# Generalized Fractional Integral Operators Involving Mittag-Leffler Function

Authors: Hafte Amsalu; D. L. Suthar
Journal: Abstract and Applied Analysis
 (2018)

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## Abstract

The aim of this paper is to study various properties of Mittag-Leffler (M-L) function. Here we establish two theorems which give the image of this M-L function under the generalized fractional integral operators involving Fox’sH-function as kernel. Corresponding assertions in terms of Euler, Mellin, Laplace, Whittaker, and K-transforms are also presented. On account of general nature of M-L function a number of results involving special functions can be obtained merely by giving particular values for the parameters.


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## Body

## 1. Introduction and Preliminaries

M-L Function. In 1903, Mittag-Leffler [1] introduced the function Eλ(z), defined by(1)Eλz=∑n=0∞1Γλn+1znλ∈C;Rλ>0.A further, two-index generalization of this function was given by Wiman [2] as(2)Eλ,βz=∑n=0∞1Γλn+βznλ,β∈C,where R(λ)>0 and R(β)>0.By means of the series representation a generalization of M-L function (2) is introduced by Prabhakar [3] as(3)Eλ,βγz=∑n=0∞γnΓλn+βn!zn,where λ,β,γ∈C(R(λ)>0). Further, it is an entire function of order [R(λ)]-1.Generalized Fractional Integral Operator. Now, we recall the definition of generalized fractional integral operators involving Fox’s H-function as kernel, defined by Saxena and Kumbhat [4] means of the following equations:(4)Rx,rμ,αfx=rx-μ-rα-1∫0xtμxr-trαHp,qm,nkU∣ap,Apbq,Bqftdt,(5)Kx,rε,αfx=rxε∫x∞t-ε-rα-1tr-xrαHp,qm,nkV∣ap,Apbq,Bqftdt,where U and V represent the expressions (6)trxrτ1-trxrυ,xrtrτ1-xrtrυ,respectively, with τ,υ>0. The sufficient conditions of operators are given below:(i)
1≤p, q<∞, p-1+q-1=1;(ii)
Rμ+rτbj/Bj>-q-1; Rα+rυbj/Bj>-q-1;Rε+α+rτbj/Bj>-p-1, (j=1,…,m);(iii)
f(x)∈LP0,∞;(iv)
arg⁡k<λπ/2, λ>0,whereλ=∑j=1m‍Bj-∑j=m+1q‍Bj+∑j=1n‍Aj-∑j=n+1p‍Aj>0.An interest in the study of the fractional calculus associated with the Mittag-Leffler function andH-function, its application in the form of differential, and integral equations of, in particular, fractional orders (see [5–10]).H-Function. Symbol Hp,qm,nx stands for well known Fox H-function [11], in operator (4) and (5) defined in terms of Mellin-Barnes type contour integral as follows:(7)Hp,qm,nz=Hp,qm,nz∣ap,Apbq,Bq=12πi∫Lχszsds,where(8)χs=∏j=1mΓbj+Bjs∏i=1nΓ1-ai-Ais∏i=n+1pΓai+Ais∏j=m+1qΓ1-bj-Bjs,m,n,p,q∈N0 with 1≤m≤q, 0≤n≤p, Ai,Bj∈R+, ai,bj∈R, or C, i=1,2,…,p; j=1,2,…,q such that Ai(bj+k)≠Bj(ai-l-1)(k,l∈N0;i=1,2,…,n;j=1,2,…,m).For the conditions of analytically continuations together with the convergence conditions ofH-function, one can see [12, 13]. Throughout the present paper, we assume that these conditions are satisfied by the function.


