# Certain Properties of GeneralizedM-Series under Generalized Fractional Integral Operators

Authors: D. L. Suthar; Fasil Gidaf; Mitku Andualem
Journal: Journal of Mathematics
 (2021)

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## Abstract

The aim of this study is to introduce new (presumed) generalized fractional integral operators involvingI-function as a kernel. In addition, two theorems have been developed under these operators that provide an image formula for this generalized M-series and also to study the different properties of the generalized M-series. The corresponding assertions in terms of Euler and Laplace transform methods are presented. Due to the general nature of the I-function and the generalized M-series, a number of results involving special functions can be achieved only by making appropriate values for the parameters.


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## Body

## 1. Introduction

Recently, in a short note, Sharma and Jain [1] introduced and studied a new special function called as generalized M-series, which is a particular case of the Wright generalized hypergeometric function ψpq. ([2], p. 56, equation (1.11.14)) and Fox’s H-function [3–5]. The generalized M-series is important because its basic cases are followed by the Mittag-Leffler function and hypergeometric function, and all these functions have actually discovered key implementations in solving problems in applied sciences, chemistry, physics, and biology. A number of researchers [6–10] have also investigated the structure, implementations, and various directions of extensions of the fractional integration and differentiation in detail. The series is defined for z,φ,ς∈ℂ, ℜφ>0, and αi,βj∈ℝ−∞,∞, αi:i=1,2,…,p;βj≠0:j=1,2,…,q as(1)Mφ,ςpqz=Mφ,ςpqα1,…,αp;β1,…,βq;z=∑k=0∞α1k…αpkβ1k…βqkzkΓφk+ς,where αjk,βjk are showing the results for Pochhammer symbols. The series (1) is defined when none of the parameters βjs,j=1,…,q is a negative integer or zero; if any numerator parameter αj is a negative integer or zero, then the series terminates to a polynomial in z. The series in (1) is convergent for all z if p≤q, it is convergent for z<ϑ=φφ if , and it is divergent if p>q+1. When p=q+1 and z<ϑ, the series can converge on conditions depending on the parameters ([2], for the general theory of the Wright function). The summation of the convergent series is denoted by the symbol Mφ,ςpq..Some essential special cases of the generalizedM-series are mentioned in the following:(1)
Forφ=ς=1, the generalized M-series is the generalized hypergeometric function [11, 12].(2)Fpqα1,…,αpβ1,…,βq;z=M1,1pqα1,…,αp;β1,…,βq;z=∑k=0∞α1k…αpkβ1k…βqkzkk!.(2)
Whenp=q=0 and ς=1, we have(3)Eφz=Mφ,100−;−;z=∑k=0∞zkΓφk+1,φ>0,Where the symbolEφz denotes the Mittag-Leffler function [13].(3)
Again, forp=q=0, we have(4)Eφ,ςz=Mφ,ς00−;−;z=∑k=0∞zkΓφk+ς,φ>0,ς>0,Where the symbolEφ,ςz denotes the two-index Mittag-Leffler function introduced by Wiman [14].(4)
Furthermore, if we putp=q=1, α1=σ∈ℂ,β1=1, the generalized M-series reduces to the generalized Mittag-Leffler function [12, 15] as follows:(5)Eφ,ςσz=Mφ,ς11σ;1;z=∑k=0∞σk1kzkΓφk+ς=∑k=0∞σkΓφk+ςzkk!.In the present study, our aim is to study some fundamental properties of generalizedM-series defined by (1), for which, we consider the two generalized fractional integral operators involving the I-function as kernel, which is described in the next section.


