# Generalized Fractional Integral Operators andM-Series

Authors: A. M. Khan; R. K. Kumbhat; Amit Chouhan; Anita Alaria
Journal: Journal of Mathematics
 (2016)

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## Abstract

Two fractional integral operators associated with FoxH-function due to Saxena and Kumbhat are applied to M-series, which is an extension of both Mittag-Leffler function and generalized hypergeometric function pFq. The Mellin and Whittaker transforms are obtained for these compositional operators with M-series. Further some interesting properties have been established including power function and Riemann-Liouville fractional integral operators. The results are expressed in terms of H-function, which are in compact form suitable for numerical computation. Special cases of the results are also pointed out in the form of lemmas and corollaries.


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## Body

## 1. Introduction and Preliminaries

The subject of fractional calculus, which deals with investigations of integrals and derivatives, has gained importance and popularity during the last four decades. It is mainly due to its vast potential demonstrated applications in fields of science and engineering. Different extensions of various fractional integrations operators are studied by Kalla [1, 2], McBride [3], Kilbas [4, 5], Kiryakova [6], Purohit and Kalla [7], Kumbhat and Khan [8], and so forth.Saxena and Kumbhat [9] defined the fractional integration operators by means of the following equations:(1)Rx,rη,αfx=rx-η-rα-1∫0xtηxr-trαftHP,QM,NkU∣aP,APbQ,BQdt,(2)Kx,rδ,αfx=rxδ∫x∞t-δ-rα-1tr-xrαftHP,QM,NkV∣aP,APbQ,BQdt,where U and V represent the expressions tr/xrm1-tr/xrn and xr/trm1-xr/trn, respectively. M, N, P, and Q are positive integers such that 1≤M≤Q, 0≤N≤P, arg⁡k<1/2λπ, λ>0, for aP,bQ∈C, AP,BQ∈0,∞, and r, m, and n are positive numbers.The conditions of the validity of these operators are as follows:(i)
1≤p,q<∞,p-1+q-1=1.(ii)
Reη+rmbj/Bj>-1/q,Reα+rnbj/Bj>-1/q,Reδ+α+mrbj/Bj>-1/p, j=1,…,M.(iii)
fx∈L0,∞.The last condition ensures thatRx,rη,αfx and Kx,rδ,αfx both exist and also ensures that both belong to L0,∞. In (1) and (2) HP,QM,N(x) denotes H-function introduced and defined by Fox [10] via a Mellin-Barnes type integral as(3)HP,QM,Nz≡HP,QM,Nz∣ai,αi1,Pbj,βj1,Q=12πi∫LHP,QM,Nsz-sdswith(4)HP,QM,Ns≡HP,QM,Nai,αi1,Pbj,βj1,Q∣s=∏j=1MΓbj+βjs∏i=1NΓ1-ai-αis∏i=N+1PΓai+αis∏j=M+1QΓ1-bj-βjs.Asymptotic expansions and analytic continuations together with the convergence conditions of H-function have been discussed by Braaksma [11].Sharma and Jain [12] introduced the generalized M-series as the function defined by means of the power series:(5)pαMqβa1,a2,…,ap;b1,b2,…,bq;z=pαMqβz=pαMqβaj1p;bj1q;z=∑n=0∞a1n⋯apnb1n⋯bqnznΓαn+β,z,α,β∈C,Rα>0,where ajn, and bjn are the known Pochhammer symbols. Series (5) is defined when none of the parameters bj’s, j=1,2,…,q, is a negative integer or zero; if any numerator parameter aj is a negative integer or zero, then the series terminates to a polynomial in z. The series in (5) is convergent for all z if p≤q, it is convergent for z<δ, if p=q+1, and it is divergent, if p>q+1. When p=q+1 and z=δ, the series can converge on conditions depending on the parameters. Properties of M-series are further studied by Saxena [13], Chouhan and Saraswat [14], and so forth.The generalized Mittag-Leffler function [15] is obtained from (5) for p=q=1; a=γ∈C; b=1, as(6)Eα,βγz=∑m=0∞γmΓαm+βzmm!=∑m=0∞γm1mzmΓαm+β=M1α1βγ;1;z.The right sided Riemann-Liouville fractional integral operator Ia+λ is defined by Samko et al. [16] for Re(λ)>0, as(7)Ia+λfx=1Γλ∫axx-tλ-1ftdt.Particularly due to Chouhan and Saraswat [14] we have(8)I0+λtρ-1x=ΓρΓρ+λxρ+λ-1.Chouhan and Saraswat [14] established the following relation for α>0, β>0, γ>0, and a∈R:(9)I0+αtγ-1pβMqγaj1p;bj1q;atβx=xα+γ-1pβMqα+γaj1p;bj1q;axβ.Due to Whittaker transform (Whittaker and Watson [17]), the following result holds:(10)∫0∞e-t/2tσ-1Wλ,μtdt=Γ1/2+μ+σΓ1/2-μ+σΓ1-λ+σ,where Reμ±σ>-1/2 and Wλ,μ· is the Whittaker function [17, 18] defined as (11)Wλ,μt=e-t/2Γ-2μt1/2+μΓ1/2-μ-λF1112+μ-λ;1+2μ;t+e-t/2Γ2μt1/2-μΓ1/2+μ-λF1112-μ-λ;1-2μ;t.


