…bqinΓkinαi+βitnn!rx.By interchanging the order of integration and summation, we reduce the right side of (57) to(58)=∏i=1r∑n=0∞a1in…apinεinτi,kib1in…bqinΓkinαi+βi1n!rI0,xμ,μ′,v,v′,δtγ+nr-1x.By applying Lemma 1 (see (27)) in (58), we get(59)=∏i=1r∑n=0∞a1in…apinεinτi,kib1in…bqinΓkinαi+βi1n!r(60)×Γγ+nrΓγ+δ-μ-μ′-v+nrΓγ+v′-μ′+nrΓγ+v′+nrΓγ+δ-μ-μ′+nrΓγ+δ-μ′-v+nrxγ+nr+δ-μ-μ′-1.By applying (44) and (45) in (60) we get(61)=xγ+δ-μ-μ′-1∏i=1rki1-βi/kiΓb1i…ΓbqiΓεi/kiΓa1i…Γapi×∑n=0∞Γa1i+n⋯Γapi+nΓεi/ki+nτiΓb1i+n⋯Γbqi+nΓnαi/ki+βi/ki1n!r(62)×Γγ+nrΓγ+δ-μ-μ′-v+nrΓγ+v′-μ′+nrΓγ+v′+nrΓγ+δ-μ-μ′+nrΓγ+δ-μ′-v+nrkinτi-αi/kixnrInterpreting the right-hand side of the above equation, in view of definition (52), we arrive at result (55).Theorem 6.
LetR(δ)>0, R(γ)<1+min⁡R-v,Rμ+μ′-δ,Rμ+v′-δ, then the generalized fractional integration Ix,∞μ,μ′,v,v′,δ of the product of S-function is given by(63)Ix,∞μ,μ′,v,v′,δtγ-1∏i=1rSpi,qiαi,βi,εi,τi,kia1i,…,api;b1i,…,bqi;t-1x=xγ+δ-μ-μ′-1∏i=1rki1-βi/kiΓb1i…ΓbqiΓεi/kiΓa1i…Γapip+1r+3ψq+1r+3·a11,1…a1r,1,…,ap1,1…apr,1,ε1k1,τ1,…,εrkr,τr,1-γ-v,r,1-γ-δ+μ+μ′,r,1-γ-δ+μ+v′,rb11,1…b1r,1,…,bq1,1…bqr,1,β1k1,α1k1,…,βrkr,αrkr,1-γ,r,1-γ-δ+μ+μ′+v′,r,1-γ+μ-v,rk1τ1-α1/k1…krτr-αr/krxr.Proof.
The proof of Theorem6 is a similar manner of Theorem 5.


#### 3.1.1. Special Cases

Now, we present some special cases of Theorems5 and 6 given as follows.If we putμ=μ+v,μ′=v′=0,v=-δ,δ=μ, then we obtain the relationship(64)I0,xμ,μ′,v,v′,δx=I0,xμ,v,δfxand (65)Ix,∞μ,μ′,v,v′,δx=Ix,∞μ,v,δfxwhich are defined in (13) and (15) as Saigo fractional integral operator.Corollary 7.
LetR(μ)>0, R(γ)>max⁡0,Rv-δ, then the generalized fractional integration I0,xμ,v,δ of the product of S-function is given by(66)I0,xμ,v,δtγ-1∏i=1rSpi,qiαi,βi,εi,τi,kia1i,…,api;b1i,…,bqi;tx=xγ-v-1∏i=1rki1-βi/kiΓb1i…ΓbqiΓεi/kiΓa1i…Γapi×p+1r+2ψq+1r+2a11,1…a1r,1,…,ap1,1…apr,1,ε1k1,τ1,…,εrkr,τr,γ,r,γ+δ-v,rb11,1…b1r,1,…,bq1,1…bqr,1,β1k1,α1k1,…,βrkr,αrkr,γ-v,r,γ+δ+μ,rk1τ1-α1/k1…krτr-αr/krxr.Corollary 8.
LetR(μ)>0, R(γ)<1+min⁡R(v),Rδ, then the generalized fractional integration Ix,∞μ,v,δ of the product of S-function is given by(67)Ix,∞μ,v,δtγ-1∏i=1rSpi,qiαi,βi,εi,τi,kia1i,…,api;b1i,…,bqi;t-1x=xγ-v-1∏i=1rki1-βi/kiΓb1i…ΓbqiΓεi/kiΓa1i…Γapi×p+1r+2ψq+1r+2a11,1…a1r,1,…,ap1,1…apr,1,ε1k1,τ1,…,εrkr,τr,1-γ+v,r,1-γ+δ,rb11,1…b1r,1,…,bq1,1…bqr,1,β1k1,α1k1,…,βrkr,αrkr,1-γ,r,1-γ+δ+μ+v,rk1τ1-α1/k1…krτr-αr/krxr.Further, if we setv=0, in Corollaries 7 and 8, then Saigo fractional integrals reduce to the following Erdélyi-Kober type fractional integral operators.Corollary 9.
LetR(μ)>0, R(γ)>-Rδ, then the generalized fractional integration I0,xμ,δ of the product of S-function is given by(68)I0,xμ,δtγ-1∏i=1rSpi,qiαi,βi,εi,τi,kia1i,…,api;b1i,…,bqi;tx=xγ-1∏i=1rki1-βi/kiΓb1i…ΓbqiΓεi/kiΓa1i…Γapi×p+1r+1ψq+1r+1a11,1…a1r,1,…,ap1,1…apr,1,ε1k1,τ1,…,εrkr,τr,γ+δ,rb11,1…b1r,1,…,bq1,1…bqr,1,β1k1,α1k1,…,βrkr,αrkr,γ+δ+μ,rk1τ1-α1/k1…krτr-αr/krxr.Corollary 10.
LetR(μ)>0, R(γ)<1+Rδ, then the generalized fractional integration Ix,∞μ,δ of the product of S-function is given by(69)Ix,∞μ,δtγ-1∏i=1rSpi,qiαi,βi,εi,τi,kia1i,…,api;b1i,…,bqi;t-1x=xγ-1∏i=1rki1-βi/kiΓb1i…ΓbqiΓεi/kiΓa1i…Γapi×p+1r+1ψq+1r+1a11,1…a1r,1,…,ap1,1…apr,1,ε1k1,τ1,…,εrkr,τr,1-γ+δ,rb11,1…b1r,1,…,bq1,1…bqr,1,β1k1,α1k1,…,βrkr,αrkr,1-γ+δ+μ,rk1τ1-α1/k1…krτr-αr/krxr.Further, if we setv=-μ, in Corollaries 9 and 10, then Saigo fractional integrals reduce to the Riemann-Liouville and the Weyl type fractional integral operators as the following results.Corollary 11.
LetR(μ)>0, R(γ)>0, then the generalized fractional integration I0,xμ of the product of S-function is given by(70)I0,xμtγ-1∏i=1rSpi,qiαi,βi,εi,τi,kia1i,…,api;b1i,…,bqi;tx=xγ+μ-1∏i=1rki1-βi/kiΓb1i…ΓbqiΓεi/kiΓa1i…Γapi×p+1r+1ψq+1r+1a11,1…a1r,1,…,ap1,1…apr,1,ε1k1,τ1,…,εrkr,τr,γ,rb11,1…b1r,1,…,bq1,1…bqr,1,β1k1,α1k1,…,βrkr,αrkr,γ+μ,rk1τ1-α1/k1…krτr-αr/krxr.Corollary 12.
Let1-R(γ)>Rμ>0, then the generalized fractional integration Ix,∞μ of the product of S-function is given by(71)Ix,∞μtγ-1∏i=1rSpi,qiαi,βi,εi,τi,kia1i,…,api;b1i,…,bqi;t-1x=xγ+μ-1∏i=1rki1-βi/kiΓb1i…ΓbqiΓεi/kiΓa1i…Γapi×p+1r+1ψq+1r+1a11,1…a1r,1,…,ap1,1…apr,1,ε1k1,τ1,…,εrkr,τr,1-γ-μ,rb11,1…b1r,1,…,bq1,1…bqr,1,β1k1,α1k1,…,βrkr,αrkr,1-γ,rk1τ1-α1/k1…krτr-αr/krxr.If we put r=1, then the results in (55), (63), and (66) to (71) reduce to the following form.Corollary 13.
LetR(δ)>0, R(γ)>max⁡0,Rμ+μ′+v-δ,Rμ′-v′, then the generalized fractional integration I0,xμ,μ′,v,v′,δ of the S-function is given by(72)I0,xμ,μ′,v,v′,δtγ-1Spi,qiαi,βi,εi,τi,kia1,…,ap;b1,…,bq;tx=xγ+δ-μ-μ′-1k1-β/kΓb1…ΓbqΓε/kΓa1…Γapp+4ψq+4a1,1…ap,1,εk,τ,γ,1,γ+δ-μ-μ′-v,1,γ-μ′+v′,1b1,1…bq,1,βk,αk,γ+v′,1,γ+δ-μ-μ′,1,γ+δ-μ′-v,1kτ-α/kx.Corollary 14.
LetR(δ)>0, R(γ)<1+min⁡R-v,Rμ+μ′-δ,Rμ+v′-δ, then the generalized fractional integration Ix,∞μ,μ′,v,v′,δ of the S-function is given by(73)Ix,∞μ,μ′,v,v′,δtγ-1Spi,qiαi,βi,εi,τi,kia1,…,ap;b1,…,bq;t-1x=xγ+δ-μ-μ′-1k1-β/kΓb1…ΓbqΓε/kΓa1…Γapp+4ψq+4a1,1…ap,1,εk,τ,1-γ-v,1,1-γ-δ+μ+μ′,1,1-γ-δ+μ+v′,1b1,1…bq,1,βk,αk,1-γ,1,1-γ-δ+μ+μ′+v′,1,1-γ+μ-v,1kτ-α/kx.Corollary 15.
LetR(μ)>0, R(γ)>max⁡0,Rv-δ, then the generalized fractional integration I0,xμ,v,δ of the S-function is given by(74)I0,xμ,v,δtγ-1Spi,qiαi,βi,εi,τi,kia1,…,ap;b1,…,bq;tx=xγ-v-1k1-β/kΓb1…ΓbqΓε/kΓa1…Γap×p+3ψq+3a1,1…ap,1,εk,τ,γ,1,γ+δ-v,1b1,1…bq,1,βk,αk,γ-v,1,γ+δ+μ,1kτ-α/kx.Corollary 16.
LetR(μ)>0, R(γ)<1+min⁡R(v),Rδ, then the generalized fractional integration Ix,∞μ,v,δ of the S-function is given by(75)Ix,∞μ,v,δtγ-1Spi,qiαi,βi,εi,τi,kia1,…,ap;b1,…,bq;t-1x=xγ-v-1k1-β/kΓb1…ΓbqΓε/kΓa1…Γap×p+3ψq+3a1,1…ap,1,εk,τ,1-γ+v,1,1-γ+δ,1b1,1…bq,1,βk,αk,1-γ,1,1-γ+δ+μ+v,1kτ-α/kx.Corollary 17.
LetR(μ)>0, R(γ)>-Rδ, then the generalized fractional integration I0,xμ,δ of the S-function is given by(76)I0,xμ,δtγ-1Spi,qiαi,βi,εi,τi,kia1,…,ap;b1,…,bq;tx=xγ-1k1-β/kΓb1…ΓbqΓε/kΓa1…Γap×p+2ψq+2a1,1…ap,1,εk,τ,γ+δ,1b1,1…bq,1,βk,αk,γ+δ+μ,1kτ-α/kx.Corollary 18.
LetR(μ)>0, R(γ)<1+Rδ, then the generalized fractional integration Ix,∞μ,δ of the S-function is given by(77)Ix,∞μ,δtγ-1Spi,qiαi,βi,εi,τi,kia1,…,ap;b1,…,bq;t-1x=xγ-1k1-β/kΓb1…ΓbqΓε/kΓa1…Γap×p+2ψq+2a1,1…ap,1,εk,τ,1-γ+δ,1b1,1…bq,1,βk,αk,1-γ+δ+μ,1kτ-α/kx.Corollary 19.
LetR(μ)>0, R(γ)>0, then the generalized fractional integration I0,xμ of the S-function is given by(78)I0,xμtγ-1Spi,qiαi,βi,εi,τi,kia1,…,ap;b1,…,bq;tx=xγ-1k1-β/kΓb1…ΓbqΓε/kΓa1…Γap×p+2ψq+2a1,1…ap,1,εk,τ,γ,1b1,1…bq,1,βk,αk,γ+μ,1kτ-α/kx.Corollary 20.
Let1-R(γ)>Rμ>0, then the generalized fractional integration Ix,∞μ of the S-function is given by(79)Ix,∞μtγ-1Spi,qiαi,βi,εi,τi,kia1,…,ap;b1,…,bq;t-1x=xγ+μ-1k1-β/kΓb1…ΓbqΓε/kΓa1…Γap×p+2ψq+2a1,1…ap,1,εk,τ,1-γ-μ,1b1,1…bq,1,βk,αk,1-γ,1kτ-α/kx.