## 2. Generalized Fractional Integral Operators

In this section, we are introducing new (presumed) generalized fractional integral operators involvingI-function as kernel, which are the extensions of Saxena and Kumbhat operators [16, 17]:(6)S0,x;rμ,ϑfx=rx−μ−rϑ−1∫0xtμxr−trϑ×Ipi,qi:rm,nλU|aj,Aj1,n;aji,Ajin+1,pibj,Bj1,m;bji,Bjim+1,qiftdtand(7)Sx,∞;rε,ϑfx=rxε∫x∞t−ε−rϑ−1tr−xrϑ×Ipi,qi:rm,nλV|aj,Aj1,n;aji,Ajin+1,pibj,Bj1,m;bji,Bjim+1,qiftdt,where(8)U=trxrτ1−trxrvandV=xrtrτ1−xrtrv;τ,ν>0.The sufficient conditions of these operators are(i)
1≤p,q<∞,p−1+q−1=1;(ii)
ℜμ+rτbj/Bj>−q−1;ℜϑ+rτbj/Bj>−q−1;(9)ℜε+ϑ+rτbjBj>−p−1;j=1,2,…,m.(iii)
fx∈Lp0,∞(iv)
argλ≤πΘ/2,Θ>0(10)Θ=∑j=1mAi+∑j=1nBi−max1≤i≤r∑j=n+1piAji+∑j=m+1qiBji.where the I-function, which is more general than Fox’s H-function, is defined by Saxena [18], by means of the following Mellin-Barnes type contour integral:(11)Iz=Ipi,qi;rm,nz|aj,Aj1,n;aji,Ajin+1,pibj,Bj1,m;bji,Bjim+1,qi=12πω∫Lφζzζdζ,where ω=−1 and(12)φζ=∏j=1mΓbj−Bjζ∏j=1nΓ1−aj+Ajζ∑j=1r∏j=m+1qiΓ1−bji+Bjiζ∏j=n+1piaji−Ajiζ,pi,qii=1,…,r,m,n are the integers satisfying 0≤n≤pi,0≤m≤qi;Aj,Bj,Aji,Bji are the real and positive numbers, and aj,bj,aji,bji are the complex numbers. L is a suitable contour of the Mellin-Barnes type running from γ−iφ to γ+iφ (γ is real) in the complex ζ-plane. Details regarding existence conditions and various parametric restrictions of I-function are provided by Saxena [18].Forr=1, (11) reduces to Fox’ H-function:(13)Ipi,qi;1m,nz|aj,Aj1,n;aji,Ajin+1,pibj,Bj1,m;bji,Bjim+1,qi=Hpi,qim,nz|aj,Aj1,n;aj,Ajn+1,pbj,Bj1,m;bj,Bjm+1,q.