## 2. Operators andM-Series

In this section, we established image formulas forM-series (5), involving R-K operators (1) and (2), in terms of Fox H-function. The results are shown in Theorems 3 and 5. We first derive the following two lemmas in order to prove Theorems 3 and 5.Lemma 1.
Iffx∈LP0,∞, 1≤p≤2, arg⁡k<1/2λπ, λ>0, Reη+rmbj/Bj>q-1, Reα+rnbj/Bj>-q-1,  j=1,2,…,M,p-1+q-1=1, ρ∈C, Reρ>0, x>0, then(12)Rx,rη,αtρ-1x=xρ-1HP+2,Q+1M,N+2k∣1-η+ρr,m,-α,n,aP,AP-η-ρr-α,m+n,bQ,BQ.Proof.
By (1), we have(13)Rx,rη,αtρ-1x=rx-η-rα-1∫0xtηxr-trαtρ-1HP,QM,NkU∣aP,APbQ,BQdtusing (3); then, by changing the order of integration valid under the conditions stated with the theorem and solving inner integral with respect to “t,” we get (14)Rx,rη,αtρ-1x=xρ-12πi∫LHP,QM,NsΓη+ρ/r-msΓα+1-nsΓη+ρ/r+α+1-m+nsk-sds;hence, by virtue of H-function definition (3) and (4), we finally obtain RHS of (12).Lemma 2.
Iffx∈LP0,∞, 1≤p≤2, arg⁡k<1/2λπ, λ>0, Reα+rnbj/Bj>-q-1, Reδ+α+rmbj/Bj>-p-1, p-1+q-1=1,  j=1,2,…,M, ρ∈C, Reρ<1, x>0, then(15)Kx,rδ,αtρ-1x=xρ-1HP+2,Q+1M,N+2k∣1-δ-ρ+1r,m,-α,n,aP,APρ-δ-1r-α,m+n,bQ,BQ.Proof.
By (2), we have(16)Kx,rδ,αtρ-1x=rxδ∫x∞t-δ-rα-1tr-xrαtρ-1HP,QM,NkV∣aP,APbQ,BQdtusing (3); then, by changing the order of integration valid under the conditions stated with the theorem and solving inner integration with respect to “t,” we get(17)Kx,rδ,αtρ-1x=xρ-12πi∫LHP,QM,NsΓδ-ρ+1/r-msΓα+1-nsΓδ-ρ+1/r+α+1-m+nsk-sds;hence, by virtue of H-function definition (3) and (4), we finally obtain RHS of (15).Theorem 3.
With all assumptions and conditions on parameters, as stated in Lemma1 with v, σ∈CRev>0, a>0, the following property holds true: (18)Rx,rη,αtρ-1Mp1vq1σatvx=∑i=0∞a1i⋯ap1ib1i⋯bq1iaiΓvi+σxρ+vi-1HP+2,Q+1M,N+2k∣1-η+ρ+vir,m,-α,n,aP,AP-η-ρ-vir-α,m+n,bQ,BQ.Proof.
Using (1) and (5), we get(19)Rx,rη,αtρ-1Mp1vq1σatvx=∑i=0∞a1i⋯ap1ib1i⋯bq1iaiΓvi+σRx,rη,αtρ-1+vix;finally, by virtue of (12), we obtained RHS of (18).Corollary 4.
With all assumptions and conditions on parameters, as stated in Theorem3 with γ∈C, the following result holds:(20)Rx,rη,αtρ-1Ev,σγatvx=∑i=0∞γiΓvi+σaii!xρ+vi-1HP+2,Q+1M,N+2k∣1-η+ρ+vir,m-α,naP,AP-η-ρ-vir-α,m+nbQ,BQ.Theorem 5.
With all assumptions and conditions on parameters, as stated in Lemma2 with v, σ∈CRev>0, a>0, the following property holds true: (21)Kx,rδ,αt-ρMp1vq1σat-vx=∑i=0∞a1i⋯ap1ib1i⋯bq1iaiΓvi+σx-ρ-viHP+2,Q+1M,N+2k∣1-δ+ρ+vir,m-α,naP,AP-ρ-δ-vir-α,m+nbQ,BQ.Proof.
By using (2) and (5) and then changing the order of summation, we get(22)Kx,rδ,αt-ρMp1vq1σat-vx=∑i=0∞a1i⋯ap1ib1i⋯bq1iaiΓvi+σKx,rδ,αt-ρ-vix;finally, by virtue of (15), we arrived at RHS of (21).Corollary 6.
With all assumptions and conditions on parameters, as stated in Theorem5 with γ∈C, the following result holds:(23)Kx,rδ,αt-ρEv,σγat-vx=∑i=0∞γiΓvi+σaii!x-ρ-viHP+2,Q+1M,N+2k∣1-δ+ρ+vir,m,-α,n,aP,AP-ρ-δ-vir-α,m+n,bQ,BQ.