### 3.2. Left- and Right-Sided Generalized Fractional Differentiation of S-Function

In this section, we establish image formulas for the product of S-function involving left- and right-sided operators of Marichev-Saigo-Meada fractional derivative operators (6) and (8), respectively, in terms of the generalized Wright function. These formulas are given by the following theorems.Theorem 21.
LetR(δ)>0, R(γ)>max⁡0,Rδ-μ-μ′-v′,Rv-μ, then the generalized fractional differentiation D0,xμ,μ′,v,v′,δ of the product of S-function is given by(80)D0,xμ,μ′,v,v′,δtγ-1∏i=1rSpi,qiαi,βi,εi,τi,kia1i,…,api;b1i,…,bqi;tx=xγ-δ+μ+μ′-1∏i=1rki1-βi/kiΓb1i…ΓbqiΓεi/kiΓa1i…Γapip+1r+3ψq+1r+3·a11,1…a1r,1,…,ap1,1…apr,1,ε1k1,τ1,…,εrkr,τr,γ,r,γ-δ+μ+μ′+v′,r,γ-v+μ,rb11,1…b1r,1,…,bq1,1…bqr,1,β1k1,α1k1,…,βrkr,αrkr,γ-v,r,γ-δ+μ+μ′,r,γ-δ+μ+v′,rk1τ1-α1/k1…krτr-αr/krxr.Proof.
Using (43) we can rewrite left side of (80) as follows:(81)D0,xμ,μ′,v,v′,δtγ-1∏i=1rSpi,qiαi,βi,εi,τi,kia1i,…,api;b1i,…,bqi;tx(82)=D0,xμ,μ′,v,v′,δtγ-1∏i=1r∑n=0∞a1in…apinεinτi,kib1in…bqinΓkinαi+βitnn!rx.By interchanging the order of differentiation and summation, we reduce the right side of (82) to(83)=∏i=1r∑n=0∞a1in…apinεinτi,kib1in…bqinΓkinαi+βi1n!rD0,xμ,μ′,v,v′,δtγ+nr-1x.By applying Lemma 1 (see (29)) in (83), we obtain(84)=∏i=1r∑n=0∞a1in…apinεinτi,kib1in…bqinΓkinαi+βi1n!r(85)×Γγ+nrΓγ-δ+μ+μ′+v′+nrΓγ-v+μ+nrΓγ-v+nrΓγ-δ+μ+μ′+nrΓγ-δ+μ+v′+nrxγ+nr-δ+μ+μ′-1.By applying (44) and (45) in (85), we get(86)=xγ-δ+μ+μ′-1∏i=1rki1-βi/kiΓb1i…ΓbqiΓεi/kiΓa1i…Γapi×∑n=0∞Γa1i+n…Γapi+nΓεi/ki+nτiΓb1i+n…Γbqi+nΓnαi/ki+βi/ki1n!r(87)×Γγ+nrΓγ-δ+μ+μ′+v′+nrΓγ-v+μ+nrΓγ-v+nrΓγ-δ+μ+μ′+nrΓγ-δ+μ+v′+nrkinτi-αi/kixnr.Interpreting the right-hand side of the above equation, in view of definition (52), we arrive at result (80).Theorem 22.
LetR(δ)>0, R(γ)<1+min⁡Rv′,Rδ-μ-μ′,Rδ-μ′-v, then the generalized fractional differentiation Dx,∞μ,μ′,v,v′,δ of the product of S-function is given by(88)Dx,∞μ,μ′,v,v′,δtγ-1∏i=1rSpi,qiαi,βi,εi,τi,kia1i,…,api;b1i,…,bqi;t-1x=xγ-δ+μ+μ′-1∏i=1rki1-βi/kiΓb1i…ΓbqiΓεi/kiΓa1i…Γapip+1r+3ψq+1r+3·a11,1…a1r,1,…,ap1,1…apr,1,ε1k1,τ1,…,εrkr,τr,1-γ+v′,r,1-γ+δ-μ-μ′,r,1-γ+δ-μ′-v,rb11,1…b1r,1,…,bq1,1…bqr,1,β1k1,α1k1,…,βrkr,αrkr,1-γ,r,1-γ+δ-μ-μ′-v,r,1-γ-μ′+v′,,rk1τ1-α1/k1…krτr-αr/krxr.Proof.
The proof of Theorem22 is a similar manner of Theorem 21.


#### 3.2.1. Special Cases

Now, we present some special cases of Theorems21 and 22 given as follows.If we putμ=μ+v,μ′=v′=0,v=-δ,δ=μ, then we obtain the relationship(89)D0,xμ,μ′,v,v′,δx=D0,xμ,v,δfxand (90)Dx,∞μ,μ′,v,v′,δx=Dx,∞μ,v,δfxwhich are defined in (17) and (18) as Saigo fractional derivative operator.Corollary 23.
LetR(μ)>0, R(γ)>-min⁡0,Rμ+v+δ, then the generalized fractional differentiation D0,xμ,v,δ of the product of S-function is given by(91)D0,xμ,v,δtγ-1∏i=1rSpi,qiαi,βi,εi,τi,kia1i,…,api;b1i,…,bqi;tx=xγ+v-1∏i=1rki1-βi/kiΓb1i…ΓbqiΓεi/kiΓa1i…Γapi×p+1r+2ψq+1r+2a11,1…a1r,1,…,ap1,1…apr,1,ε1k1,τ1,…,εrkr,τr,γ,r,γ+δ+μ+v,rb11,1…b1r,1,…,bq1,1…bqr,1,β1k1,α1k1,…,βrkr,αrkr,γ+v,r,γ+δ,rk1τ1-α1/k1…krτr-αr/krxr.Corollary 24.
LetR(μ)>0, R(γ)<1+min⁡R-v,Rδ+μ, then the generalized fractional differentiation Dx,∞μ,v,δ of the product of S-function is given by(92)Dx,∞μ,v,δtγ-1∏i=1rSpi,qiαi,βi,εi,τi,kia1i,…,api;b1i,…,bqi;t-1x=xγ+v-1∏i=1rki1-βi/kiΓb1i…ΓbqiΓεi/kiΓa1i…Γapi×p+1r+2ψq+1r+2a11,1…a1r,1,…,ap1,1…apr,1,ε1k1,τ1,…,εrkr,τr,1-γ-v,r,1-γ+μ+δ,rb11,1…b1r,1,…,bq1,1…bqr,1,β1k1,α1k1,…,βrkr,αrkr,1-γ,r,1-γ+δ-v,rk1τ1-α1/k1…krτr-αr/krxr.Further, if we setv=0, Corollaries 23 and 24 and then Saigo fractional differential formulas reduce to the following Erdélyi-Kober type fractional differential formulas.Corollary 25.
LetR(μ)>0, R(γ)>-Rδ+μ, then the generalized fractional differentiation D0,xμ,δ of the product of S-function is given by(93)D0,xμ,δtγ-1∏i=1rSpi,qiαi,βi,εi,τi,kia1i,…,api;b1i,…,bqi;tx=xγ-1∏i=1rki1-βi/kiΓb1i…ΓbqiΓεi/kiΓa1i…Γapi×p+1r+1ψq+1r+1a11,1…a1r,1,…,ap1,1…apr,1,ε1k1,τ1,…,εrkr,τr,γ+δ+μ,rb11,1…b1r,1,…,bq1,1…bqr,1,β1k1,α1k1,…,βrkr,αrkr,γ+δ,rk1τ1-α1/k1…krτr-αr/krxr.Corollary 26.
LetR(μ)>0, R(γ)<1+Rδ+μ, then the generalized fractional differentiation Dx,∞μ,δ of the product of S-function is given by(94)Dx,∞μ,δtγ-1∏i=1rSpi,qiαi,βi,εi,τi,kia1i,…,api;b1i,…,bqi;t-1x=xγ-1∏i=1rki1-βi/kiΓb1i…ΓbqiΓεi/kiΓa1i…Γapi×p+1r+1ψq+1r+1a11,1…a1r,1,…,ap1,1…apr,1,ε1k1,τ1,…,εrkr,τr,1-γ+μ+δ,rb11,1…b1r,1,…,bq1,1…bqr,1,β1k1,α1k1,…,βrkr,αrkr,1-γ+δ,rk1τ1-α1/k1…krτr-αr/krxr.Further, if we setv=-μ, in Corollaries 9 and 10 then Saigo fractional derivative formulas reduce to the Riemann-Liouville and the Weyl type fractional derivative formulas as the following results.Corollary 27.
LetR(μ)>0, R(γ)>0, and then the generalized fractional differentiation D0,xμ of the product of S-function is given by(95)D0,xμtγ-1∏i=1rSpi,qiαi,βi,εi,τi,kia1i,…,api;b1i,…,bqi;tx=xγ-μ-1∏i=1rki1-βi/kiΓb1i…ΓbqiΓεi/kiΓa1i…Γapi×p+1r+1ψq+1r+1a11,1…a1r,1,…,ap1,1…apr,1,ε1k1,τ1,…,εrkr,τr,γ,rb11,1…b1r,1,…,bq1,1…bqr,1,β1k1,α1k1,…,βrkr,αrkr,γ-μ,rk1τ1-α1/k1…krτr-αr/krxr.Corollary 28.
LetR(γ)<1+Rμ, then the generalized fractional differentiation Dx,∞μ of the product of S-function is given by(96)Dx,∞μtγ-1∏i=1rSpi,qiαi,βi,εi,τi,kia1i,…,api;b1i,…,bqi;t-1x=xγ-μ-1∏i=1rki1-βi/kiΓb1i…ΓbqiΓεi/kiΓa1i…Γapi×p+1r+1ψq+1r+1a11,1…a1r,1,…,ap1,1…apr,1,ε1k1,τ1,…,εrkr,τr,1-γ+μ,rb11,1…b1r,1,…,bq1,1…bqr,1,β1k1,α1k1,…,βrkr,αrkr,1-γ,rk1τ1-α1/k1…krτr-αr/krxr.If we putr=1, then the results in (80), (88), and (91) to (96) reduce to the following form.Corollary 29.
LetR(δ)>0, R(γ)>max⁡0,Rδ-μ-μ′-v′,Rv-μ, then the generalized fractional differentiation D0,xμ,μ′,v,v′,δ of the S-function is given by(97)D0,xμ,μ′,v,v′,δtγ-1Spi,qiαi,βi,εi,τi,kia1,…,ap;b1,…,bq;tx=xγ-δ+μ+μ′-1k1-β/kΓb1…ΓbqΓε/kΓa1…Γapp+4ψq+4a1,1…ap,1,εk,τ,γ,1,γ-δ+μ+μ′+v′,1,γ-v+μ,1b1,1…bq,1,βk,αk,γ-v,1,γ-δ+μ+μ′,1,γ-δ+μ+v′,1kτ-α/kx.Corollary 30.
LetR(δ)>0, R(γ)<1+min⁡Rv′,Rδ-μ-μ′,Rδ-μ′-v, then the generalized fractional differentiation Dx,∞μ,μ′,v,v′,δ of the S-function is given by(98)Dx,∞μ,μ′,v,v′,δtγ-1Spi,qiαi,βi,εi,τi,kia1,…,ap;b1,…,bq;t-1x=xγ-δ+μ+μ′-1k1-β/kΓb1…ΓbqΓε/kΓa1…Γapp+4ψq+4a1,1…ap,1,εk,τ,1-γ+v′,1,1-γ+δ-μ-μ′,1,1-γ+δ-μ′-v,1b1,1…bq,1,βk,αk,1-γ,1,1-γ+δ-μ-μ′-v,1,1-γ-μ′+v′,,1kτ-α/kx.Corollary 31.
LetR(μ)>0, R(γ)>-min⁡0,Rμ+v+δ, then the generalized fractional differentiation D0,xμ,v,δ of the S-function is given by(99)D0,xμ,v,δtγ-1Spi,qiαi,βi,εi,τi,kia1,…,ap;b1,…,bq;tx=xγ+v-1k1-β/kΓb1…ΓbqΓε/kΓa1…Γap×p+3ψq+3a1,1…ap,1,εk,τ,γ,1,γ+δ+μ+v,1b1,1…bq,1,βk,αk,γ+v,1,γ+δ,1kτ-α/kx.Corollary 32.
LetR(μ)>0, R(γ)<1+min⁡R-v,Rδ+μ, then the generalized fractional differentiation Dx,∞μ,v,δ of the S-function is given by(100)Dx,∞μ,v,δtγ-1Spi,qiαi,βi,εi,τi,kia1,…,ap;b1,…,bq;t-1x=xγ+v-1k1-β/kΓb1…ΓbqΓε/kΓa1…Γap×p+3ψq+3a1,1…ap,1,εk,τ,1-γ-v,1,1-γ+μ+δ,1b1,1…bq,1,βk,αk,1-γ,1,1-γ+δ-v,1kτ-α/kx.Corollary 33.
LetR(μ)>0, R(γ)>-Rδ+μ, then the generalized fractional differentiation D0,xμ,δ of the S-function is given by(101)D0,xμ,δtγ-1Spi,qiαi,βi,εi,τi,kia1,…,ap;b1,…,bq;tx=xγ-1k1-β/kΓb1…ΓbqΓε/kΓa1…Γap×p+2ψq+2a1,1…ap,1,εk,τ,γ+δ+μ,1b1,1…bq,1,βk,αk,γ+δ,1kτ-α/kx.Corollary 34.
LetR(μ)>0, R(γ)<1+Rδ+μ, then the generalized fractional differentiation Dx,∞μ,δ of the S-function is given by(102)Dx,∞μ,δtγ-1Spi,qiαi,βi,εi,τi,kia1,…,ap;b1,…,bq;t-1x=xγ-1k1-β/kΓb1…ΓbqΓε/kΓa1…Γap×p+2ψq+2a1,1…ap,1,εk,τ,1-γ+μ+δ,1b1,1…bq,1,βk,αk,1-γ+δ,1kτ-α/kx.Corollary 35.
LetR(μ)>0, R(γ)>0, and then the generalized fractional differentiation D0,xμ of the S-function is given by(103)D0,xμtγ-1Spi,qiαi,βi,εi,τi,kia1,…,ap;b1,…,bq;tx=xγ-μ-1k1-β/kΓb1…ΓbqΓε/kΓa1…Γap×p+2ψq+2a1,1…ap,1,εk,τ,γ,1b1,1…bq,1,βk,αk,γ-μ,1kτ-α/kxCorollary 36.
LetR(γ)<1+Rμ, then the generalized fractional differentiation Dx,∞μ of the S-function is given by(104)Dx,∞μtγ-1Spi,qiαi,βi,εi,τi,kia1,…,ap;b1,…,bq;t-1x=xγ-μ-1k1-β/kΓb1…ΓbqΓε/kΓa1…Γap×p+2ψq+2a1,1…ap,1,εk,τ,1-γ+μ,1b1,1…bq,1,βk,αk,1-γ,1kτ-α/kx.