## 2. Images of M-L Function Involving the Generalized Fractional Integral Operators

In this section, we consider two generalized fractional integral operators involving the Fox’sH-function as the kernels and derived the following theorems.Theorem 1.
Letλ,β,ϑ,γ∈C, x>0, Rλ>0, Rϑ>0, f(x)∈LP0,∞, 1≤p≤2, arg⁡k<λπ/2, λ>0, a∈C; then the fractional integration Rx,rμ,α of the product of M-L function exists, under the condition(9)p-1+q-1=1;Rμ+rτbjBj>-q-1;Rα+rυbjBj>-q-1;then there holds the following formula:(10)Rx,rμ,αtϑ-1Eλ,βγatνx=xϑ-1∑n=0∞γnΓλn+βn!axνn×Hp+2,q+1m,n+2k∣ap,Ap,1-μ+ϑ+1+νnr,τ,-α,υ-μ+ϑ+1+νnr-α,τ+υ,bq,Bq.Proof.
Letl be the left-hand side of (10); using (3) and (4), we have(11)l=rx-μ-rα-1∫0xtμ+ϑ-1xr-trα12πi∫LχskUsdsγnΓλn+βn!axνndt.Changing the order of the integration valid under the condition given with the theorem, we obtain (12)l=rx-μ-rα-1∑n=0∞γnanΓλn+βn!×12πi∫Lχsksxrα-rτs∫0xtμ+ϑ+νn+rτs-11-trxrα+υsdtds.Let the substitution tr/xr=w; then t=xw1/r in the above term; we get (13)=xϑ-1∑n=0∞γnanΓλn+βn!xvn2πi∫Lχsksxνs×∫01w1/rμ+ϑ+νn+rτs-11-wα+υsdwds.Using beta function for (13), the inner integral reduces to(14)=xϑ-1∑n=0∞γnΓλn+βn!axνn12πi∫Lχsks×Γμ+ϑ+νn/r+τsΓα+1+υsΓμ+ϑ+νn/r+α+1+τ+υsds.Interpreting the right-hand side of (14), in view of the definition (7), we arrive at the result (10).Theorem 2.
Letλ,β,ϑ,γ∈C, x>0, Rλ>0, Rϑ<1, f(x)∈LP0,∞, 1≤p≤2, arg⁡k<λπ/2, λ>0, and a∈C; then the fractional integration Kx,rε,α of the product of M-L function exists, under the condition(15)p-1+q-1=1,Rα+rυbjBj>-q-1,Rε+α+rτbjBj>-p-1and then the following formula holds: (16)Kx,rε,αt-ϑEλ,βγat-νx=x-ϑ∑n=0∞γnΓλn+βn!ax-νn×Hp+2,q+1m,n+2k∣ap,Ap,1-ε+ϑ+νnr,τ,-α,υ-α-ε+ϑ+νnr,τ+υ,bq,Bq.Proof.
Let℘ be the left-hand side of (16); using (3) and (5), we have(17)℘=rxε∫x∞t-ε-ϑ-rα-1tr-xrα×12πi∫LχskV-sds∑n=0∞γnΓλn+βn!ax-νndt.Changing the order of the integration valid under the condition given with the theorem statement, we obtain(18)℘=rxε∑n=0∞γnanΓλn+βn!12πi∫Lχsk-sx-rτs×∫x∞t-ε-ϑ-νn+rτs-11-xrtrα-υsdtds.Letting the substitution xr/tr=u, then t=x/u1/r in the above term and, using beta function, we get (19)=x-ϑ∑n=0∞γnΓλn+βn!ax-νn12πi∫Lχsk-s×Γε+ϑ+νn/r-τsΓα+1-υsΓε+ϑ+νn/r+α+1-τ+υsds.Interpreting the right-hand side of (19), in view of definition (7), we arrive at the result (16).