## 3. Mellin and Whittaker Transforms

In this section Mellin and Whittaker transforms of the results established in Theorems3 and 5 have been obtained.Theorem 7.
Iffx∈LP(0,∞), 1≤p≤2, arg⁡k<1/2λπ, λ>0, Reη+rmbj/Bj>q-1, Reα+rnbj/Bj>-q-1, j=1,2,…,M, p-1+q-1=1, Res>-Rev and 0<x<1, Reρ>0, Rev>0, a>0, then(24)MRx,rη,αtρ-1Mp1vq1σatvs=∑i=0∞a1i⋯ap1ib1i⋯bq1iaiΓvi+σHP+2,Q+1M,N+2k∣1-η+ρ+vir,m,-α,n,aP,AP-η-ρ-vir-α,m+n,bQ,BQ1s+ρ+vi-1,where M[fx] is Mellin transform of fx.Proof.
From (18), it follows that (25)MRx,rη,αtρ-1Mp1vq1σatvs=∑i=0∞a1i⋯ap1ib1i⋯bq1iaiΓvi+σHP+2,Q+1M,N+2k∣1-η+ρ+vir,m-α,naP,AP-η-ρ-vir-α,m+nbQ,BQMtρ+vi-1;the theorem readily follows on evaluating the Mellin transform of tρ+vi-1 by means of the formula given by Erdelyi [18].Theorem 8.
Iffx∈LP0,∞, 1≤p≤2, arg⁡k<1/2λπ, λ>0, Reα+rnbj/Bj>-q-1, Reδ+α+rmbj/Bj>-p-1, p-1+q-1=1, j=1,2,…,M; ρ,ν,σ∈C, a>0, Reν>0, Re1-ρ<1, Res)>-Re(ν, 0<x<1, then(26)MKx,rδ,αt-ρMp1vq1σat-vs=∑i=0∞a1i⋯ap1ib1i⋯bq1iaiΓvi+σHP+2,Q+1M,N+2k∣1-δ+ρ+vir,m,-α,n,aP,AP-ρ-δ-vir-α,m+n,bQ,BQ1s-ρ-δ-vi,where M[fx] is Mellin transform of fx.Proof.
From (21), it follows that(27)MKx,rδ,αt-ρMp1vq1δat-vs=∑i=0∞a1i⋯ap1ib1i⋯bq1iaiΓvi+σHP+2,Q+1M,N+2k∣1-2δ+ρ+vir,m,-α,n,aP,AP-ρ-2δ-vir-α,m+n,bQ,BQMt-ρ-δ-vi.Theorem 8 readily follows on evaluating the Mellin transform of t-ρ-δ-vi by means of the formula given by Erdelyi [18].Theorem 9.
With all assumptions and conditions on parameters, as stated in Theorem3 with Reμ±σ+ρ+vi-1>-1/2, the following result holds: (28)∫0∞e-ϕt/2tσ-1wλ,μϕtRx,rη,αtρ-1Mp1vq1σatvdt=ϕ1-ρ-σ∑i=0∞a1i⋯ap1ib1i⋯bq1iaϕ-viΓvi+σΓ1/2+μ+σ+ρ+vi-1Γ1/2-μ+σ+ρ+vi-1Γvi+σ+ρ-λHP+2,Q+1M,N+2k∣1-η+ρ+vir,m,-α,n,aP,AP-η-ρ-vir-α,m+n,bQ,BQ.Proof.
From (18), it follows that (29)I≡∫0∞e-ϕt/2tσ-1wλ,μϕtRx,rη,αtρ-1Mp1vq1σatvdt=∑i=0∞a1i⋯ap1ib1i⋯bq1iaiΓvi+σHP+2,Q+1M,N+2k∣1-η+ρ+vir,m,-α,n,aP,AP-η-ρ-vir-α,m+n,bQ,BQ∫0∞e-ϕt/2tσ+ρ+vi-1-1wλ,μϕtdt;letting ϕt=z⇒dt=dz/ϕ, we obtained(30)I=∑i=0∞a1i⋯ap1ib1i⋯bq1iaiΓvi+σHP+2,Q+1M,N+2k∣1-η+ρ+vir,m,-α,n,aP,AP-η-ρ-vir-α,m+n,bQ,BQϕ1-ρ-σ-vi∫0∞e-z/2zσ+ρ+vi-1-1wλ,μzdz;finally by virtue of Whittaker transform (10) we obtained (28).Theorem 10.
With all assumptions and conditions on parameters, as stated in Theorem3 with Reμ±σ-ρ-vi>-1/2, the following result holds: (31)∫0∞e-ϕt/2tσ-1wλ,μϕtKx,rδ,αt-ρMp1vq1σat-vdt=ϕρ-σ∑i=0∞a1i⋯ap1ib1i⋯bq1iaϕviΓvi+σΓ1/2+μ+σ-ρ-viΓ1/2-μ+σ-ρ-viΓ1-λ+σ-ρ-viHP+2,Q+1M,N+2k∣1-η+ρ+vir,m,-α,n,aP,AP-η-ρ-vir-α,m+n,bQ,BQ.Proof.
From (21), it follows that (32)I≡∫0∞e-ϕt/2tσ-1wλ,μϕtKx,rδ,αt-ρMp1vq1σat-vdt=∑i=0∞a1i⋯ap1ib1i⋯bq1iaiΓvi+σHP+2,Q+1M,N+2k∣1-δ+ρ+vir,m-α,naP,AP-ρ-δ-vir-α,m+nbQ,BQ∫0∞e-ϕt/2tσ-ρ-vi-1wλ,μϕtdt.
Lettingϕt=z⇒dt=dz/ϕ, we obtained(33)I=∑i=0∞a1i⋯ap1ib1i⋯bq1iaiΓvi+σHP+2,Q+1M,N+2k∣1-η+ρ+vir,m,-α,n,aP,AP-η-ρ-vir-α,m+n,bQ,BQϕρ-σ+vi∫0∞e-z/2zσ-ρ-vi-1-1wλ,μzdz;
finally, by virtue of Whittaker transform (10), we obtained (31).