## 3.1. Left- and Right-Sided Generalized Fractional Integration of S-Function

In this section, we establish image formulas for the product of S-function involving left- and right-sided operators of Marichev-Saigo-Meada fractional integral operators (1) and (3), respectively, in terms of the generalized Wright function. These formulas are given by the following theorems.Theorem 5.
LetR(δ)>0, R(γ)>max⁡0,Rμ+μ′+v-δ,Rμ′-v′, then the generalized fractional integration I0,xμ,μ′,v,v′,δ of the product of S-function is given by(55)I0,xμ,μ′,v,v′,δtγ-1∏i=1rSpi,qiαi,βi,εi,τi,kia1i,…,api;b1i,…,bqi;tx=xγ+δ-μ-μ′-1∏i=1rki1-βi/kiΓb1i…ΓbqiΓεi/kiΓa1i…Γapip+1r+3ψq+1r+3·a11,1…a1r,1,…,ap1,1…apr,1,ε1k1,τ1,…,εrkr,τr,γ,r,γ+δ-μ-μ′-v,r,γ-μ′+v′,rb11,1…b1r,1,…,bq1,1…bqr,1,β1k1,α1k1,…,βrkr,αrkr,γ+v′,r,γ+δ-μ-μ′,r,γ+δ-μ′-v,rk1τ1-α1/k1…krτr-αr/krxr.Proof.
On using (43), writing the function in the series form, the left-hand side of (55) leads to(56)I0,xμ,μ′,v,v′,δtγ-1∏i=1rSpi,qiαi,βi,εi,τi,kia1i,…,api;b1i,…,bqi;tx(57)=I0,xμ,μ′,v,v′,δtγ-1∏i=1r∑n=0∞a1in…apinεinτi,kib1in…bqinΓkinαi+βitnn!rx.By interchanging the order of integration and summation, we reduce the right side of (57) to(58)=∏i=1r∑n=0∞a1in…apinεinτi,kib1in…bqinΓkinαi+βi1n!rI0,xμ,μ′,v,v′,δtγ+nr-1x.By applying Lemma 1 (see (27)) in (58), we get(59)=∏i=1r∑n=0∞a1in…apinεinτi,kib1in…bqinΓkinαi+βi1n!r(60)×Γγ+nrΓγ+δ-μ-μ′-v+nrΓγ+v′-μ′+nrΓγ+v′+nrΓγ+δ-μ-μ′+nrΓγ+δ-μ′-v+nrxγ+nr+δ-μ-μ′-1.By applying (44) and (45) in (60) we get(61)=xγ+δ-μ-μ′-1∏i=1rki1-βi/kiΓb1i…ΓbqiΓεi/kiΓa1i…Γapi×∑n=0∞Γa1i+n⋯Γapi+nΓεi/ki+nτiΓb1i+n⋯Γbqi+nΓnαi/ki+βi/ki1n!r(62)×Γγ+nrΓγ+δ-μ-μ′-v+nrΓγ+v′-μ′+nrΓγ+v′+nrΓγ+δ-μ-μ′+nrΓγ+δ-μ′-v+nrkinτi-αi/kixnrInterpreting the right-hand side of the above equation, in view of definition (52), we arrive at result (55).Theorem 6.
LetR(δ)>0, R(γ)<1+min⁡R-v,Rμ+μ′-δ,Rμ+v′-δ, then the generalized fractional integration Ix,∞μ,μ′,v,v′,δ of the product of S-function is given by(63)Ix,∞μ,μ′,v,v′,δtγ-1∏i=1rSpi,qiαi,βi,εi,τi,kia1i,…,api;b1i,…,bqi;t-1x=xγ+δ-μ-μ′-1∏i=1rki1-βi/kiΓb1i…ΓbqiΓεi/kiΓa1i…Γapip+1r+3ψq+1r+3·a11,1…a1r,1,…,ap1,1…apr,1,ε1k1,τ1,…,εrkr,τr,1-γ-v,r,1-γ-δ+μ+μ′,r,1-γ-δ+μ+v′,rb11,1…b1r,1,…,bq1,1…bqr,1,β1k1,α1k1,…,βrkr,αrkr,1-γ,r,1-γ-δ+μ+μ′+v′,r,1-γ+μ-v,rk1τ1-α1/k1…krτr-αr/krxr.Proof.
The proof of Theorem6 is a similar manner of Theorem 5.


### 3.1.1. Special Cases

Now, we present some special cases of Theorems5 and 6 given as follows.If we putμ=μ+v,μ′=v′=0,v=-δ,δ=μ, then we obtain the relationship(64)I0,xμ,μ′,v,v′,δx=I0,xμ,v,δfxand (65)Ix,∞μ,μ′,v,v′,δx=Ix,∞μ,v,δfxwhich are defined in (13) and (15) as Saigo fractional integral operator.Corollary 7.
LetR(μ)>0, R(γ)>max⁡0,Rv-δ, then the generalized fractional integration I0,xμ,v,δ of the product of S-function is given by(66)I0,xμ,v,δtγ-1∏i=1rSpi,qiαi,βi,εi,τi,kia1i,…,api;b1i,…,bqi;tx=xγ-v-1∏i=1rki1-βi/kiΓb1i…ΓbqiΓεi/kiΓa1i…Γapi×p+1r+2ψq+1r+2a11,1…a1r,1,…,ap1,1…apr,1,ε1k1,τ1,…,εrkr,τr,γ,r,γ+δ-v,rb11,1…b1r,1,…,bq1,1…bqr,1,β1k1,α1k1,…,βrkr,αrkr,γ-v,r,γ+δ+μ,rk1τ1-α1/k1…krτr-αr/krxr.Corollary 8.
LetR(μ)>0, R(γ)<1+min⁡R(v),Rδ, then the generalized fractional integration Ix,∞μ,v,δ of the product of S-function is given by(67)Ix,∞μ,v,δtγ-1∏i=1rSpi,qiαi,βi,εi,τi,kia1i,…,api;b1i,…,bqi;t-1x=xγ-v-1∏i=1rki1-βi/kiΓb1i…ΓbqiΓεi/kiΓa1i…Γapi×p+1r+2ψq+1r+2a11,1…a1r,1,…,ap1,1…apr,1,ε1k1,τ1,…,εrkr,τr,1-γ+v,r,1-γ+δ,rb11,1…b1r,1,…,bq1,1…bqr,1,β1k1,α1k1,…,βrkr,αrkr,1-γ,r,1-γ+δ+μ+v,rk1τ1-α1/k1…krτr-αr/krxr.Further, if we setv=0, in Corollaries 7 and 8, then Saigo fractional integrals reduce to the following Erdélyi-Kober type fractional integral operators.Corollary 9.
LetR(μ)>0, R(γ)>-Rδ, then the generalized fractional integration I0,xμ,δ of the product of S-function is given by(68)I0,xμ,δtγ-1∏i=1rSpi,qiαi,βi,εi,τi,kia1i,…,api;b1i,…,bqi;tx=xγ-1∏i=1rki1-βi/kiΓb1i…ΓbqiΓεi/kiΓa1i…Γapi×p+1r+1ψq+1r+1a11,1…a1r,1,…,ap1,1…apr,1,ε1k1,τ1,…,εrkr,τr,γ+δ,rb11,1…b1r,1,…,bq1,1…bqr,1,β1k1,α1k1,…,βrkr,αrkr,γ+δ+μ,rk1τ1-α1/k1…krτr-αr/krxr.Corollary 10.
LetR(μ)>0, R(γ)<1+Rδ, then the generalized fractional integration Ix,∞μ,δ of the product of S-function is given by(69)Ix,∞μ,δtγ-1∏i=1rSpi,qiαi,βi,εi,τi,kia1i,…,api;b1i,…,bqi;t-1x=xγ-1∏i=1rki1-βi/kiΓb1i…ΓbqiΓεi/kiΓa1i…Γapi×p+1r+1ψq+1r+1a11,1…a1r,1,…,ap1,1…apr,1,ε1k1,τ1,…,εrkr,τr,1-γ+δ,rb11,1…b1r,1,…,bq1,1…bqr,1,β1k1,α1k1,…,βrkr,αrkr,1-γ+δ+μ,rk1τ1-α1/k1…krτr-αr/krxr.Further, if we setv=-μ, in Corollaries 9 and 10, then Saigo fractional integrals reduce to the Riemann-Liouville and the Weyl type fractional integral operators as the following results.Corollary 11.
LetR(μ)>0, R(γ)>0, then the generalized fractional integration I0,xμ of the product of S-function is given by(70)I0,xμtγ-1∏i=1rSpi,qiαi,βi,εi,τi,kia1i,…,api;b1i,…,bqi;tx=xγ+μ-1∏i=1rki1-βi/kiΓb1i…ΓbqiΓεi/kiΓa1i…Γapi×p+1r+1ψq+1r+1a11,1…a1r,1,…,ap1,1…apr,1,ε1k1,τ1,…,εrkr,τr,γ,rb11,1…b1r,1,…,bq1,1…bqr,1,β1k1,α1k1,…,βrkr,αrkr,γ+μ,rk1τ1-α1/k1…krτr-αr/krxr.Corollary 12.
Let1-R(γ)>Rμ>0, then the generalized fractional integration Ix,∞μ of the product of S-function is given by(71)Ix,∞μtγ-1∏i=1rSpi,qiαi,βi,εi,τi,kia1i,…,api;b1i,…,bqi;t-1x=xγ+μ-1∏i=1rki1-βi/kiΓb1i…ΓbqiΓεi/kiΓa1i…Γapi×p+1r+1ψq+1r+1a11,1…a1r,1,…,ap1,1…apr,1,ε1k1,τ1,…,εrkr,τr,1-γ-μ,rb11,1…b1r,1,…,bq1,1…bqr,1,β1k1,α1k1,…,βrkr,αrkr,1-γ,rk1τ1-α1/k1…krτr-αr/krxr.If we put r=1, then the results in (55), (63), and (66) to (71) reduce to the following form.Corollary 13.
LetR(δ)>0, R(γ)>max⁡0,Rμ+μ′+v-δ,Rμ′-v′, then the generalized fractional integration I0,xμ,μ′,v,v′,δ of the S-function is given by(72)I0,xμ,μ′,v,v′,δtγ-1Spi,qiαi,βi,εi,τi,kia1,…,ap;b1,…,bq;tx=xγ+δ-μ-μ′-1k1-β/kΓb1…ΓbqΓε/kΓa1…Γapp+4ψq+4a1,1…ap,1,εk,τ,γ,1,γ+δ-μ-μ′-v,1,γ-μ′+v′,1b1,1…bq,1,βk,αk,γ+v′,1,γ+δ-μ-μ′,1,γ+δ-μ′-v,1kτ-α/kx.Corollary 14.
LetR(δ)>0, R(γ)<1+min⁡R-v,Rμ+μ′-δ,Rμ+v′-δ, then the generalized fractional integration Ix,∞μ,μ′,v,v′,δ of the S-function is given by(73)Ix,∞μ,μ′,v,v′,δtγ-1Spi,qiαi,βi,εi,τi,kia1,…,ap;b1,…,bq;t-1x=xγ+δ-μ-μ′-1k1-β/kΓb1…ΓbqΓε/kΓa1…Γapp+4ψq+4a1,1…ap,1,εk,τ,1-γ-v,1,1-γ-δ+μ+μ′,1,1-γ-δ+μ+v′,1b1,1…bq,1,βk,αk,1-γ,1,1-γ-δ+μ+μ′+v′,1,1-γ+μ-v,1kτ-α/kx.Corollary 15.
LetR(μ)>0, R(γ)>max⁡0,Rv-δ, then the generalized fractional integration I0,xμ,v,δ of the S-function is given by(74)I0,xμ,v,δtγ-1Spi,qiαi,βi,εi,τi,kia1,…,ap;b1,…,bq;tx=xγ-v-1k1-β/kΓb1…ΓbqΓε/kΓa1…Γap×p+3ψq+3a1,1…ap,1,εk,τ,γ,1,γ+δ-v,1b1,1…bq,1,βk,αk,γ-v,1,γ+δ+μ,1kτ-α/kx.Corollary 16.
LetR(μ)>0, R(γ)<1+min⁡R(v),Rδ, then the generalized fractional integration Ix,∞μ,v,δ of the S-function is given by(75)Ix,∞μ,v,δtγ-1Spi,qiαi,βi,εi,τi,kia1,…,ap;b1,…,bq;t-1x=xγ-v-1k1-β/kΓb1…ΓbqΓε/kΓa1…Γap×p+3ψq+3a1,1…ap,1,εk,τ,1-γ+v,1,1-γ+δ,1b1,1…bq,1,βk,αk,1-γ,1,1-γ+δ+μ+v,1kτ-α/kx.Corollary 17.
LetR(μ)>0, R(γ)>-Rδ, then the generalized fractional integration I0,xμ,δ of the S-function is given by(76)I0,xμ,δtγ-1Spi,qiαi,βi,εi,τi,kia1,…,ap;b1,…,bq;tx=xγ-1k1-β/kΓb1…ΓbqΓε/kΓa1…Γap×p+2ψq+2a1,1…ap,1,εk,τ,γ+δ,1b1,1…bq,1,βk,αk,γ+δ+μ,1kτ-α/kx.Corollary 18.
LetR(μ)>0, R(γ)<1+Rδ, then the generalized fractional integration Ix,∞μ,δ of the S-function is given by(77)Ix,∞μ,δtγ-1Spi,qiαi,βi,εi,τi,kia1,…,ap;b1,…,bq;t-1x=xγ-1k1-β/kΓb1…ΓbqΓε/kΓa1…Γap×p+2ψq+2a1,1…ap,1,εk,τ,1-γ+δ,1b1,1…bq,1,βk,αk,1-γ+δ+μ,1kτ-α/kx.Corollary 19.
LetR(μ)>0, R(γ)>0, then the generalized fractional integration I0,xμ of the S-function is given by(78)I0,xμtγ-1Spi,qiαi,βi,εi,τi,kia1,…,ap;b1,…,bq;tx=xγ-1k1-β/kΓb1…ΓbqΓε/kΓa1…Γap×p+2ψq+2a1,1…ap,1,εk,τ,γ,1b1,1…bq,1,βk,αk,γ+μ,1kτ-α/kx.Corollary 20.
Let1-R(γ)>Rμ>0, then the generalized fractional integration Ix,∞μ of the S-function is given by(79)Ix,∞μtγ-1Spi,qiαi,βi,εi,τi,kia1,…,ap;b1,…,bq;t-1x=xγ+μ-1k1-β/kΓb1…ΓbqΓε/kΓa1…Γap×p+2ψq+2a1,1…ap,1,εk,τ,1-γ-μ,1b1,1…bq,1,βk,αk,1-γ,1kτ-α/kx.