## 3. Integral Transforms of Fractional Integral Involving M-L Function

In this section, Mellin, Laplace, Euler, Whittaker, andK-transforms of the results established in Theorems 1 and 2 have been obtained.Euler Transform (Sneddon [14]). The Euler transform of a function f(t) is defined as(20)Bft;a,b=∫01ta-11-tb-1ftdt,a,b∈C,Ra>0,Rb>0.Theorem 3.
Letλ,β,ϑ,γ,c,d∈C, Rc>0, Rd>0, Rϑ>0, Rλ>0, p-1+q-1=1; f(x)∈LP0,∞, 1≤p≤2, arg⁡k<λπ/2, λ>0, p-1+q-1=1; Rμ+rτbj/Bj>-q-1; Rα+rυbj/Bj>-q-1; (j=1,…,m); then(21)BRx,rμ,αtϑ-1Eλ,βγatν;c,d=Γd∑n=0∞γnanΓλn+βn!Γc+ϑ-1+νnΓc+d+ϑ-1+νn×Hp+2,q+1m,n+2k∣ap,Ap,1-μ+ϑ+1+νnr,τ,-α,υ-μ+ϑ+1+νnr-α,τ+υ,bq,Bq.Proof.
Using (10) and (20) gives(22)BRx,rμ,αtϑ-1Eλ,βγatν;c,d=∑n=0∞γnΓλn+βann!×Hp+2,q+1m,n+2k∣ap,Ap,1-μ+ϑ+1+νnr,τ,-α,υ-μ+ϑ+1+νnr-α,τ+υ,bq,Bq×∫01tc+ϑ+νn-1-11-td-1dt(23)=∑n=0∞γnΓλn+βann!Γc+ϑ+νn-1ΓdΓc+d+ϑ+νn-1×Hp+2,q+1m,n+2k∣ap,Ap,1-μ+ϑ+1+νnr,τ,-α,υ-μ+ϑ+1+νnr-α,τ+υ,bq,Bq.Now, we obtain the result (23). This completes the proof of the theorem.Theorem 4.
Letλ,β,ϑ,γ,c,d∈C, a>0, Rc>0, Rd>0, Rλ>0, R1-ϑ<1, p-1+q-1=1; f(x)∈LP0,∞, 1≤p≤2, arg⁡k<λπ/2,λ>0, p-1+q-1=1; Rε+α+rτbj/Bj>-p-1; Rα+rυbj/Bj>-q-1; (j=1,…,m); then(24)BKx,rε,αt-ϑEλ,βγat-v;c,d=Γd∑n=0∞γnΓλn+βn!an×Γc-ϑ-νnΓc+d-ϑ-νnHp+2,q+1m,n+2k∣ap,Ap,1-ε+ϑ+νnr,τ,-α,υ-α-ε+ϑ+νnr,τ+υ,bq,Bq.Proof.
In similar manner, in proof of Theorem3, we obtain the result (24).Mellin Transform (Debnath and Bhatta [15]). The Mellin transform of a function f(t) is defined as(25)Mfts=∫0∞ts-1ftdt,Rs>0.Theorem 5.
All conditions follow from that stated in Theorem1 with Rs>Rν; the following result holds: (26)MRx,rμ,αtϑ-1Eλ,βγatνs=∑n=0∞γnΓλn+βn!an×Hp+2,q+1m,n+2k∣ap,Ap,1-μ+ϑ+1+νnr,τ,-α,υ-μ+ϑ+1+νnr-α,τ+υ,bq,Bq1s+ϑ+νn-1.Proof.
From (10) and (25), it gives(27)
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All conditions follow from what is stated in Theorem2 with R1-ϑ<1, Rs>Rν; the following result holds:(28)MKx,rε,αt-ϑEλ,βγat-νs=∑n=0∞γnΓλn+βn!an×Hp+2,q+1m,n+2k∣ap,Ap,1-ε+ϑ+νnr,τ,-α,υ-α-ε+ϑ+νnr,τ+υ,bq,Bq1s-ε-ϑ-νn.Proof.
In similar manner, in proof of Theorem5, we obtain the result (28).Laplace Transform (Sneddon [14]). The Laplace transform of a function f(t), denoted by F(s), is defined by the equation(29)Fs=Lfs=Lft;s=∫0∞e-stftdt,Rs>0.Provided the integral (29) is convergent and that the function, f(t), is continuous for t>0 and of exponential order as t→∞, (29) may be symbolically written as(30)Fs=Lft;sorft=L-1Fs;t.The following result is well known:(31)∫0∞e-sttp-1dt=Γpsp,Rp>1,Rs>1.Theorem 7.
All conditions follow from what is stated in Theorem1 with Rs>0 and Rϑ+νn>0; the following result holds:(32)LRx,rμ,αtϑ-1Eλ,βγatv;s=s-ϑ∑n=0∞γnΓλn+βn!as-νnΓϑ+νn×Hp+2,q+1m,n+2k∣ap,Ap,1-μ+ϑ+1+νnr,τ,-α,υ-μ+ϑ+1+νnr-α,τ+υ,bq,Bq.Proof.
we can develop similar line by using result of Laplace integral (31).Theorem 8.
All conditions follow from what is stated in Theorem2 with Rs>0 and R1-ϑ-νn>0; the following result holds:(33)LKx,rε,αt-ϑEλ,βγat-vs=s1-ϑ∑n=0∞γnΓλn+βn!as-νnΓ1-ϑ-νn×Hp+2,q+1m,n+2k∣ap,Ap,1-ε+ϑ+νnr,τ,-α,υ-α-ε+ϑ+νnr,τ+υ,bq,Bq.Proof.
