# On a Unified Mittag-Leffler Function and Associated Fractional Integral Operator

Authors: Yanyan Zhang; Ghulam Farid; Zabidin Salleh; Ayyaz Ahmad
Journal: Mathematical Problems in Engineering
 (2021)

---

## Abstract

The aim of this paper is to unify the extended Mittag-Leffler function and generalizedQ function and define a unified Mittag-Leffler function. Both the extended Mittag-Leffler function and generalized Q function can be obtained from the unified Mittag-Leffler function. The Laplace, Euler beta, and Whittaker transformations are applied for this function, and generalized formulas are obtained. These formulas reproduce integral transformations of various deduced Mittag-Leffler functions and Q function. Also, the convergence of this unified Mittag-Leffler function is proved, and an associated fractional integral operator is constructed.


---

## Body

## 1. Introduction

The exponential function naturally exists in the solution of differential equations and plays a very vital role in solving real-world problems modeled in the form of differential mathematical systems. At the same time, the Mittag-Leffler function provides assistance in the formulation of solutions of complicated fractional dynamical systems. The aim of this paper is to unify two types of functions, namely, an extended generalized Mittag-Leffler function given in (7) and the Q function given in (8). We study Laplace, Euler beta, and Whittaker transformations of extended generalized Mittag-Leffler function given in (7) and the Q function given in (8) in the compact formulas. Also we will define a compact form of fractional integral operator.First, we give some basic definitions and notations which will be helpful to understand later definitions. These include the Laplace transform, Euler beta transform, Whittaker transform, gamma functionΓ, beta function B, p-beta function Bp, Mittag-Leffler function Eα,β, extended Mittag-Leffler function Eα,β,γλ,r,k,θ, the fractional integral operator associated with extended Mittag-Leffler function and generalized Q function Qα,β,γ,δ,μ,νλ,ρ,θ,k,n.Definition 1.
(see [1]). Laplace transform of an integrable function f on 0,∞ is defined as follows:(1)Lft=∫0∞e−stftdt,where s∈ℂ is the variable of the transform.Definition 2.
(see [2]). The Euler beta transform of a function f is defined by the following definite integral:(2)Bft;a,b=∫01ta−11−tb−1ftdt,where a and b are any complex number with ℜa>0 and ℜb>0.Definition 3.
(see [2]). The Whittaker transform is defined by the following improper integral:(3)∫0∞e−t/2tν−1ωλ,μtdt=Γ1/2+μ+νΓ1/2−μ+νΓ1−λ+ν,where ℜμ±ν>−1/2 and ωλ,μ is the Whittaker confluent hypergeometric function.Definition 4.
(see [3]). The gamma function is defined by the following improper integral:(4)Γz=∫0∞e−ttz−1dt,where ℜz>0.Definition 5.
(see [2]). The beta function is defined by a definite integral and is given by(5)Bm,n=∫01tm−11−tn−1dt,where ℜm,ℜn>0.Definition 6.
(see [4]). The Mittag-Leffler function with two parameters is defined by the following series:(6)Eα,βz=∑l=0∞zlΓαl+β,where ℜα>0.Definition 7.
(see [5]). An extended and generalized Mittag-Leffler function Eα,β,γδ,μ,k,ν is defined by the following series:(7)Eα,β,γλ,r,k,θz;p=∑l=0∞Bpλ+lk,θ−λθlkzlBλ,θ−λγlrΓαl+β,where z,α,β,γ,θ,λ∈ℂ,ℜα,ℜβ,ℜγ,ℜθ,ℜλ>0 with p≥0, r>0, 0<k≤r+ℜα, and θlk=Γθ+lk/Γθ.Definition 8.
(see [2]). A generalized Q function Qα,β,γ,δ,μ,νλ,ρ,θ,k,n is defined by the following series:(8)Qα,β,γ,δ,μ,νλ,ρ,θ,k,nz;a¯,b¯=∑l=0∞∏i=1nBbi,lλρlθklzl∏i=1nBai,lγδlμνlΓαl+β,where a¯=a1,a2,…,an,b¯=b1,b2,…,bn,α,β,γ,δ,μ,ν,λ,ρ,θ,ai,bi∈ℂ, minℜα,ℜβ,ℜγ,ℜθ,ℜλ,ℜδ,ℜρ>0,k∈0,1∪ℕ.The Mittag-Leffler function takes place naturally similar to that of the exponential function in the solutions of fractional integro-differential equations having the arbitrary order. The Mittag-Leffler functions have to gain more recognition due to their wide applications in diverse fields [5–9]. They are used to define new fractional integral operators, and the fractional integral operators are used to generalize mathematical inequalities, see [5, 8, 10–14].Our motivation is to introduce the unified Mittag-Leffler function. In this paper, we unify the extended Mittag-Leffler function (7) and generalized Q function (8) in a single function named unified Mittag-Leffler function. We studied the Laplace, Euler beta, and Whittaker transformation of the unified Mittag-Leffler function and obtained the compact formulas which reproduce integral transformations of Mittag-Leffler function and generalized Q function. Furthermore, the convergence of unified Mittag-Leffler function is proved, associated fractional integral operator is defined, and its boundedness is provided.In the next section, we give the definition of unified Mittag-Leffler function and deduce extended generalized Mittag-Leffler function and generalizedQ functions.