## 3. Images of GeneralizedM-Series under the Generalized Fractional Integral Operators

In this section, we established the image formula for the generalizedM-series (1) under the generalized fractional integral operators (6) and (7) in terms of the I-function as the kernel. The results are given in Theorems 1 and 2.Theorem 1.
Leta>0,x>0;φ,ς,η,ξ∈ℂ,ℜφ>0,ℜξ>0,ℜη>0,ℜς>0,1≤p≤2, then(14)S0,x;rμ,ϑtη−1Mφ,ςpqatξx=xη−1∑k=0∞α1k…αpkβ1k…βqkaxξkΓφk+ς×Ip+2,q+1;rm,n+2λ|aj,Aj1,n;aji,Ajin+1,pi,1−μ+η+ξkr,τ,−ϑ,v−ϑ−μ+η+ξkr,τ+v,bj,ςj1,m;bji,ςjim+1,qi,provided the conditions, stated with operator (6), are satisfied.Proof.
We assumeΩ1 be the on the left-hand side of (14); using the definition of generalized M-series (1) and the generalized fractional integral operator (6) on the left-hand side of (14), we have(15)Ω1=rx−μ−rϑ−1∫0xtμ+η−1xr−trϑ12πω∫lφζλUζdζ×∑k=0∞α1k…αpkβ1k…βqkatξkΓφk+ςdt.
Now, by changing the order of the integration which is valid under the given with theorem, we get(16)Ω1=rx−μ−rϑ−1∑k=0∞α1k…αpkβ1k…βqkakΓφk+ς×12πω∫Lφζλζxrϑ−rτζ∫0xtμ+η+ξk+rτζ−11−trxrϑ+vζdtdζ.
Let the substitutiontr/xr=w and then t=xw1/r in (16), we get(17)Ω1=xη−1∑k=0∞α1k…αpkβ1k…βqkaxξkΓφk+ς×12πω∫Lφζλζ∫01wμ+η+ξk/r+τζ−11−wϑ+vζdwdζ.
Using the definition of the well-known beta function in the inner integral, we have(18)Ω1=xη−1∑k=0∞α1k…αpkβ1k…βqkaxξkΓk+ς12πω∫LφζλζΓμ+η+ξk/r+τζΓ1+ϑ+vζΓ1+ϑ+μ+η+ξk/r+τ+vζdζ.
Interpreting the right-hand side of (18), in view of the definition (11), we arrive at the result (14).Theorem 2.
Leta>0,x>0;φ,ς,η,ξ∈ℂ,ℜφ>0,ℜξ>0,ℜη>0,ℜς>0,1≤p≤2, then(19)Sx,∞;rε,ϑt−ηMφ,ςpqatξx=x−η∑k=0∞α1k…αpkβ1k…βqka/xξkΓφk+ς×Ip+2,q+1;rm,n+2λ|aj,Aj1,n;aji,Ajin+1,pi,1−ε+η+ξkr,τ,−ϑ,v−ϑ−ε+η+ξkr,τ+v,bj,ςj1,m;bji,ςjim+1,qi,provided the conditions, stated with operator (7), are satisfied.Proof.
On the left-hand side of (19), let Ω2, using (1) and (7) on the left-hand side of (19), we have(20)Ω2=rxε∫x∞t−ε−η−rϑ−1tr−xrϑ12πω∫LφζλVζdζ×∑k=0∞α1k…αpkβ1k…βqka/tξkΓφk+ςdt.
Now, by changing the order of the integration which is valid under the given stated theorem, we get(21)Ω2=rxε∑k=0∞α1k…αpkβ1k…βqkakΓφk+ς×12πω∫Lφζλζxrτζ∫x∞t−ε−η−ξk−rτζ−11−xrtrϑ+vζdtdζ.
Let the replacementxr/tr=w and then t=x/w1/r in (21), we get(22)Ω2=x−η−1∑k=0∞α1k…αpkβ1k…βqkakx−ξkΓφk+ς×12πω∫Lφζλζ∫01wε+η+ξk/r+τζ−11−wϑ+vζdwdζ.
By beta function, we have(23)Ω2=x−η+rφ∑k=0∞α1k…αpkβ1k…βqkax−ξkΓφk+ς12πω∫Lφζλζ×Γε+η+ξk/r+τζΓ1+ϑ+vζΓε+η+ξk/r+1+ϑ+τ+vζdζ.
Interpreting the right-hand side of (23), in view of the definition (11), we arrive at the result (19).


## 4. Special Cases

(1)
If we putφ=ς=1 in Theorems 1 and 2, we obtain the following interesting results on the right, and it is known as the generalized hypergeometric function.Corollary 1.
Forφ=ς=1, equation (14) reduces in the following form:(24)S0,x;rμ,ϑtη−1M1,1pqα1,…,αp;β1,…,βq;atξx=xη−1Fpqα1,…,αpβ1,…,βq;axξ×Ip+2,q+1;rm,n+2λ|aj,Aj1,n;aji,Ajin+1,pi,1−μ+η+ξkr,τ,−ϑ,v−ϑ−μ+η+ξkr,τ+v,bj,ςj1,m;bji,ςjim+1,qi.Corollary 2.
Forφ=ς=1, equation (19) reduces in the following form:(25)Sx,∞;rε,ϑt−ηM1,1pqα1,…,αp;β1,…,βq;atξx=x−ηFpqα1,…,αpβ1,…,βq;axξ×Ip+2,q+1;rm,n+2λ|aj,Aj1,n;aji,Ajin+1,pi,1−ε+η+ξkr,τ,−ϑ,v−ϑ−ε+η+ξkr,τ+v,bj,ςj1,m;bji,ςjim+1,qi.(2)
If we putp=q=0 in Theorems 1 and 2, we obtain the following interesting results on the right, and it is known as the two-index Mittag-Leffler function.Corollary 3.
Forp=q=0, equation (16) reduces in the following form:(26)S0,x;rμ,ϑtη−1Mφ,ς00−;−;atξx=xη−1Eφ,ςaxξ×Ip+2,q+1;rm,n+2λ|aj,Aj1,n;aji,Ajin+1,pi,1−μ+η+ξkr,τ,−ϑ,v−ϑ−μ+η+ξkr,τ+v,bj,ςj1,m;bji,ςjim+1,qi.Corollary 4.
Forp=q=0, equation (19) reduces in the following form:(27)Sx,∞;rε,ϑt−ηMφ,ς00−;−;atξx=x−ηEφ,ςaxξ×Ip+2,q+1;rm,n+2λ|aj,Aj1,n;aji,Ajin+1,pi,1−ε+η+ξkr,τ,−ϑ,v−ϑ−ε+η+ξkr,τ+v,bj,ςj1,m;bji,ςjim+1,qi.(3)
If we putp=q=1, α1=σ∈ℂ,β1=1 in Theorems 1 and 2, we obtain the following interesting results on the right, and it is known as the generalized Mittag-Leffler function.Corollary 5.
Forp=q=1, a1=ξ∈ℂ,b1=1, equation (14) reduces in the following form:(28)S0,x;rμ,ϑtη−1Mφ,ς11σ;1;atξx=xη−1Eφ,ςσaxξ×Ip+2,q+1;rm,n+2λ|aj,Aj1,n;aji,Ajin+1,pi,1−μ+η+ξkr,τ,−ϑ,v−ϑ−μ+η+ξkr,τ+v,bj,ςj1,m;bji,ςjim+1,qi.Corollary 6.
Forp=q=1, α1=σ∈ℂ,β1=1, equation (19) reduces in the following form:(29)Sx,∞;rε,ϑt−ηMφ,ς11σ;1;atξx=x−ηEφ,ςσaxξ×Ip+2,q+1;rm,n+2λ|aj,Aj1,n;aji,Ajin+1,pi,1−ε+η+ξkr,τ,−ϑ,v−ϑ−ε+η+ξkr,τ+v,bj,ςj1,m;bji,ςjim+1,qi.