## 3.1.1. Special Cases

Now, we present some special cases of Theorems5 and 6 given as follows.If we putμ=μ+v,μ′=v′=0,v=-δ,δ=μ, then we obtain the relationship(64)I0,xμ,μ′,v,v′,δx=I0,xμ,v,δfxand (65)Ix,∞μ,μ′,v,v′,δx=Ix,∞μ,v,δfxwhich are defined in (13) and (15) as Saigo fractional integral operator.Corollary 7.
LetR(μ)>0, R(γ)>max⁡0,Rv-δ, then the generalized fractional integration I0,xμ,v,δ of the product of S-function is given by(66)I0,xμ,v,δtγ-1∏i=1rSpi,qiαi,βi,εi,τi,kia1i,…,api;b1i,…,bqi;tx=xγ-v-1∏i=1rki1-βi/kiΓb1i…ΓbqiΓεi/kiΓa1i…Γapi×p+1r+2ψq+1r+2a11,1…a1r,1,…,ap1,1…apr,1,ε1k1,τ1,…,εrkr,τr,γ,r,γ+δ-v,rb11,1…b1r,1,…,bq1,1…bqr,1,β1k1,α1k1,…,βrkr,αrkr,γ-v,r,γ+δ+μ,rk1τ1-α1/k1…krτr-αr/krxr.Corollary 8.
LetR(μ)>0, R(γ)<1+min⁡R(v),Rδ, then the generalized fractional integration Ix,∞μ,v,δ of the product of S-function is given by(67)Ix,∞μ,v,δtγ-1∏i=1rSpi,qiαi,βi,εi,τi,kia1i,…,api;b1i,…,bqi;t-1x=xγ-v-1∏i=1rki1-βi/kiΓb1i…ΓbqiΓεi/kiΓa1i…Γapi×p+1r+2ψq+1r+2a11,1…a1r,1,…,ap1,1…apr,1,ε1k1,τ1,…,εrkr,τr,1-γ+v,r,1-γ+δ,rb11,1…b1r,1,…,bq1,1…bqr,1,β1k1,α1k1,…,βrkr,αrkr,1-γ,r,1-γ+δ+μ+v,rk1τ1-α1/k1…krτr-αr/krxr.Further, if we setv=0, in Corollaries 7 and 8, then Saigo fractional integrals reduce to the following Erdélyi-Kober type fractional integral operators.Corollary 9.
LetR(μ)>0, R(γ)>-Rδ, then the generalized fractional integration I0,xμ,δ of the product of S-function is given by(68)I0,xμ,δtγ-1∏i=1rSpi,qiαi,βi,εi,τi,kia1i,…,api;b1i,…,bqi;tx=xγ-1∏i=1rki1-βi/kiΓb1i…ΓbqiΓεi/kiΓa1i…Γapi×p+1r+1ψq+1r+1a11,1…a1r,1,…,ap1,1…apr,1,ε1k1,τ1,…,εrkr,τr,γ+δ,rb11,1…b1r,1,…,bq1,1…bqr,1,β1k1,α1k1,…,βrkr,αrkr,γ+δ+μ,rk1τ1-α1/k1…krτr-αr/krxr.Corollary 10.
LetR(μ)>0, R(γ)<1+Rδ, then the generalized fractional integration Ix,∞μ,δ of the product of S-function is given by(69)Ix,∞μ,δtγ-1∏i=1rSpi,qiαi,βi,εi,τi,kia1i,…,api;b1i,…,bqi;t-1x=xγ-1∏i=1rki1-βi/kiΓb1i…ΓbqiΓεi/kiΓa1i…Γapi×p+1r+1ψq+1r+1a11,1…a1r,1,…,ap1,1…apr,1,ε1k1,τ1,…,εrkr,τr,1-γ+δ,rb11,1…b1r,1,…,bq1,1…bqr,1,β1k1,α1k1,…,βrkr,αrkr,1-γ+δ+μ,rk1τ1-α1/k1…krτr-αr/krxr.Further, if we setv=-μ, in Corollaries 9 and 10, then Saigo fractional integrals reduce to the Riemann-Liouville and the Weyl type fractional integral operators as the following results.Corollary 11.
LetR(μ)>0, R(γ)>0, then the generalized fractional integration I0,xμ of the product of S-function is given by(70)I0,xμtγ-1∏i=1rSpi,qiαi,βi,εi,τi,kia1i,…,api;b1i,…,bqi;tx=xγ+μ-1∏i=1rki1-βi/kiΓb1i…ΓbqiΓεi/kiΓa1i…Γapi×p+1r+1ψq+1r+1a11,1…a1r,1,…,ap1,1…apr,1,ε1k1,τ1,…,εrkr,τr,γ,rb11,1…b1r,1,…,bq1,1…bqr,1,β1k1,α1k1,…,βrkr,αrkr,γ+μ,rk1τ1-α1/k1…krτr-αr/krxr.Corollary 12.
Let1-R(γ)>Rμ>0, then the generalized fractional integration Ix,∞μ of the product of S-function is given by(71)Ix,∞μtγ-1∏i=1rSpi,qiαi,βi,εi,τi,kia1i,…,api;b1i,…,bqi;t-1x=xγ+μ-1∏i=1rki1-βi/kiΓb1i…ΓbqiΓεi/kiΓa1i…Γapi×p+1r+1ψq+1r+1a11,1…a1r,1,…,ap1,1…apr,1,ε1k1,τ1,…,εrkr,τr,1-γ-μ,rb11,1…b1r,1,…,bq1,1…bqr,1,β1k1,α1k1,…,βrkr,αrkr,1-γ,rk1τ1-α1/k1…krτr-αr/krxr.If we put r=1, then the results in (55), (63), and (66) to (71) reduce to the following form.Corollary 13.
LetR(δ)>0, R(γ)>max⁡0,Rμ+μ′+v-δ,Rμ′-v′, then the generalized fractional integration I0,xμ,μ′,v,v′,δ of the S-function is given by(72)I0,xμ,μ′,v,v′,δtγ-1Spi,qiαi,βi,εi,τi,kia1,…,ap;b1,…,bq;tx=xγ+δ-μ-μ′-1k1-β/kΓb1…ΓbqΓε/kΓa1…Γapp+4ψq+4a1,1…ap,1,εk,τ,γ,1,γ+δ-μ-μ′-v,1,γ-μ′+v′,1b1,1…bq,1,βk,αk,γ+v′,1,γ+δ-μ-μ′,1,γ+δ-μ′-v,1kτ-α/kx.Corollary 14.
LetR(δ)>0, R(γ)<1+min⁡R-v,Rμ+μ′-δ,Rμ+v′-δ, then the generalized fractional integration Ix,∞μ,μ′,v,v′,δ of the S-function is given by(73)Ix,∞μ,μ′,v,v′,δtγ-1Spi,qiαi,βi,εi,τi,kia1,…,ap;b1,…,bq;t-1x=xγ+δ-μ-μ′-1k1-β/kΓb1…ΓbqΓε/kΓa1…Γapp+4ψq+4a1,1…ap,1,εk,τ,1-γ-v,1,1-γ-δ+μ+μ′,1,1-γ-δ+μ+v′,1b1,1…bq,1,βk,αk,1-γ,1,1-γ-δ+μ+μ′+v′,1,1-γ+μ-v,1kτ-α/kx.Corollary 15.
LetR(μ)>0, R(γ)>max⁡0,Rv-δ, then the generalized fractional integration I0,xμ,v,δ of the S-function is given by(74)I0,xμ,v,δtγ-1Spi,qiαi,βi,εi,τi,kia1,…,ap;b1,…,bq;tx=xγ-v-1k1-β/kΓb1…ΓbqΓε/kΓa1…Γap×p+3ψq+3a1,1…ap,1,εk,τ,γ,1,γ+δ-v,1b1,1…bq,1,βk,αk,γ-v,1,γ+δ+μ,1kτ-α/kx.Corollary 16.
LetR(μ)>0, R(γ)<1+min⁡R(v),Rδ, then the generalized fractional integration Ix,∞μ,v,δ of the S-function is given by(75)Ix,∞μ,v,δtγ-1Spi,qiαi,βi,εi,τi,kia1,…,ap;b1,…,bq;t-1x=xγ-v-1k1-β/kΓb1…ΓbqΓε/kΓa1…Γap×p+3ψq+3a1,1…ap,1,εk,τ,1-γ+v,1,1-γ+δ,1b1,1…bq,1,βk,αk,1-γ,1,1-γ+δ+μ+v,1kτ-α/kx.Corollary 17.
LetR(μ)>0, R(γ)>-Rδ, then the generalized fractional integration I0,xμ,δ of the S-function is given by(76)I0,xμ,δtγ-1Spi,qiαi,βi,εi,τi,kia1,…,ap;b1,…,bq;tx=xγ-1k1-β/kΓb1…ΓbqΓε/kΓa1…Γap×p+2ψq+2a1,1…ap,1,εk,τ,γ+δ,1b1,1…bq,1,βk,αk,γ+δ+μ,1kτ-α/kx.Corollary 18.
LetR(μ)>0, R(γ)<1+Rδ, then the generalized fractional integration Ix,∞μ,δ of the S-function is given by(77)Ix,∞μ,δtγ-1Spi,qiαi,βi,εi,τi,kia1,…,ap;b1,…,bq;t-1x=xγ-1k1-β/kΓb1…ΓbqΓε/kΓa1…Γap×p+2ψq+2a1,1…ap,1,εk,τ,1-γ+δ,1b1,1…bq,1,βk,αk,1-γ+δ+μ,1kτ-α/kx.Corollary 19.
LetR(μ)>0, R(γ)>0, then the generalized fractional integration I0,xμ of the S-function is given by(78)I0,xμtγ-1Spi,qiαi,βi,εi,τi,kia1,…,ap;b1,…,bq;tx=xγ-1k1-β/kΓb1…ΓbqΓε/kΓa1…Γap×p+2ψq+2a1,1…ap,1,εk,τ,γ,1b1,1…bq,1,βk,αk,γ+μ,1kτ-α/kx.Corollary 20.
Let1-R(γ)>Rμ>0, then the generalized fractional integration Ix,∞μ of the S-function is given by(79)Ix,∞μtγ-1Spi,qiαi,βi,εi,τi,kia1,…,ap;b1,…,bq;t-1x=xγ+μ-1k1-β/kΓb1…ΓbqΓε/kΓa1…Γap×p+2ψq+2a1,1…ap,1,εk,τ,1-γ-μ,1b1,1…bq,1,βk,αk,1-γ,1kτ-α/kx.


## 3.2. Left- and Right-Sided Generalized Fractional Differentiation of S-Function

In this section, we establish image formulas for the product of S-function involving left- and right-sided operators of Marichev-Saigo-Meada fractional derivative operators (6) and (8), respectively, in terms of the generalized Wright function. These formulas are given by the following theorems.Theorem 21.
LetR(δ)>0, R(γ)>max⁡0,Rδ-μ-μ′-v′,Rv-μ, then the generalized fractional differentiation D0,xμ,μ′,v,v′,δ of the product of S-function is given by(80)D0,xμ,μ′,v,v′,δtγ-1∏i=1rSpi,qiαi,βi,εi,τi,kia1i,…,api;b1i,…,bqi;tx=xγ-δ+μ+μ′-1∏i=1rki1-βi/kiΓb1i…ΓbqiΓεi/kiΓa1i…Γapip+1r+3ψq+1r+3·a11,1…a1r,1,…,ap1,1…apr,1,ε1k1,τ1,…,εrkr,τr,γ,r,γ-δ+μ+μ′+v′,r,γ-v+μ,rb11,1…b1r,1,…,bq1,1…bqr,1,β1k1,α1k1,…,βrkr,αrkr,γ-v,r,γ-δ+μ+μ′,r,γ-δ+μ+v′,rk1τ1-α1/k1…krτr-αr/krxr.Proof.
Using (43) we can rewrite left side of (80) as follows:(81)D0,xμ,μ′,v,v′,δtγ-1∏i=1rSpi,qiαi,βi,εi,τi,kia1i,…,api;b1i,…,bqi;tx(82)=D0,xμ,μ′,v,v′,δtγ-1∏i=1r∑n=0∞a1in…apinεinτi,kib1in…bqinΓkinαi+βitnn!rx.By interchanging the order of differentiation and summation, we reduce the right side of (82) to(83)=∏i=1r∑n=0∞a1in…apinεinτi,kib1in…bqinΓkinαi+βi1n!rD0,xμ,μ′,v,v′,δtγ+nr-1x.By applying Lemma 1 (see (29)) in (83), we obtain(84)=∏i=1r∑n=0∞a1in…apinεinτi,kib1in…bqinΓkinαi+βi1n!r(85)×Γγ+nrΓγ-δ+μ+μ′+v′+nrΓγ-v+μ+nrΓγ-v+nrΓγ-δ+μ+μ′+nrΓγ-δ+μ+v′+nrxγ+nr-δ+μ+μ′-1.By applying (44) and (45) in (85), we get(86)=xγ-δ+μ+μ′-1∏i=1rki1-βi/kiΓb1i…ΓbqiΓεi/kiΓa1i…Γapi×∑n=0∞Γa1i+n…Γapi+nΓεi/ki+nτiΓb1i+n…Γbqi+nΓnαi/ki+βi/ki1n!r(87)×Γγ+nrΓγ-δ+μ+μ′+v′+nrΓγ-v+μ+nrΓγ-v+nrΓγ-δ+μ+μ′+nrΓγ-δ+μ+v′+nrkinτi-αi/kixnr.Interpreting the right-hand side of the above equation, in view of definition (52), we arrive at result (80).Theorem 22.
LetR(δ)>0, R(γ)<1+min⁡Rv′,Rδ-μ-μ′,Rδ-μ′-v, then the generalized fractional differentiation Dx,∞μ,μ′,v,v′,δ of the product of S-function is given by(88)Dx,∞μ,μ′,v,v′,δtγ-1∏i=1rSpi,qiαi,βi,εi,τi,kia1i,…,api;b1i,…,bqi;t-1x=xγ-δ+μ+μ′-1∏i=1rki1-βi/kiΓb1i…ΓbqiΓεi/kiΓa1i…Γapip+1r+3ψq+1r+3·a11,1…a1r,1,…,ap1,1…apr,1,ε1k1,τ1,…,εrkr,τr,1-γ+v′,r,1-γ+δ-μ-μ′,r,1-γ+δ-μ′-v,rb11,1…b1r,1,…,bq1,1…bqr,1,β1k1,α1k1,…,βrkr,αrkr,1-γ,r,1-γ+δ-μ-μ′-v,r,1-γ-μ′+v′,,rk1τ1-α1/k1…krτr-αr/krxr.Proof.
The proof of Theorem22 is a similar manner of Theorem 21.