## 4. Properties of Integral Operators

Here we give some formal properties of the operators as consequences of Theorems3 and 5. These properties show compositions of power function and Riemann-Liouville fractional integral operator (7) with operators (1) and (2).Theorem 11.
With all assumptions and conditions on parameters, as stated in Theorem3 along with Reβ+ρ>0, the following property holds:(34)xβRx,rη,αtρ-1Mp1vq1σatvx=Rx,rη-β,αtβ+ρ-1Mp1vq1σatvx.Proof.
From (18), the LHS of (34) follows as (35)xβRx,rη,αtρ-1Mp1vq1σatvx=∑i=0∞a1i⋯ap1ib1i⋯bq1iaiΓvi+σxβ+ρ+vi-1HP+2,Q+1M,N+2k∣1-η+ρ+vir,m,-α,n,aP,AP-η-ρ-vir-α,m+n,bQ,BQ;again by (18) the RHS of (34) follows as (36)Rx,rη-β,αtβ+ρ-1Mp1vq1σatvx=∑i=0∞a1i⋯ap1ib1i⋯bq1iaiΓvi+σxρ+β+vi-1HP+2,Q+1M,N+2k∣1-η-β+ρ+β+vir,m-α,naP,AP-η+β-ρ-β-vir-α,m+nbQ,BQ.Apparently, Theorem 11 readily follows from (35) and (36).Theorem 12.
With all assumptions and conditions on parameters, as stated in Theorem5 along with Re1-ρ-β<1, the following property holds:(37)x-βKx,rδ,αt-ρMp1vq1σat-vx=Kx,rδ-β,αt-ρ-βMp1vq1σat-vx.Proof.
From (21), the LHS of (37) follows as (38)x-βKx,rδ,αt-ρMp1vq1σat-vx=∑i=0∞a1i⋯ap1ib1i⋯bq1iaiΓvi+σx-ρ-β-viHP+2,Q+1M,N+2k∣1-δ+ρ+vir,m-α,naP,AP-ρ-δ-vir-α,m+nbQ,BQ;again by (21) the RHS of (37) follows as (39)Kx,rδ-β,αt-ρ-βMp1vq1σat-vx=∑i=0∞a1i⋯ap1ib1i⋯bq1iaiΓvi+σx-ρ-β-viHP+2,Q+1M,N+2k∣1-δ-β+ρ+β+vir,m-α,naP,AP-ρ-β-δ+β-vir-α,m+nbQ,BQ.Apparently, Theorem 12 readily follows from (38) and (39).Theorem 13.
With all assumptions and conditions on parameters, as stated in Theorem3 along with λ∈C, Reλ>0, the following property holds: (40)I0+λRx,rη,αtσ-1Mp1vq1σatvx=Rx,rη,αI0+λtσ-1Mp1vq1σatvx.Proof.
Applying right sided Riemann-Liouville fractional integral operatorI0+λ to (18) and using (8), we obtained(41)I0+λRx,rη,αtσ-1Mp1vq1σatvx=∑i=0∞a1i⋯ap1ib1i⋯bq1iaiΓvi+σ+λxσ+λ+vi-1HP+2,Q+1M,N+2k∣1-η+σ+vir,m,-α,n,aP,AP-η-σ-vir-α,m+n,bQ,BQ;again, by virtue of relations (9) and (18), we obtained(42)Rx,rη,αI0+λtσ-1Mp1vq1σatvx=∑i=0∞a1i⋯ap1ib1i⋯bq1iaiΓvi+σ+λxσ+λ+vi-1HP+2,Q+1M,N+2k∣1-η+σ+vir,m,-α,n,aP,AP-η-σ-vir-α,m+n,bQ,BQ.Theorem 13 readily follows from (41) and (42).Theorem 14.