## 5. Certain Integral Transforms

In this section, with the aid of the results developed in the prior segment, we will provide some very important outcomes of several theorems connected with the transforms of Euler and Laplace. To this end, we would like to define these transforms first.Definition 1.
The well-known Euler transform (e.g., [19]) of a function ft is defined as(30)Bft;c,d=∫01tc−11−td−1ftdt;c,d∈ℂ,ℜc>0,ℜd>0.Definition 2.
The Laplace transform (e.g., [19]) of the function ft is defined, as usual, by(31)Lft;s=∫0∞e−stftdt;ℜs>0.
This section would establish the following fascinating outcomes in the form of theorems. As these findings are direct implications of Definitions1 and 2 and Theorems 1 and 2, they are provided without evidence here.Theorem 3.
The Euler transform of the Theorem1 gives the following result:(32)BS0,x;rμ,ϑtη−1Mφ,ςpqatξ;c,d=∑k=0∞α1k…αpkβ1k…βqkakΓφk+ςBc+η−1+ξk,d×Ip+2,q+1;rm,n+2λ|aj,Aj1,n;aji,Ajin+1,pi,1−μ+η+ξkr,τ,−ϑ,v−ϑ−μ+η+ξkr,τ+v,bj,ςj1,m;bji,ςjim+1,qi,provided that the conditions mentioned with the operator and Euler transform are satisfied.Theorem 4.
The Euler transform of the Theorem2 gives the following result:(33)BSx,∞;rε,ϑt−ηMφ,ςpqatξ;c,d=∑k=0∞α1k…αpkβ1k…βqkakΓφk+ςBc−η−ξk,d×Ip+2,q+1;rm,n+2λ|aj,Aj1,n;aji,Ajin+1,pi,1−ε+η+ξkr,τ,−ϑ,v−ϑ−ε+η+ξkr,τ+v,bj,ςj1,m;bji,ςjim+1,qi,provided that the conditions mentioned with the operator and Euler transform are satisfied.Theorem 5.
The Laplace transform of the Theorem1 gives the following result:(34)LS0,x;rμ,ϑtη−1Mφ,ςpqatξ;s=s−η∑k=0∞α1k…αpkβ1k…βqkΓη+ξkΓφk+ςas−ξk×Ip+2,q+1;rm,n+2λ|aj,Aj1,n;aji,Ajin+1,pi,1−μ+η+ξkr,τ,−ϑ,v−ϑ−μ+η+ξkr,τ+v,bj,ςj1,m;bji,ςjim+1,qi,provided that the conditions mentioned with the operator and Laplace transform are satisfied.Theorem 6.
The Laplace transform of the Theorem2 gives the following result:(35)LSx,∞;rε,ϑt−ηMφ,ςpqatξ;s=sη−1∑k=0∞α1k…αpkβ1k…βqkΓ1−η−ξkΓφk+ςasξk×Ip+2,q+1;rm,n+2λ|aj,Aj1,n;aji,Ajin+1,pi,1−ε+η+ξkr,τ,−ϑ,v−ϑ−ε+η+ξkr,τ+v,bj,ςj1,m;bji,ςjim+1,qi,provided that the conditions mentioned with the operator and Laplace transform are satisfied.