### 3.2.1. Special Cases

Now, we present some special cases of Theorems21 and 22 given as follows.If we putμ=μ+v,μ′=v′=0,v=-δ,δ=μ, then we obtain the relationship(89)D0,xμ,μ′,v,v′,δx=D0,xμ,v,δfxand (90)Dx,∞μ,μ′,v,v′,δx=Dx,∞μ,v,δfxwhich are defined in (17) and (18) as Saigo fractional derivative operator.Corollary 23.
LetR(μ)>0, R(γ)>-min⁡0,Rμ+v+δ, then the generalized fractional differentiation D0,xμ,v,δ of the product of S-function is given by(91)D0,xμ,v,δtγ-1∏i=1rSpi,qiαi,βi,εi,τi,kia1i,…,api;b1i,…,bqi;tx=xγ+v-1∏i=1rki1-βi/kiΓb1i…ΓbqiΓεi/kiΓa1i…Γapi×p+1r+2ψq+1r+2a11,1…a1r,1,…,ap1,1…apr,1,ε1k1,τ1,…,εrkr,τr,γ,r,γ+δ+μ+v,rb11,1…b1r,1,…,bq1,1…bqr,1,β1k1,α1k1,…,βrkr,αrkr,γ+v,r,γ+δ,rk1τ1-α1/k1…krτr-αr/krxr.Corollary 24.
LetR(μ)>0, R(γ)<1+min⁡R-v,Rδ+μ, then the generalized fractional differentiation Dx,∞μ,v,δ of the product of S-function is given by(92)Dx,∞μ,v,δtγ-1∏i=1rSpi,qiαi,βi,εi,τi,kia1i,…,api;b1i,…,bqi;t-1x=xγ+v-1∏i=1rki1-βi/kiΓb1i…ΓbqiΓεi/kiΓa1i…Γapi×p+1r+2ψq+1r+2a11,1…a1r,1,…,ap1,1…apr,1,ε1k1,τ1,…,εrkr,τr,1-γ-v,r,1-γ+μ+δ,rb11,1…b1r,1,…,bq1,1…bqr,1,β1k1,α1k1,…,βrkr,αrkr,1-γ,r,1-γ+δ-v,rk1τ1-α1/k1…krτr-αr/krxr.Further, if we setv=0, Corollaries 23 and 24 and then Saigo fractional differential formulas reduce to the following Erdélyi-Kober type fractional differential formulas.Corollary 25.
LetR(μ)>0, R(γ)>-Rδ+μ, then the generalized fractional differentiation D0,xμ,δ of the product of S-function is given by(93)D0,xμ,δtγ-1∏i=1rSpi,qiαi,βi,εi,τi,kia1i,…,api;b1i,…,bqi;tx=xγ-1∏i=1rki1-βi/kiΓb1i…ΓbqiΓεi/kiΓa1i…Γapi×p+1r+1ψq+1r+1a11,1…a1r,1,…,ap1,1…apr,1,ε1k1,τ1,…,εrkr,τr,γ+δ+μ,rb11,1…b1r,1,…,bq1,1…bqr,1,β1k1,α1k1,…,βrkr,αrkr,γ+δ,rk1τ1-α1/k1…krτr-αr/krxr.Corollary 26.
LetR(μ)>0, R(γ)<1+Rδ+μ, then the generalized fractional differentiation Dx,∞μ,δ of the product of S-function is given by(94)Dx,∞μ,δtγ-1∏i=1rSpi,qiαi,βi,εi,τi,kia1i,…,api;b1i,…,bqi;t-1x=xγ-1∏i=1rki1-βi/kiΓb1i…ΓbqiΓεi/kiΓa1i…Γapi×p+1r+1ψq+1r+1a11,1…a1r,1,…,ap1,1…apr,1,ε1k1,τ1,…,εrkr,τr,1-γ+μ+δ,rb11,1…b1r,1,…,bq1,1…bqr,1,β1k1,α1k1,…,βrkr,αrkr,1-γ+δ,rk1τ1-α1/k1…krτr-αr/krxr.Further, if we setv=-μ, in Corollaries 9 and 10 then Saigo fractional derivative formulas reduce to the Riemann-Liouville and the Weyl type fractional derivative formulas as the following results.Corollary 27.
LetR(μ)>0, R(γ)>0, and then the generalized fractional differentiation D0,xμ of the product of S-function is given by(95)D0,xμtγ-1∏i=1rSpi,qiαi,βi,εi,τi,kia1i,…,api;b1i,…,bqi;tx=xγ-μ-1∏i=1rki1-βi/kiΓb1i…ΓbqiΓεi/kiΓa1i…Γapi×p+1r+1ψq+1r+1a11,1…a1r,1,…,ap1,1…apr,1,ε1k1,τ1,…,εrkr,τr,γ,rb11,1…b1r,1,…,bq1,1…bqr,1,β1k1,α1k1,…,βrkr,αrkr,γ-μ,rk1τ1-α1/k1…krτr-αr/krxr.Corollary 28.
LetR(γ)<1+Rμ, then the generalized fractional differentiation Dx,∞μ of the product of S-function is given by(96)Dx,∞μtγ-1∏i=1rSpi,qiαi,βi,εi,τi,kia1i,…,api;b1i,…,bqi;t-1x=xγ-μ-1∏i=1rki1-βi/kiΓb1i…ΓbqiΓεi/kiΓa1i…Γapi×p+1r+1ψq+1r+1a11,1…a1r,1,…,ap1,1…apr,1,ε1k1,τ1,…,εrkr,τr,1-γ+μ,rb11,1…b1r,1,…,bq1,1…bqr,1,β1k1,α1k1,…,βrkr,αrkr,1-γ,rk1τ1-α1/k1…krτr-αr/krxr.If we putr=1, then the results in (80), (88), and (91) to (96) reduce to the following form.Corollary 29.
LetR(δ)>0, R(γ)>max⁡0,Rδ-μ-μ′-v′,Rv-μ, then the generalized fractional differentiation D0,xμ,μ′,v,v′,δ of the S-function is given by(97)D0,xμ,μ′,v,v′,δtγ-1Spi,qiαi,βi,εi,τi,kia1,…,ap;b1,…,bq;tx=xγ-δ+μ+μ′-1k1-β/kΓb1…ΓbqΓε/kΓa1…Γapp+4ψq+4a1,1…ap,1,εk,τ,γ,1,γ-δ+μ+μ′+v′,1,γ-v+μ,1b1,1…bq,1,βk,αk,γ-v,1,γ-δ+μ+μ′,1,γ-δ+μ+v′,1kτ-α/kx.Corollary 30.
LetR(δ)>0, R(γ)<1+min⁡Rv′,Rδ-μ-μ′,Rδ-μ′-v, then the generalized fractional differentiation Dx,∞μ,μ′,v,v′,δ of the S-function is given by(98)Dx,∞μ,μ′,v,v′,δtγ-1Spi,qiαi,βi,εi,τi,kia1,…,ap;b1,…,bq;t-1x=xγ-δ+μ+μ′-1k1-β/kΓb1…ΓbqΓε/kΓa1…Γapp+4ψq+4a1,1…ap,1,εk,τ,1-γ+v′,1,1-γ+δ-μ-μ′,1,1-γ+δ-μ′-v,1b1,1…bq,1,βk,αk,1-γ,1,1-γ+δ-μ-μ′-v,1,1-γ-μ′+v′,,1kτ-α/kx.Corollary 31.
LetR(μ)>0, R(γ)>-min⁡0,Rμ+v+δ, then the generalized fractional differentiation D0,xμ,v,δ of the S-function is given by(99)D0,xμ,v,δtγ-1Spi,qiαi,βi,εi,τi,kia1,…,ap;b1,…,bq;tx=xγ+v-1k1-β/kΓb1…ΓbqΓε/kΓa1…Γap×p+3ψq+3a1,1…ap,1,εk,τ,γ,1,γ+δ+μ+v,1b1,1…bq,1,βk,αk,γ+v,1,γ+δ,1kτ-α/kx.Corollary 32.
LetR(μ)>0, R(γ)<1+min⁡R-v,Rδ+μ, then the generalized fractional differentiation Dx,∞μ,v,δ of the S-function is given by(100)Dx,∞μ,v,δtγ-1Spi,qiαi,βi,εi,τi,kia1,…,ap;b1,…,bq;t-1x=xγ+v-1k1-β/kΓb1…ΓbqΓε/kΓa1…Γap×p+3ψq+3a1,1…ap,1,εk,τ,1-γ-v,1,1-γ+μ+δ,1b1,1…bq,1,βk,αk,1-γ,1,1-γ+δ-v,1kτ-α/kx.Corollary 33.
LetR(μ)>0, R(γ)>-Rδ+μ, then the generalized fractional differentiation D0,xμ,δ of the S-function is given by(101)D0,xμ,δtγ-1Spi,qiαi,βi,εi,τi,kia1,…,ap;b1,…,bq;tx=xγ-1k1-β/kΓb1…ΓbqΓε/kΓa1…Γap×p+2ψq+2a1,1…ap,1,εk,τ,γ+δ+μ,1b1,1…bq,1,βk,αk,γ+δ,1kτ-α/kx.Corollary 34.
LetR(μ)>0, R(γ)<1+Rδ+μ, then the generalized fractional differentiation Dx,∞μ,δ of the S-function is given by(102)Dx,∞μ,δtγ-1Spi,qiαi,βi,εi,τi,kia1,…,ap;b1,…,bq;t-1x=xγ-1k1-β/kΓb1…ΓbqΓε/kΓa1…Γap×p+2ψq+2a1,1…ap,1,εk,τ,1-γ+μ+δ,1b1,1…bq,1,βk,αk,1-γ+δ,1kτ-α/kx.Corollary 35.
LetR(μ)>0, R(γ)>0, and then the generalized fractional differentiation D0,xμ of the S-function is given by(103)D0,xμtγ-1Spi,qiαi,βi,εi,τi,kia1,…,ap;b1,…,bq;tx=xγ-μ-1k1-β/kΓb1…ΓbqΓε/kΓa1…Γap×p+2ψq+2a1,1…ap,1,εk,τ,γ,1b1,1…bq,1,βk,αk,γ-μ,1kτ-α/kxCorollary 36.
LetR(γ)<1+Rμ, then the generalized fractional differentiation Dx,∞μ of the S-function is given by(104)Dx,∞μtγ-1Spi,qiαi,βi,εi,τi,kia1,…,ap;b1,…,bq;t-1x=xγ-μ-1k1-β/kΓb1…ΓbqΓε/kΓa1…Γap×p+2ψq+2a1,1…ap,1,εk,τ,1-γ+μ,1b1,1…bq,1,βk,αk,1-γ,1kτ-α/kx.