In a similar manner, in proof of Theorem7, we obtain the result (33).Whittaker Transform (Whittaker and Watson [17]). Due to Whittaker transform, the following result holds:(34)∫0∞e-t/2tζ-1Wχ,ωtdt=Γ1/2+ω+ζΓ1/2-ω+ζΓ1-χ+ζ,where Rω±ζ>-1/2 and Wχ,ωt is the Whittaker confluent hypergeometric function:(35)Wω,ζz=Γ-2ωΓ1/2-χ-ωMχ,ωz+Γ2ωΓ1/2+χ+ωMχ,-ωz,where Mχ,ωz is defined by(36)Mχ,ωz=z1/2+ωe-1/2z1F112+ω-χ;2ω+1;z.Theorem 9.
Following what is stated in Theorem1 for conditions on parameters, with Rω±ϑ+ζ+νn-1>1/2, then the following result holds:(37)∫0∞e-φt/2tζ-1Wχ,ωφtRx,rμ,αtϑ-1Eλ,βγatνdt=φ1-ϑ-ζ∑n=0∞γnΓλn+βn!aφ-ν×Γω+ϑ+ζ+νn-1/2Γϑ-ω+ζ+νn-1/2Γϑ-χ+ζ+νn×Hp+2,q+1m,n+2k∣ap,Ap,1-μ+ϑ+1+νnr,τ,-α,υ-μ+ϑ+1+νnr-α,τ+υ,bq,Bq.Proof.
Using (10) and (34), it gives(38)∫0∞e-φt/2tζ-1Wχ,ωφtRx,rμ,αtϑ-1Eλ,βγatνdt=∑n=0∞γnanΓλn+βn!×Hp+2,q+1m,n+2k∣ap,Ap,1-μ+ϑ+1+νnr,τ,-α,υ-μ+ϑ+1+νnr-α,τ+υ,bq,Bq×∫0∞e-φt/2tϑ+ζ+νn-1-1Wχ,ωφtdt.Assume that t=k, ⇒dt=dk/φ; we get(39)=∑n=0∞γnanΓλn+βn!Hp+2,q+1m,n+2k∣ap,Ap,1-μ+ϑ+1+νnr,τ,-α,υ-μ+ϑ+1+νnr-α,τ+υ,bq,Bq×φ1-ϑ-ζ-νn∫0∞e-k/2kϑ+ζ+vn-1-1Wχ,ωkdk.Interpreting the right-hand side of (39), using (34), we arrive at the result (37).Theorem 10.
Following what is stated in Theorem2 for conditions on parameters, with Rω±-ϑ+ζ-vn-1>1/2, then the following result holds:(40)∫0∞e-φt/2tζ-1Wχ,ωφtKx,rε,αt-ϑEλ,βγat-νdt=φϑ-ζ∑n=0∞γnΓλn+βn!aφν×Γω-ϑ+ζ-νn+1/2Γ-ϑ-ω+ζ-νn+1/2Γ1-ϑ-χ+ζ-νn×Hp+2,q+1m,n+2k∣ap,Ap,1-ε+ϑ+νnr,τ,-α,υ-α-ε+ϑ+νnr,τ+υ,bq,Bq.Proof.
In a similar manner, in proof of Theorem9, we obtain the result (40).K-Transform (Erdélyi et al. [18]). This transform is defined by the following integral equation:(41)Rυfx;p=gp;υ=∫0∞px1/2Kυpxfxdx,where Rp>0;Kυx is the Bessel function of the second kind defined by ([18], p. 332) (42)Kυz=π2z1/2W0,υ2z,where W0,υ· is the Whittaker function defined in Erdélyi et al. [18].The following result given in Mathai et al. ([16], p. 54, eq. 2.37) will be used in evaluating the integrals:(43)∫0∞tρ-1Kυaxdx=2ρ-2a-ρΓρ±υ2;Ra>0;Rρ±υ>0.Theorem 11.
Following what is stated in Theorem1 for conditions on parameters, with Rω>0;Rρ+ϑ+νn-1±l>0, then the following result holds:(44)∫0∞tρ-1KlωtRx,rμ,αtϑ-1Eλ,βγatνdt=2ρ+ϑ-3ω1-ρ-ϑ∑n=0∞γnΓλn+βn!a2ωνΓρ+ϑ+νn-1±l2×Hp+2,q+1m,n+2k∣ap,Ap,1-μ+ϑ+1+νnr,τ,-α,υ-μ+ϑ+1+νnr-α,τ+υ,bq,Bq.Proof.
Using (10) and (44), it gives (45)∫0∞tρ-1KlωtRx,rμ,αtϑ-1pKqμ,ξ,γatνdt=∑n=0∞γnanΓλn+βn!×Hp+2,q+1m,n+2k∣ap,Ap,1-μ+ϑ+1+νnr,τ,-α,υ-μ+ϑ+1+νnr-α,τ+υ,bq,Bq×∫0∞tρ+ϑ+νn-1-1Klωtdt,and we get (46)=∑n=0∞γnanΓλn+βn!Hp+2,q+1m,n+2k∣ap,Ap,1-μ+ϑ+1+νnr,τ,-α,υ-μ+ϑ+1+νnr-α,τ+υ,bq,Bq×2ρ+ϑ+νn-3ω1-ρ-ϑ-νnΓρ+ϑ+νn-1±l2.Interpreting the right-hand side of (46), we arrive at the result (44).Theorem 12.
Following what is stated in Theorem2 for conditions on parameters, with Rω>0;Rρ-ϑ-νn±l>0, then the following result holds:(47)∫0∞tρ-1KlωtKx,rε,αt-ϑEλ,βγat-νdt=2ρ-ϑ-2ωϑ-ρ∑n=0∞γnΓλn+βn!aω2νΓρ-ϑ-νn±l2×Hp+2,q+1m,n+2k∣ap,Ap,1-ε+ϑ+νnr,τ,-α,υ-α-ε+ϑ+νnr,τ+υ,bq,Bq.Proof.
In a similar manner, in proof of Theorem11, we obtain the result (47).