With all assumptions and conditions on parameters, as stated in Theorem5 along with λ∈C, Reλ>0, Re1-σ<1, the following property holds:(43)I0+λKx,rδ,αt-σMp1-vq11-σat-vx=Kx,rδ,αI0+λt-σMp1-vq11-σat-vx.Proof.
Applying right sided Riemann-Liouville fractional operatorI0+λ to (21) after suitable changes on parameters, we get(44)I0+λKx,rδ,αt-σMp1-vq11-σat-vx=∑i=0∞a1i⋯ap1ib1i⋯bq1iaiΓ-vi+1-σHP+2,Q+1M,N+2k∣1-δ+σ-vir,m,-α,n,aP,AP-σ-δ+vir-α,m+n,bQ,BQI0+λx-σ-vi=∑i=0∞a1i⋯ap1ib1i⋯bq1iaiΓ-vi+1-σ+λHP+2,Q+1M,N+2k∣1-δ+σ-vir,m,-α,n,aP,AP-σ-δ+vir-α,m+n,bQ,BQx-σ-vi+λ;again, by virtue of relations (9) and (21), we get(45)Kx,rδ,αI0+λt-σMp1-vq11-σat-vx=Kx,rδ,αxλ-σMp1-vq1λ+1-σat-vx=∑i=0∞a1i⋯ap1ib1i⋯bq1iaiΓ-vi+λ+1-σxλ-σ-viHP+2,Q+1M,N+2k∣1-δ+σ-vir,m,-α,n,aP,AP-σ-δ+vir-α,m+n,bQ,BQ;hence, from (44) and (45), Theorem 14 readily follows.


## 5. Conclusions

Recently, fractional operator’s theory was recognized to be good tool for modeling complex problems, kinetic equations, fractional reaction, fractional diffusion equations, and so forth. In this work, the authors investigated and studied two fractional integral operators associated with FoxH-function due to Saxena and Kumbhat which are applied to M-series. We obtain series expansion of the images of the M-series through these fractional operators. Besides some interesting properties of these operators in Section 4, the authors also discussed their behavior under Mellin and Whittaker transforms and results are given in better realistic series solutions which converge rapidly. Results derived in this paper are very significant and may find applications in the solution of fractional order differential equations that are arising in certain areas of turbulence, propagation of seismic waves, and diffusion processes. On account of general nature of H-function and M-series a number of results involving special functions can be obtained merely by specializing the parameters.



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