## 3.2.1. Special Cases

Now, we present some special cases of Theorems21 and 22 given as follows.If we putμ=μ+v,μ′=v′=0,v=-δ,δ=μ, then we obtain the relationship(89)D0,xμ,μ′,v,v′,δx=D0,xμ,v,δfxand (90)Dx,∞μ,μ′,v,v′,δx=Dx,∞μ,v,δfxwhich are defined in (17) and (18) as Saigo fractional derivative operator.Corollary 23.
LetR(μ)>0, R(γ)>-min⁡0,Rμ+v+δ, then the generalized fractional differentiation D0,xμ,v,δ of the product of S-function is given by(91)D0,xμ,v,δtγ-1∏i=1rSpi,qiαi,βi,εi,τi,kia1i,…,api;b1i,…,bqi;tx=xγ+v-1∏i=1rki1-βi/kiΓb1i…ΓbqiΓεi/kiΓa1i…Γapi×p+1r+2ψq+1r+2a11,1…a1r,1,…,ap1,1…apr,1,ε1k1,τ1,…,εrkr,τr,γ,r,γ+δ+μ+v,rb11,1…b1r,1,…,bq1,1…bqr,1,β1k1,α1k1,…,βrkr,αrkr,γ+v,r,γ+δ,rk1τ1-α1/k1…krτr-αr/krxr.Corollary 24.
LetR(μ)>0, R(γ)<1+min⁡R-v,Rδ+μ, then the generalized fractional differentiation Dx,∞μ,v,δ of the product of S-function is given by(92)Dx,∞μ,v,δtγ-1∏i=1rSpi,qiαi,βi,εi,τi,kia1i,…,api;b1i,…,bqi;t-1x=xγ+v-1∏i=1rki1-βi/kiΓb1i…ΓbqiΓεi/kiΓa1i…Γapi×p+1r+2ψq+1r+2a11,1…a1r,1,…,ap1,1…apr,1,ε1k1,τ1,…,εrkr,τr,1-γ-v,r,1-γ+μ+δ,rb11,1…b1r,1,…,bq1,1…bqr,1,β1k1,α1k1,…,βrkr,αrkr,1-γ,r,1-γ+δ-v,rk1τ1-α1/k1…krτr-αr/krxr.Further, if we setv=0, Corollaries 23 and 24 and then Saigo fractional differential formulas reduce to the following Erdélyi-Kober type fractional differential formulas.Corollary 25.
LetR(μ)>0, R(γ)>-Rδ+μ, then the generalized fractional differentiation D0,xμ,δ of the product of S-function is given by(93)D0,xμ,δtγ-1∏i=1rSpi,qiαi,βi,εi,τi,kia1i,…,api;b1i,…,bqi;tx=xγ-1∏i=1rki1-βi/kiΓb1i…ΓbqiΓεi/kiΓa1i…Γapi×p+1r+1ψq+1r+1a11,1…a1r,1,…,ap1,1…apr,1,ε1k1,τ1,…,εrkr,τr,γ+δ+μ,rb11,1…b1r,1,…,bq1,1…bqr,1,β1k1,α1k1,…,βrkr,αrkr,γ+δ,rk1τ1-α1/k1…krτr-αr/krxr.Corollary 26.
LetR(μ)>0, R(γ)<1+Rδ+μ, then the generalized fractional differentiation Dx,∞μ,δ of the product of S-function is given by(94)Dx,∞μ,δtγ-1∏i=1rSpi,qiαi,βi,εi,τi,kia1i,…,api;b1i,…,bqi;t-1x=xγ-1∏i=1rki1-βi/kiΓb1i…ΓbqiΓεi/kiΓa1i…Γapi×p+1r+1ψq+1r+1a11,1…a1r,1,…,ap1,1…apr,1,ε1k1,τ1,…,εrkr,τr,1-γ+μ+δ,rb11,1…b1r,1,…,bq1,1…bqr,1,β1k1,α1k1,…,βrkr,αrkr,1-γ+δ,rk1τ1-α1/k1…krτr-αr/krxr.Further, if we setv=-μ, in Corollaries 9 and 10 then Saigo fractional derivative formulas reduce to the Riemann-Liouville and the Weyl type fractional derivative formulas as the following results.Corollary 27.
LetR(μ)>0, R(γ)>0, and then the generalized fractional differentiation D0,xμ of the product of S-function is given by(95)D0,xμtγ-1∏i=1rSpi,qiαi,βi,εi,τi,kia1i,…,api;b1i,…,bqi;tx=xγ-μ-1∏i=1rki1-βi/kiΓb1i…ΓbqiΓεi/kiΓa1i…Γapi×p+1r+1ψq+1r+1a11,1…a1r,1,…,ap1,1…apr,1,ε1k1,τ1,…,εrkr,τr,γ,rb11,1…b1r,1,…,bq1,1…bqr,1,β1k1,α1k1,…,βrkr,αrkr,γ-μ,rk1τ1-α1/k1…krτr-αr/krxr.Corollary 28.
LetR(γ)<1+Rμ, then the generalized fractional differentiation Dx,∞μ of the product of S-function is given by(96)Dx,∞μtγ-1∏i=1rSpi,qiαi,βi,εi,τi,kia1i,…,api;b1i,…,bqi;t-1x=xγ-μ-1∏i=1rki1-βi/kiΓb1i…ΓbqiΓεi/kiΓa1i…Γapi×p+1r+1ψq+1r+1a11,1…a1r,1,…,ap1,1…apr,1,ε1k1,τ1,…,εrkr,τr,1-γ+μ,rb11,1…b1r,1,…,bq1,1…bqr,1,β1k1,α1k1,…,βrkr,αrkr,1-γ,rk1τ1-α1/k1…krτr-αr/krxr.If we putr=1, then the results in (80), (88), and (91) to (96) reduce to the following form.Corollary 29.
LetR(δ)>0, R(γ)>max⁡0,Rδ-μ-μ′-v′,Rv-μ, then the generalized fractional differentiation D0,xμ,μ′,v,v′,δ of the S-function is given by(97)D0,xμ,μ′,v,v′,δtγ-1Spi,qiαi,βi,εi,τi,kia1,…,ap;b1,…,bq;tx=xγ-δ+μ+μ′-1k1-β/kΓb1…ΓbqΓε/kΓa1…Γapp+4ψq+4a1,1…ap,1,εk,τ,γ,1,γ-δ+μ+μ′+v′,1,γ-v+μ,1b1,1…bq,1,βk,αk,γ-v,1,γ-δ+μ+μ′,1,γ-δ+μ+v′,1kτ-α/kx.Corollary 30.
LetR(δ)>0, R(γ)<1+min⁡Rv′,Rδ-μ-μ′,Rδ-μ′-v, then the generalized fractional differentiation Dx,∞μ,μ′,v,v′,δ of the S-function is given by(98)Dx,∞μ,μ′,v,v′,δtγ-1Spi,qiαi,βi,εi,τi,kia1,…,ap;b1,…,bq;t-1x=xγ-δ+μ+μ′-1k1-β/kΓb1…ΓbqΓε/kΓa1…Γapp+4ψq+4a1,1…ap,1,εk,τ,1-γ+v′,1,1-γ+δ-μ-μ′,1,1-γ+δ-μ′-v,1b1,1…bq,1,βk,αk,1-γ,1,1-γ+δ-μ-μ′-v,1,1-γ-μ′+v′,,1kτ-α/kx.Corollary 31.
LetR(μ)>0, R(γ)>-min⁡0,Rμ+v+δ, then the generalized fractional differentiation D0,xμ,v,δ of the S-function is given by(99)D0,xμ,v,δtγ-1Spi,qiαi,βi,εi,τi,kia1,…,ap;b1,…,bq;tx=xγ+v-1k1-β/kΓb1…ΓbqΓε/kΓa1…Γap×p+3ψq+3a1,1…ap,1,εk,τ,γ,1,γ+δ+μ+v,1b1,1…bq,1,βk,αk,γ+v,1,γ+δ,1kτ-α/kx.Corollary 32.
LetR(μ)>0, R(γ)<1+min⁡R-v,Rδ+μ, then the generalized fractional differentiation Dx,∞μ,v,δ of the S-function is given by(100)Dx,∞μ,v,δtγ-1Spi,qiαi,βi,εi,τi,kia1,…,ap;b1,…,bq;t-1x=xγ+v-1k1-β/kΓb1…ΓbqΓε/kΓa1…Γap×p+3ψq+3a1,1…ap,1,εk,τ,1-γ-v,1,1-γ+μ+δ,1b1,1…bq,1,βk,αk,1-γ,1,1-γ+δ-v,1kτ-α/kx.Corollary 33.
LetR(μ)>0, R(γ)>-Rδ+μ, then the generalized fractional differentiation D0,xμ,δ of the S-function is given by(101)D0,xμ,δtγ-1Spi,qiαi,βi,εi,τi,kia1,…,ap;b1,…,bq;tx=xγ-1k1-β/kΓb1…ΓbqΓε/kΓa1…Γap×p+2ψq+2a1,1…ap,1,εk,τ,γ+δ+μ,1b1,1…bq,1,βk,αk,γ+δ,1kτ-α/kx.Corollary 34.
LetR(μ)>0, R(γ)<1+Rδ+μ, then the generalized fractional differentiation Dx,∞μ,δ of the S-function is given by(102)Dx,∞μ,δtγ-1Spi,qiαi,βi,εi,τi,kia1,…,ap;b1,…,bq;t-1x=xγ-1k1-β/kΓb1…ΓbqΓε/kΓa1…Γap×p+2ψq+2a1,1…ap,1,εk,τ,1-γ+μ+δ,1b1,1…bq,1,βk,αk,1-γ+δ,1kτ-α/kx.Corollary 35.
LetR(μ)>0, R(γ)>0, and then the generalized fractional differentiation D0,xμ of the S-function is given by(103)D0,xμtγ-1Spi,qiαi,βi,εi,τi,kia1,…,ap;b1,…,bq;tx=xγ-μ-1k1-β/kΓb1…ΓbqΓε/kΓa1…Γap×p+2ψq+2a1,1…ap,1,εk,τ,γ,1b1,1…bq,1,βk,αk,γ-μ,1kτ-α/kxCorollary 36.
LetR(γ)<1+Rμ, then the generalized fractional differentiation Dx,∞μ of the S-function is given by(104)Dx,∞μtγ-1Spi,qiαi,βi,εi,τi,kia1,…,ap;b1,…,bq;t-1x=xγ-μ-1k1-β/kΓb1…ΓbqΓε/kΓa1…Γap×p+2ψq+2a1,1…ap,1,εk,τ,1-γ+μ,1b1,1…bq,1,βk,αk,1-γ,1kτ-α/kx.


## 4. Integral Transform Formulas of the Product of S-Function

In this section, we establish some theorems involving the results obtained in previous sections pertaining with the integral transform as like the Beta transform and the Laplace transform.


### 4.1. Beta Transform

The Beta transform [33] of the function f(z) is defined as(105)Bfz;g.h=∫01zg-11-zh-1fzdz,g,h∈C,Rg>0,Rh>0.Theorem 37.
Letμ,μ′,v,v′,δ,γ,αi,βi,εi,τi∈C, ki∈R, pi,qi∈N, Rαi>kiR(τi), x>0, and pi<qi+1, where (i=1,…,r), such that R(δ)>0, R(γ)>max⁡0,Rμ+μ′+v-δ,Rμ′-v′, then the following fractional integral holds true:(106)BI0,xμ,μ′,v,v′,δtγ-1∏i=1rSpi,qiαi,βi,εi,τi,kia1i,…api;b1i,…bqi;tzx:g,h=xγ+δ-μ-μ′-1Γh∏i=1rki1-βi/kiΓb1i…ΓbqiΓεi/kiΓa1i…Γapi×p+1r+4ψq+1r+4a11,1…a1r,1,…,ap1,1…apr,1,ε1k1,τ1,…,εrkr,τr,γ,r,γ+δ-μ-μ′-v,r,γ-μ′+v′,r,g,rb11,1…b1r,1,…,bq1,1…bqr,1,β1k1,α1k1,…,βrkr,αrkr,γ+v′,r,γ+δ-μ-μ′,r,γ+δ-μ′-v,r,g+h,rk1τ1-α1/k1…krτr-αr/krxr.Proof.
On using (105), the left-hand side of (106) leads to(107)BI0,xμ,μ′,v,v′,δtγ-1∏i=1r∑n=0∞a1in…apinεinτi,kib1in…bqinΓkinαi+βitznn!rx,g,h.Using definition of Beta transform right side of (107) becomes(108)=∫0∞zg-11-zh-1(109)×I0,xμ,μ′,v,v′,δtγ-1∏i=1r∑n=0∞a1in…apinεinτi,kib1in…bqinΓkinαi+βitznn!rdz.By interchanging the order of integration and summation we reduce the right side of (109) to(110)=∏i=1r∑n=0∞a1in…apinεinτi,kib1in…bqinΓkinαi+βi1n!r(111)×I0,xμ,μ′,v,v′,δtγ+nr-1x∫0∞zg+nr-11-zh-1dz.By applying definition of Beta transform in (111), we get(112)=xγ+δ-μ-μ′-1∏i=1r∑n=0∞a1i…apinεinτi,kib1in…bqinΓkinαi+βi1n!r(113)×Γγ+nrΓγ+δ-μ-μ′-v+nrΓγ+v′-μ′+nrΓg+nrΓhΓγ+v′+nrΓγ+δ-μ-μ′+nrΓγ+δ-μ′-v+nrΓg+h+nrxnr,and using (44) and (45) in (113) we get(114)=xγ+δ-μ-μ′-1Γh∏i=1rki1-βi/kiΓb1i…ΓbqiΓεi/kiΓa1i…Γapi×∑n=0∞Γa1i+n…Γapi+nΓεi/ki+nτiΓb1i+n…Γbqi+nΓnαi/ki+βi/ki1n!rΓγ+nrΓγ+v′+nr(115)×Γγ+δ-μ-μ′-v+nrΓγ+v′-μ′+nrΓg+nrΓγ+δ-μ-μ′+nrΓγ+δ-μ′-v+nrΓg+h+nrkinτi-αi/kixnr.Interpreting the right-hand side of the above equation, in view of definition (52), we arrive at result (106).Theorem 38.
Letμ,μ′,v,v′,δ,γ,αi,βi,εi,τi∈C, ki∈R, pi,qi∈N, Rαi>kiR(τi), x>0, and pi<qi+1, where (i=1,…,r), such that R(δ)>0, R(γ)<1+min⁡R-v,Rμ+μ′-δ,Rμ+v′-δ, then the following fractional integral holds true:(116)BIx,∞μ,μ′,v,v′,δtγ-1∏i=1rSpi,qiαi,βi,εi,τi,kia1i⋯api;b1i⋯bqi;zt-1x:g,h=xγ+δ-μ-μ′-1Γh∏i=1rki1-βi/kiΓb1i…ΓbqiΓεi/kiΓa1i…Γapi×p+1r+4ψq+1r+4a11,1…a1r,1,…,ap1,1…apr,1,ε1k1,τ1,…,εrkr,τr,1-γ-v,r,1-γ-δ+μ+μ′,r,1-γ-δ+μ+v′,r,g,rb11,1…b1r,1,…,bq1,1…bqr,1,β1k1,α1k1,…,βrkr,αrkr,1-γ,r,1-γ-δ+μ+μ′+v′,r,1-γ+μ-v,r,g+h,rk1τ1-α1/k1…krτr-αr/krxr.Proof.
The proof of Theorem38 is a similar manner of Theorem 37.Theorem 39.
Letμ,μ′,v,v′,δ,γ,αi,βi,εi,τi∈C, ki∈R, pi,qi∈N, Rαi>kiR(τi), x>0, and pi<qi+1, where (i=1,…,r), such that R(δ)>0, R(γ)>max⁡0,Rδ-μ-μ′-v′,Rv-μ, then the following fractional derivative formula holds true:(117)BD0,xμ,μ′,v,v′,δtγ-1∏i=1rSpi,qiαi,βi,εi,τi,kia1i,….api;b1i,…,bqi;tzx:g,h=xγ-δ+μ+μ′-1Γh∏i=1rki1-βi/kiΓb1i…ΓbqiΓεi/kiΓa1i…Γapi×p+1r+4ψq+1r+4a11,1…a1r,1,…,ap1,1…apr,1,ε1k1,τ1,…,εrkr,τr,γ,r,γ-δ+μ+μ′+v′,r,γ-v+μ,r,g,rb11,1…b1r,1,…,bq1,1…bqr,1,β1k1,α1k1,…,βrkr,αrkr,γ-v,r,γ-δ+μ+μ′,r,γ-δ+μ+v′,r,g+h,rk1τ1-α1/k1…krτr-αr/krxr.Proof.
On using (105), the left-hand side of (117) leads to(118)BD0,xμ,μ′,v,v′,δtγ-1∏i=1r∑n=0∞a1in…apinεinτi,kib1in…bqinΓkinαi+βitznn!rx:g,h.Using definition of Beta transform right side of (118) becomes(119)=∫0∞zg-11-zh-1(120)×D0,xμ,μ′,v,v′,δtγ-1∏i=1r∑n=0∞a1in…apinεinτi,kib1in…bqinΓkinαi+βitznn!rdz.By interchanging the order of integration and summation we reduce the right side of (120) to(121)=∏i=1r∑n=0∞a1in…apinεinτi,kib1in…bqinΓkinαi+βi1n!r(122)×D0,xμ,μ′,v,v′,δtγ+nr-1x∫0∞zg+nr-11-zh-1dz.By applying definition of Beta transform in (122), we get(123)=xγ-δ+μ+μ′-1∏i=1r∑n=0∞a1i…apinεinτi,kib1in…bqinΓkinαi+βi1n!r(124)×Γγ+nrΓγ-δ+μ+μ′+v′+nrΓγ-v+μ+nrΓg+nrΓhΓγ-v+nrΓγ-δ+μ+μ′+nrΓγ-δ+μ+v′+nrΓg+h+nrxnr,and using (44) and (45) in (124) we get(125)=xγ-δ+μ+μ′-1Γh∏i=1rki1-βi/kiΓb1i…ΓbqiΓεi/kiΓa1i…Γapi×∑n=0∞Γa1i+n…Γapi+nΓεi/ki+nτiΓb1i+n…Γbqi+nΓnαi/ki+βi/ki1n!r×Γγ+nrΓγ-δ+μ+μ′+v′+nrΓγ-v+μ+nrΓg+nrΓγ-v+nrΓγ-δ+μ+μ′+nrΓγ-δ+μ+v′+nrΓg+h+nr×kinτi-αi/kixnr.Interpreting the right-hand side of the above equation, in view of definition (52), we arrive at result (117).Theorem 40.
Letμ,μ′,v,v′,δ,γ,αi,βi,εi,τi∈C, ki∈R, pi,qi∈N, Rαi>kiR(τi), x>0, and pi<qi+1, where (i=1,…,r), such that R(δ)>0, R(γ)<1+min⁡Rv′,Rδ-μ-μ′,Rδ-μ′-v, then the following fractional derivative formula holds true:(126)BDx,∞μ,μ′,v,v′,δtγ-1∏i=1rSpi,qiαi,βi,εi,τi,kia1i,…,api;b1i,…,bqi;zt-1x:g,h=xγ-δ+μ+μ′-1Γh∏i=1rki1-βi/kiΓb1i…ΓbqiΓεi/kiΓa1i…Γapi×p+1r+4ψq+1r+4a11,1…a1r,1,…,ap1,1…apr,1,ε1k1,τ1,…,εrkr,τr,1-γ+v′,r,1-γ+δ-μ-μ′,r,1-γ+δ-μ′-v,r,g,rb11,1…b1r,1,…,bq1,1…bqr,1,β1k1,α1k1,…,βrkr,αrkr,1-γ,r,1-γ+δ-μ-μ′-v,r,1-γ-μ′+v′,,r,g+h,rk1τ1-α1/k1…krτr-αr/krxr.Proof.
The proof of Theorem40 is a similar manner of Theorem 39.