## 4. Properties of Integral Operators

Here, we established some properties of the operators as consequences of Theorems1 and 2. These properties show compositions of power function.Theorem 13.
Following all the conditions on parameters as stated in Theorem1 with Rψ+ϑ>0, then the following result holds true:(48)xψRx,rμ,αtϑ-1Eλ,βγatνx=Rx,rμ-ψ,αtψ+ϑ-1Eλ,βγatνx.Proof.
From (10), the left-hand side of (48), we have (49)xψRx,rμ,αtϑ-1Eλ,βγatνx=∑n=0∞γnanΓλn+βn!xϑ+ψ+νn-1×Hp+2,q+1m,n+2k∣ap,Ap,1-μ+ϑ+1+νnr,τ,-α,υ-μ+ϑ+1+νnr-α,τ+υ,bq,Bq,and again, by (10), the right-hand side of (48) follows:(50)Rx,rμ-ψ,αtψ+ϑ-1Eλ,βγatνx=∑n=0∞γnanΓλn+βn!xϑ+ψ+νn-1×Hp+2,q+1m,n+2k∣ap,Ap,1-μ+ϑ+1+νnr,τ,-α,υ-μ+ϑ+1+νnr-α,τ+υ,bq,Bq.It seems that Theorem 13 readily follows due to (49) and (50).Theorem 14.
Following all the conditions on parameters as stated in Theorem2 with Rβ+ϑ>0, then the following result holds true:(51)x-ψKx,rε,αt-ϑEλ,βγat-νx=Kx,rε-ψ,αt-ϑ-ψEλ,βγat-νx.Proof.
From (12), the left-hand side of (51), we have (52)x-ψKx,rε,αt-ϑEλ,βγat-νx=∑n=0∞γnanΓλn+βn!x-ψ-ϑ-νn×Hp+2,q+1m,n+2k∣ap,Ap,1-ε+ϑ+νnr,τ,-α,υ-α-ε+ϑ+νnr,τ+υ,bq,Bq.Again by (12), the right-hand side of (51) follows:(53)Kx,rε-ψ,αt-ϑ-ψEλ,βγat-νx=∑n=0∞γnanΓλn+βn!x-ψ-ϑ-νn×Hp+2,q+1m,n+2k∣ap,Ap,1-ε+ϑ+νnr,τ,-α,υ-α-ε+ϑ+νnr,τ+υ,bq,Bq.It seems that Theorem 14 readily follows due to (52) and (53).


## 5. Conclusions

In this article, we have investigated and studied two classes of generalized fractional integral operators involving Fox’sH-function as kernel due to Saxena and Kumbhat which are applied on M-L function. We discussed the actions of fractional integral operators under Euler, Mellin, Laplace, Whittaker, and K-transforms and results are given in better pragmatic series solutions. The majority of the results derived here are general in nature and compact forms are fairly helpful in deriving a variety of integral formulas in the theory of integral operators which arises in a range of problems of applied sciences like kinematics, diffusion equation, kinetic equation, fractal geometry, anomalous diffusion, propagation of seismic waves, turbulence, etc. We may obtain other special functions such as M-L function and Bessel-Maitland function (see, e.g., ([19–21]) as its special cases and, therefore, various unified fractional integral presentations can be obtained as special cases of our results.



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