## 6. Properties of Generalized Fractional Integral Operators

Here, we establish some properties of the operators as consequences of Theorems1 and 2. These properties show compositions of the power function.Theorem 7.
Following all the conditions on parameters as stated in Theorem1 with ℜψ+η>0, then the following result holds true:(36)xψS0,x;rμ,ϑtη−1Mφ,ςpqatξx=S0,x;rμ−ψ,ϑtψ+η−1Mφ,ςpqatξx.Proof.
Let us use (14) in the left-hand side of (36), and we get(37)xψS0,x;rμ,ϑtη−1Mφ,ςpqatξx=∑k=0∞α1k…αpkβ1k…βqkakΓφn+ςxη+ψ+ξk−1×Ip+2,q+1;rm,n+2λ|aj,Aj1,n;aji,Ajin+1,pi,1−μ+η+ξkr,τ,−ϑ,v−ϑ−μ+η+ξkr,τ+v,bj,ςj1,m;bji,ςjim+1,qi.
Again, using (14) in the right-hand side of (37), we get(38)S0,x;rμ−ψ,ϑtψ+η−1Mφ,ςpqatξx=∑k=0∞α1k…αpkβ1k…βqkakΓφk+ςxη+ψ+ξk−1×Ip+2,q+1;rm,n+2λ|aj,Aj1,n;aji,Ajin+1,pi,1−μ+η+ξkr,τ,−ϑ,v−ϑ−μ+η+ξkr,τ+v,bj,ςj1,m;bji,ςjim+1,qi.
It seems that Theorem6 readily follows due to (37) and (38).Theorem 8.
Follow all the conditions on parameters as stated in Theorem2 with ℜ1−ψ+η<1; then, the following result holds true:(39)x−ψSx,∞;rε,ϑt−ηMφ,ςpqatξx=Sx,∞;rε−ψ,ϑt−ψ−ηMφ,ςpqatξx.Proof.
From (14) in the left-hand side of (39), we get(40)x−ψSx,∞;rε,ϑt−ηMφ,ςpqatξx=∑k=0∞α1k…αpkβ1k…βqkakΓφk+ςx−η−ψ−ξk×Ip+2,q+1;rm,n+2λ|aj,Aj1,n;aji,Ajin+1,pi,1−ε+η+ξkr,τ,−ϑ,v−ϑ−ε+η+ξkr,τ+v,bj,ςj1,m;bji,ςjim+1,qi.
Again, using (14) in the right-hand side of (39), we get(41)Sx,∞;rε−ψ,ϑt−ψ−ηMφ,ςpqatξx=∑k=0∞α1k…αpkβ1k…βqkakΓφk+ςx−η−ψ−ξk×Ip+2,q+1;rm,n+2λ|aj,Aj1,n;aji,Ajin+1,pi,1−ε+η+ξkr,τ,−ϑ,v−ϑ−ε+η+ξkr,τ+v,bj,ςj1,m;bji,ςjim+1,qi.
It seems that Theorem7 readily follows due to (40) and (41).


## 7. Concluding Remark and Observations

In this study, we introduced and studied the properties of generalizedM-series under the new (presumed) generalized fractional integral operators which are defined in equations (6) and (7) and also developed some new images. The results established in this study contain various special cases, such that if we take r=1, we recover the known results recorded in [20]. Furthermore, we can present certain very interesting results in the form of several theorems associated with Mellin, Whittaker, and K-transforms. We left this as an exercise to the interested reader.



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