### 4.2. Laplace Transform

The Laplace transform [33, 34] of f(z) is defined as(127)Lfz=∫0∞e-szfzdz,Rs>0.Theorem 41.
Letμ,μ′,v,v′,δ,γ,αi,βi,εi,τi∈C, ki∈R, pi,qi∈N, Rαi>kiR(τi),x>0, and pi<qi+1, where (i=1,…,r), such that R(δ)>0, R(γ)>max⁡0,Rμ+μ′+v-δ,Rμ′-v′, then the following fractional integral holds true:(128)Lzg-1I0,xμ,μ′,v,v′,δtγ-1∏i=1rSpi,qiαi,βi,εi,τi,kia1i,…,api;b1i,…,bqi;tzx=xγ+δ-μ-μ′-1sg∏i=1rki1-βi/kiΓb1i…ΓbqiΓεi/kiΓa1i…Γapi×p+1r+4ψq+1r+3a11,1…a1r,1,…,ap1,1…apr,1,ε1k1,τ1,…,εrkr,τr,γ,r,γ+δ-μ-μ′-v,r,γ-μ′+v′,r,g,rb11,1…b1r,1,…,bq1,1…bqr,1,β1k1,α1k1,…,βrkr,αrkr,γ+v′,r,γ+δ-μ-μ′,r,γ+δ-μ′-v,rk1τ1-α1/k1…krτr-αr/krxsr.Proof.
In order to prove (128), we use (127) as(129)Lzg-1I0,xμ,μ′,v,v′,δtγ-1∏i=1rSpi,qiαi,βi,εi,τi,kia1i,…,api;b1i,…,bqi;tzx=∫0∞e-szI0,xμ,μ′,v,v′,δtγ-1∏i=1r∑n=0∞a1in…apinεinτi,kib1in…bqinΓkinαi+βiztnn!rdz,=xγ-δ+μ+μ′-1∏i=1r∑n=0∞a1in…apinεinτi,kib1in…bqinΓkinαi+βi1n!r×Γγ+nrΓγ+δ-μ-μ′-v+nrΓγ+v′-μ′+nrΓγ+v′+nrΓγ+δ-μ-μ′+nrΓγ+δ-μ′-v+nrxnr(130)×∫0∞e-szzg+nr-1dz.By interchanging the order of integration and summation and little simplification, we have(131)=xγ+δ-μ-μ′-1sg∏i=1r∑n=0∞a1in…apinεinτi,kib1in…bqinΓkinαi+βi1n!r(132)×Γγ+nrΓγ+δ-μ-μ′-v+nrΓγ+v′-μ′+nrΓg+nrΓγ+v′+nrΓγ+δ-μ-μ′+nrΓγ+δ-μ′-v+nrxsnr.By applying (44) and (45) in (132) we get(133)=xγ+δ-μ-μ′-1sg∏i=1rki1-βi/kiΓb1i…ΓbqiΓεi/kiΓa1i…Γapi×∑n=0∞Γa1i+n…Γapi+nΓεi/ki+nτiΓb1i+n…Γbqi+nΓnαi/ki+βi/ki1n!r×Γγ+nrΓγ+δ-μ-μ′-v+nrΓγ+v′-μ′+nrΓg+nrΓγ+v′+nrΓγ+δ-μ-μ′+nrΓγ+δ-μ′-v+nr(134)×kinτi-αi/kixsnr.Interpreting the right-hand side of the above equation, in view of definition (52), we arrive at result (128).Theorem 42.
Letμ,μ′,v,v′,δ,γ,αi,βi,εi,τi∈C, ki∈R, pi,qi∈N, Rαi>kiR(τi),x>0, and pi<qi+1, where (i=1,…,r), such that R(δ)>0, R(γ)<1+min⁡R-v,Rμ+μ′-δ,Rμ+v′-δ, then the following fractional integral holds true:(135)Lzg-1Ix,∞μ,μ′,v,v′,δtγ-1∏i=1rSpi,qiαi,βi,εi,τi,kia1i,…,api;b1i,…,bqi;zt-1x=xγ+δ-μ-μ′-1sg∏i=1rki1-βi/kiΓb1i…ΓbqiΓεi/kiΓa1i…Γapi×p+1r+4ψq+1r+3a11,1…a1r,1,…,ap1,1…apr,1,ε1k1,τ1,…,εrkr,τr,γ,r,γ+δ-μ-μ′-v,r,γ-μ′+v′,r,g,rb11,1…b1r,1,…,bq1,1…bqr,1,β1k1,α1k1,…,βrkr,αrkr,γ+v′,r,γ+δ-μ-μ′,r,γ+δ-μ′-v,rk1τ1-α1/k1…krτr-αr/kr1xsr.Proof.
The proof of Theorem42 is a similar manner of Theorem 41.Theorem 43.
Letμ,μ′,v,v′,δ,γ,αi,βi,εi,τi∈C, ki∈R, pi,qi∈N, Rαi>kiR(τi), x>0, and pi<qi+1, where (i=1,…,r), such that R(δ)>0R(γ)>max⁡0,Rδ-μ-μ′-v′,Rv-μ, then the following fractional derivative formula holds true:(136)Lzg-1D0,xμ,μ′,v,v′,δtγ-1∏i=1rSpi,qiαi,βi,εi,τi,kia1i,…,api;b1i,…,bqi;tzx=xγ-δ+μ+μ′-1sg∏i=1rki1-βi/kiΓb1i…ΓbqiΓεi/kiΓa1i…Γapi×p+1r+4ψq+1r+3a11,1…a1r,1,…,ap1,1…apr,1,ε1k1,τ1,…,εrkr,τr,γ,r,γ-δ+μ+μ′+v′,r,γ-v+μ,r,g,rb11,1…b1r,1,…,bq1,1…bqr,1,β1k1,α1k1,…,βrkr,αrkr,γ-v,r,γ-δ+μ+μ′,r,γ-δ+μ+v′,rk1τ1-α1/k1…krτr-αr/krxsr.Proof.
In order to prove (136), we use (127) as(137)Lzg-1D0,xμ,μ′,v,v′,δtγ-1∏i=1rSpi,qiαi,βi,εi,τi,kia1i,…,api;b1i,…,bqi;tzx(138)=∫0∞e-szD0,xμ,μ′,v,v′,δtγ-1∏i=1r∑n=0∞a1in…apinεinτi,kib1in…bqinΓkinαi+βiztnn!rdz.By interchanging the order of integration and summation, we reduce the right side of (138) to(139)=xγ-δ+μ+μ′-1∏i=1r∑n=0∞a1in…apinεinτi,kib1in…bqinΓkinαi+βi1n!r×Γγ+nrΓγ-δ+μ+μ′+v′+nrΓγ-v+μ+nrΓγ-v+nrΓγ-δ+μ+μ′+nrΓγ-δ+μ+v′+nrxnr(140)×∫0∞e-szzg+nr-1dz.After a little simplification and using (44) and (45), we obtain(141)=xγ+δ-μ-μ′-1sg∏i=1rki1-βi/kiΓb1i…ΓbqiΓεi/kiΓa1i…Γapi×∑n=0∞Γa1i+n…Γapi+nΓεi/ki+nτiΓb1i+n…Γbqi+nΓnαi/ki+βi/ki1n!r×Γγ+nrΓγ-δ+μ+μ′+v′+nrΓγ-v+μ+nrΓg+nrΓγ-v+nrΓγ-δ+μ+μ′+nrΓγ-δ+μ+v′+nr(142)×kinτi-αi/kixsnr.Interpreting the right-hand side of the above equation, in view of definition (52), we arrive at result (136).Theorem 44.
Letμ,μ′,v,v′,δ,γ,αi,βi,εi,τi∈C,ki∈R,pi,qi∈N,Rαi>kiτi,x>0, and pi<qi+1, where i=1,…,r, such that Rδ>0Rγ<1+min⁡Rv’,Rδ-μ-μ′,Rδ-μ′-v, then the following fractional derivative formula holds true:(143)Lzg-1Dx,∞μ,μ′,v,v′,δtγ-1∏i=1rSpi,qiαi,βi,εi,τi,kia1i,…,api;b1i,…,bqi;zt-1x=xγ-δ+μ+μ′-1sg∏i=1rki1-βi/kiΓb1i…ΓbqiΓεi/kiΓa1i…Γapi×p+1r+4ψq+1r+3a11,1…a1r,1,…,ap1,1…apr,1,ε1k1,τ1,…,εrkr,τr,g,r,1-γ+v′,r,1-γ+δ-μ-μ′,r,1-γ+δ-μ′-v,rb11,1…b1r,1,…,bq1,1…bqr,1,β1k1,α1k1,…,βrkr,αrkr,1-γ,r,1-γ+δ-μ-μ′-v,r,1-γ-μ′+v′,rk1τ1-α1/k1…krτr-αr/kr1xsr.


## 4.1. Beta Transform

The Beta transform [33] of the function f(z) is defined as(105)Bfz;g.h=∫01zg-11-zh-1fzdz,g,h∈C,Rg>0,Rh>0.Theorem 37.
Letμ,μ′,v,v′,δ,γ,αi,βi,εi,τi∈C, ki∈R, pi,qi∈N, Rαi>kiR(τi), x>0, and pi<qi+1, where (i=1,…,r), such that R(δ)>0, R(γ)>max⁡0,Rμ+μ′+v-δ,Rμ′-v′, then the following fractional integral holds true:(106)BI0,xμ,μ′,v,v′,δtγ-1∏i=1rSpi,qiαi,βi,εi,τi,kia1i,…api;b1i,…bqi;tzx:g,h=xγ+δ-μ-μ′-1Γh∏i=1rki1-βi/kiΓb1i…ΓbqiΓεi/kiΓa1i…Γapi×p+1r+4ψq+1r+4a11,1…a1r,1,…,ap1,1…apr,1,ε1k1,τ1,…,εrkr,τr,γ,r,γ+δ-μ-μ′-v,r,γ-μ′+v′,r,g,rb11,1…b1r,1,…,bq1,1…bqr,1,β1k1,α1k1,…,βrkr,αrkr,γ+v′,r,γ+δ-μ-μ′,r,γ+δ-μ′-v,r,g+h,rk1τ1-α1/k1…krτr-αr/krxr.Proof.
On using (105), the left-hand side of (106) leads to(107)BI0,xμ,μ′,v,v′,δtγ-1∏i=1r∑n=0∞a1in…apinεinτi,kib1in…bqinΓkinαi+βitznn!rx,g,h.Using definition of Beta transform right side of (107) becomes(108)=∫0∞zg-11-zh-1(109)×I0,xμ,μ′,v,v′,δtγ-1∏i=1r∑n=0∞a1in…apinεinτi,kib1in…bqinΓkinαi+βitznn!rdz.By interchanging the order of integration and summation we reduce the right side of (109) to(110)=∏i=1r∑n=0∞a1in…apinεinτi,kib1in…bqinΓkinαi+βi1n!r(111)×I0,xμ,μ′,v,v′,δtγ+nr-1x∫0∞zg+nr-11-zh-1dz.By applying definition of Beta transform in (111), we get(112)=xγ+δ-μ-μ′-1∏i=1r∑n=0∞a1i…apinεinτi,kib1in…bqinΓkinαi+βi1n!r(113)×Γγ+nrΓγ+δ-μ-μ′-v+nrΓγ+v′-μ′+nrΓg+nrΓhΓγ+v′+nrΓγ+δ-μ-μ′+nrΓγ+δ-μ′-v+nrΓg+h+nrxnr,and using (44) and (45) in (113) we get(114)=xγ+δ-μ-μ′-1Γh∏i=1rki1-βi/kiΓb1i…ΓbqiΓεi/kiΓa1i…Γapi×∑n=0∞Γa1i+n…Γapi+nΓεi/ki+nτiΓb1i+n…Γbqi+nΓnαi/ki+βi/ki1n!rΓγ+nrΓγ+v′+nr(115)×Γγ+δ-μ-μ′-v+nrΓγ+v′-μ′+nrΓg+nrΓγ+δ-μ-μ′+nrΓγ+δ-μ′-v+nrΓg+h+nrkinτi-αi/kixnr.Interpreting the right-hand side of the above equation, in view of definition (52), we arrive at result (106).Theorem 38.
Letμ,μ′,v,v′,δ,γ,αi,βi,εi,τi∈C, ki∈R, pi,qi∈N, Rαi>kiR(τi), x>0, and pi<qi+1, where (i=1,…,r), such that R(δ)>0, R(γ)<1+min⁡R-v,Rμ+μ′-δ,Rμ+v′-δ, then the following fractional integral holds true:(116)BIx,∞μ,μ′,v,v′,δtγ-1∏i=1rSpi,qiαi,βi,εi,τi,kia1i⋯api;b1i⋯bqi;zt-1x:g,h=xγ+δ-μ-μ′-1Γh∏i=1rki1-βi/kiΓb1i…ΓbqiΓεi/kiΓa1i…Γapi×p+1r+4ψq+1r+4a11,1…a1r,1,…,ap1,1…apr,1,ε1k1,τ1,…,εrkr,τr,1-γ-v,r,1-γ-δ+μ+μ′,r,1-γ-δ+μ+v′,r,g,rb11,1…b1r,1,…,bq1,1…bqr,1,β1k1,α1k1,…,βrkr,αrkr,1-γ,r,1-γ-δ+μ+μ′+v′,r,1-γ+μ-v,r,g+h,rk1τ1-α1/k1…krτr-αr/krxr.Proof.
The proof of Theorem38 is a similar manner of Theorem 37.Theorem 39.
Letμ,μ′,v,v′,δ,γ,αi,βi,εi,τi∈C, ki∈R, pi,qi∈N, Rαi>kiR(τi), x>0, and pi<qi+1, where (i=1,…,r), such that R(δ)>0, R(γ)>max⁡0,Rδ-μ-μ′-v′,Rv-μ, then the following fractional derivative formula holds true:(117)BD0,xμ,μ′,v,v′,δtγ-1∏i=1rSpi,qiαi,βi,εi,τi,kia1i,….api;b1i,…,bqi;tzx:g,h=xγ-δ+μ+μ′-1Γh∏i=1rki1-βi/kiΓb1i…ΓbqiΓεi/kiΓa1i…Γapi×p+1r+4ψq+1r+4a11,1…a1r,1,…,ap1,1…apr,1,ε1k1,τ1,…,εrkr,τr,γ,r,γ-δ+μ+μ′+v′,r,γ-v+μ,r,g,rb11,1…b1r,1,…,bq1,1…bqr,1,β1k1,α1k1,…,βrkr,αrkr,γ-v,r,γ-δ+μ+μ′,r,γ-δ+μ+v′,r,g+h,rk1τ1-α1/k1…krτr-αr/krxr.Proof.
On using (105), the left-hand side of (117) leads to(118)BD0,xμ,μ′,v,v′,δtγ-1∏i=1r∑n=0∞a1in…apinεinτi,kib1in…bqinΓkinαi+βitznn!rx:g,h.Using definition of Beta transform right side of (118) becomes(119)=∫0∞zg-11-zh-1(120)×D0,xμ,μ′,v,v′,δtγ-1∏i=1r∑n=0∞a1in…apinεinτi,kib1in…bqinΓkinαi+βitznn!rdz.By interchanging the order of integration and summation we reduce the right side of (120) to(121)=∏i=1r∑n=0∞a1in…apinεinτi,kib1in…bqinΓkinαi+βi1n!r(122)×D0,xμ,μ′,v,v′,δtγ+nr-1x∫0∞zg+nr-11-zh-1dz.By applying definition of Beta transform in (122), we get(123)=xγ-δ+μ+μ′-1∏i=1r∑n=0∞a1i…apinεinτi,kib1in…bqinΓkinαi+βi1n!r(124)×Γγ+nrΓγ-δ+μ+μ′+v′+nrΓγ-v+μ+nrΓg+nrΓhΓγ-v+nrΓγ-δ+μ+μ′+nrΓγ-δ+μ+v′+nrΓg+h+nrxnr,and using (44) and (45) in (124) we get(125)=xγ-δ+μ+μ′-1Γh∏i=1rki1-βi/kiΓb1i…ΓbqiΓεi/kiΓa1i…Γapi×∑n=0∞Γa1i+n…Γapi+nΓεi/ki+nτiΓb1i+n…Γbqi+nΓnαi/ki+βi/ki1n!r×Γγ+nrΓγ-δ+μ+μ′+v′+nrΓγ-v+μ+nrΓg+nrΓγ-v+nrΓγ-δ+μ+μ′+nrΓγ-δ+μ+v′+nrΓg+h+nr×kinτi-αi/kixnr.Interpreting the right-hand side of the above equation, in view of definition (52), we arrive at result (117).Theorem 40.
Letμ,μ′,v,v′,δ,γ,αi,βi,εi,τi∈C, ki∈R, pi,qi∈N, Rαi>kiR(τi), x>0, and pi<qi+1, where (i=1,…,r), such that R(δ)>0, R(γ)<1+min⁡Rv′,Rδ-μ-μ′,Rδ-μ′-v, then the following fractional derivative formula holds true:(126)BDx,∞μ,μ′,v,v′,δtγ-1∏i=1rSpi,qiαi,βi,εi,τi,kia1i,…,api;b1i,…,bqi;zt-1x:g,h=xγ-δ+μ+μ′-1Γh∏i=1rki1-βi/kiΓb1i…ΓbqiΓεi/kiΓa1i…Γapi×p+1r+4ψq+1r+4a11,1…a1r,1,…,ap1,1…apr,1,ε1k1,τ1,…,εrkr,τr,1-γ+v′,r,1-γ+δ-μ-μ′,r,1-γ+δ-μ′-v,r,g,rb11,1…b1r,1,…,bq1,1…bqr,1,β1k1,α1k1,…,βrkr,αrkr,1-γ,r,1-γ+δ-μ-μ′-v,r,1-γ-μ′+v′,,r,g+h,rk1τ1-α1/k1…krτr-αr/krxr.Proof.
The proof of Theorem40 is a similar manner of Theorem 39.


## 4.2. Laplace Transform

The Laplace transform [33, 34] of f(z) is defined as(127)Lfz=∫0∞e-szfzdz,Rs>0.Theorem 41.
Letμ,μ′,v,v′,δ,γ,αi,βi,εi,τi∈C, ki∈R, pi,qi∈N, Rαi>kiR(τi),x>0, and pi<qi+1, where (i=1,…,r), such that R(δ)>0, R(γ)>max⁡0,Rμ+μ′+v-δ,Rμ′-v′, then the following fractional integral holds true:(128)Lzg-1I0,xμ,μ′,v,v′,δtγ-1∏i=1rSpi,qiαi,βi,εi,τi,kia1i,…,api;b1i,…,bqi;tzx=xγ+δ-μ-μ′-1sg∏i=1rki1-βi/kiΓb1i…ΓbqiΓεi/kiΓa1i…Γapi×p+1r+4ψq+1r+3a11,1…a1r,1,…,ap1,1…apr,1,ε1k1,τ1,…,εrkr,τr,γ,r,γ+δ-μ-μ′-v,r,γ-μ′+v′,r,g,rb11,1…b1r,1,…,bq1,1…bqr,1,β1k1,α1k1,…,βrkr,αrkr,γ+v′,r,γ+δ-μ-μ′,r,γ+δ-μ′-v,rk1τ1-α1/k1…krτr-αr/krxsr.Proof.
In order to prove (128), we use (127) as(129)Lzg-1I0,xμ,μ′,v,v′,δtγ-1∏i=1rSpi,qiαi,βi,εi,τi,kia1i,…,api;b1i,…,bqi;tzx=∫0∞e-szI0,xμ,μ′,v,v′,δtγ-1∏i=1r∑n=0∞a1in…apinεinτi,kib1in…bqinΓkinαi+βiztnn!rdz,=xγ-δ+μ+μ′-1∏i=1r∑n=0∞a1in…apinεinτi,kib1in…bqinΓkinαi+βi1n!r×Γγ+nrΓγ+δ-μ-μ′-v+nrΓγ+v′-μ′+nrΓγ+v′+nrΓγ+δ-μ-μ′+nrΓγ+δ-μ′-v+nrxnr(130)×∫0∞e-szzg+nr-1dz.By interchanging the order of integration and summation and little simplification, we have(131)=xγ+δ-μ-μ′-1sg∏i=1r∑n=0∞a1in…apinεinτi,kib1in…bqinΓkinαi+βi1n!r(132)×Γγ+nrΓγ+δ-μ-μ′-v+nrΓγ+v′-μ′+nrΓg+nrΓγ+v′+nrΓγ+δ-μ-μ′+nrΓγ+δ-μ′-v+nrxsnr.By applying (44) and (45) in (132) we get(133)=xγ+δ-μ-μ′-1sg∏i=1rki1-βi/kiΓb1i…ΓbqiΓεi/kiΓa1i…Γapi×∑n=0∞Γa1i+n…Γapi+nΓεi/ki+nτiΓb1i+n…Γbqi+nΓnαi/ki+βi/ki1n!r×Γγ+nrΓγ+δ-μ-μ′-v+nrΓγ+v′-μ′+nrΓg+nrΓγ+v′+nrΓγ+δ-μ-μ′+nrΓγ+δ-μ′-v+nr(134)×kinτi-αi/kixsnr.Interpreting the right-hand side of the above equation, in view of definition (52), we arrive at result (128).Theorem 42.
Letμ,μ′,v,v′,δ,γ,αi,βi,εi,τi∈C, ki∈R, pi,qi∈N, Rαi>kiR(τi),x>0, and pi<qi+1, where (i=1,…,r), such that R(δ)>0, R(γ)<1+min⁡R-v,Rμ+μ′-δ,Rμ+v′-δ, then the following fractional integral holds true:(135)Lzg-1Ix,∞μ,μ′,v,v′,δtγ-1∏i=1rSpi,qiαi,βi,εi,τi,kia1i,…,api;b1i,…,bqi;zt-1x=xγ+δ-μ-μ′-1sg∏i=1rki1-βi/kiΓb1i…ΓbqiΓεi/kiΓa1i…Γapi×p+1r+4ψq+1r+3a11,1…a1r,1,…,ap1,1…apr,1,ε1k1,τ1,…,εrkr,τr,γ,r,γ+δ-μ-μ′-v,r,γ-μ′+v′,r,g,rb11,1…b1r,1,…,bq1,1…bqr,1,β1k1,α1k1,…,βrkr,αrkr,γ+v′,r,γ+δ-μ-μ′,r,γ+δ-μ′-v,rk1τ1-α1/k1…krτr-αr/kr1xsr.Proof.
The proof of Theorem42 is a similar manner of Theorem 41.Theorem 43.
Letμ,μ′,v,v′,δ,γ,αi,βi,εi,τi∈C, ki∈R, pi,qi∈N, Rαi>kiR(τi), x>0, and pi<qi+1, where (i=1,…,r), such that R(δ)>0R(γ)>max⁡0,Rδ-μ-μ′-v′,Rv-μ, then the following fractional derivative formula holds true:(136)Lzg-1D0,xμ,μ′,v,v′,δtγ-1∏i=1rSpi,qiαi,βi,εi,τi,kia1i,…,api;b1i,…,bqi;tzx=xγ-δ+μ+μ′-1sg∏i=1rki1-βi/kiΓb1i…ΓbqiΓεi/kiΓa1i…Γapi×p+1r+4ψq+1r+3a11,1…a1r,1,…,ap1,1…apr,1,ε1k1,τ1,…,εrkr,τr,γ,r,γ-δ+μ+μ′+v′,r,γ-v+μ,r,g,rb11,1…b1r,1,…,bq1,1…bqr,1,β1k1,α1k1,…,βrkr,αrkr,γ-v,r,γ-δ+μ+μ′,r,γ-δ+μ+v′,rk1τ1-α1/k1…krτr-αr/krxsr.Proof.
In order to prove (136), we use (127) as(137)Lzg-1D0,xμ,μ′,v,v′,δtγ-1∏i=1rSpi,qiαi,βi,εi,τi,kia1i,…,api;b1i,…,bqi;tzx(138)=∫0∞e-szD0,xμ,μ′,v,v′,δtγ-1∏i=1r∑n=0∞a1in…apinεinτi,kib1in…bqinΓkinαi+βiztnn!rdz.By interchanging the order of integration and summation, we reduce the right side of (138) to(139)=xγ-δ+μ+μ′-1∏i=1r∑n=0∞a1in…apinεinτi,kib1in…bqinΓkinαi+βi1n!r×Γγ+nrΓγ-δ+μ+μ′+v′+nrΓγ-v+μ+nrΓγ-v+nrΓγ-δ+μ+μ′+nrΓγ-δ+μ+v′+nrxnr(140)×∫0∞e-szzg+nr-1dz.After a little simplification and using (44) and (45), we obtain(141)=xγ+δ-μ-μ′-1sg∏i=1rki1-βi/kiΓb1i…ΓbqiΓεi/kiΓa1i…Γapi×∑n=0∞Γa1i+n…Γapi+nΓεi/ki+nτiΓb1i+n…Γbqi+nΓnαi/ki+βi/ki1n!r×Γγ+nrΓγ-δ+μ+μ′+v′+nrΓγ-v+μ+nrΓg+nrΓγ-v+nrΓγ-δ+μ+μ′+nrΓγ-δ+μ+v′+nr(142)×kinτi-αi/kixsnr.Interpreting the right-hand side of the above equation, in view of definition (52), we arrive at result (136).Theorem 44.
Letμ,μ′,v,v′,δ,γ,αi,βi,εi,τi∈C,ki∈R,pi,qi∈N,Rαi>kiτi,x>0, and pi<qi+1, where i=1,…,r, such that Rδ>0Rγ<1+min⁡Rv’,Rδ-μ-μ′,Rδ-μ′-v, then the following fractional derivative formula holds true:(143)Lzg-1Dx,∞μ,μ′,v,v′,δtγ-1∏i=1rSpi,qiαi,βi,εi,τi,kia1i,…,api;b1i,…,bqi;zt-1x=xγ-δ+μ+μ′-1sg∏i=1rki1-βi/kiΓb1i…ΓbqiΓεi/kiΓa1i…Γapi×p+1r+4ψq+1r+3a11,1…a1r,1,…,ap1,1…apr,1,ε1k1,τ1,…,εrkr,τr,g,r,1-γ+v′,r,1-γ+δ-μ-μ′,r,1-γ+δ-μ′-v,rb11,1…b1r,1,…,bq1,1…bqr,1,β1k1,α1k1,…,βrkr,αrkr,1-γ,r,1-γ+δ-μ-μ′-v,r,1-γ-μ′+v′,rk1τ1-α1/k1…krτr-αr/kr1xsr.


## 5. Consequence Results and Concluding Remarks

Marichev-Saigo-Maeda fractional integral and derivative operators have advantage that they generalize the R-L, Weyl, Erdélyi-Kober, and Saigo’s fractional integral and derivative operators; therefore, many authors called this a general operator. So, we conclude this paper by emphasizing that many other interesting image formulas can be derived as the specific cases of our leading results (Theorems5, 6, 21, and 22), involving familiar fractional integral and derivative operators as above said. Further, the S-function defined in (43) possesses the lead that a number of k-Mittag-Leffler functions, K-function, M-series, and Mittag-Leffler function happen to be the particular cases of this function. Some special cases of fractional calculus involved as above said function have been explored in the literature by a numeral of authors ([35–41]) with different arguments. Therefore, results presented in this paper are easily converted in terms of a comparable type of novel interesting integrals with diverse arguments after various suitable parametric replacements.



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