## 2. Unified Mittag-Leffler Function

We define a Mittag-Leffler function (will be called the unified Mittag-Leffler function) which unifies the functions given in (7) and (8) as follows:(9)Mα,β,γ,δ,μ,νλ,ρ,θ,k,nz;a¯,b¯,c¯,p=∑l=0∞∏i=1nBpbi,aiλρlθkl∏i=1nBci,aiγδlμνlzlΓαl+β,where a¯=a1,a2,…,an,b¯=b1,b2,…,bn,c¯=c1,c2,…,cn,ai,bi,ci∈ℂ;i=1,…,n such that ℜai,ℜbi,ℜci>0,∀i. Also let α,β,γ,δ,μ,ν,λ,ρ, θ,z∈ℂ, minℜα,ℜβ,ℜγ,ℜδ,ℜλ,ℜθ>0, and k∈0,1∪ℕ with k+ℜρ<ℜδ+ν+α, Imρ=Imδ+ν+α.Forn=1, (9) will obtain the following form:(10)Mα,β,γ,δ,μ,νλ,ρ,θ,k,1z;a¯,b¯,c¯,p=∑l=0∞Bpb1,a1λρlθklBc1,a1γδlμνlzlΓαl+β.By settingb1=c1+lk,a1=θ−λ,c1=λ,ρ=ν=0,δ>0 in (10), we will get (7). Also, by substituting ai=l,p=0 and ℜρ>0 in (9), we will obtain (8). Hence the newly defined Mittag-Leffler function provides different kinds of related functions by setting the specific values of the parameters. The functions defined in [2, 4–9, 15] are particular cases of this newly defined function.


### 2.1. Integral Transforms of Unified Mittag-Leffler Function

Now we give the integral transforms of the unified Mittag-Leffler function. These transformations include the Laplace transform, Euler beta transform, and Whittaker transform.


#### 2.1.1. Laplace Transform

First we give the Laplace transform of the unified Mittag-Leffler function.Theorem 1.
Fora¯=a1,a2,…,an,b¯=b1,b2,…,bn,c¯=c1,c2,…,cn,ai,bi,ci∈ℂ;i=1,…,n such that ℜai,ℜbi,ℜci>0,∀i. Also let α,β,γ,δ,μ,ν,λ,ρ, θ,t∈ℂ, minℜα,ℜβ,ℜγ,ℜδ,ℜλ,ℜθ>0, and k∈0,1∪ℕ with k+ℜρ<ℜδ+ν+α, Imρ=Imδ+ν+α, and the Laplace transform of unified Mittag-Leffler function is given as follows:(11)LMα,β,γ,δ,μ,νλ,ρ,θ,k,nt;a¯,b¯,c¯,p=s−1Mα,β,γ,δ,μ,νλ,ρ,θ,k,1,ns−1;a¯,b¯,c¯,p.Proof.
By the definition of the Laplace transform of a function, we have(12)LMα,β,γ,δ,μ,νλ,ρ,θ,k,nt;a¯,b¯,c¯,p=∫0∞e−st∑l=0∞∏i=1nBpbi,aiλρlθkl∏i=1nBci,aiγδlμνltlΓαl+βdt=∑l=0∞∏i=1nBpbi,aiλρlθkl∏i=1nBci,aiγδlμνl1Γαl+βl!sl+1=1s∑l=0∞∏i=1nBpbi,aiλρlθkl∏i=1nBci,aiγδlμνl1Γαl+βs−l1l.
Hence the Laplace transform of the unified Mittag-Leffler function can be given as follows:(13)LMα,β,γ,δ,μ,νλ,ρ,θ,k,nt;a¯,b¯,c¯,p=s−1Mα,β,γ,δ,μ,νλ,ρ,θ,k,1,ns−1;a¯,b¯,c¯,p.Corollary 1.
Forai=l,p=0, and ℜρ>0, the Laplace transform of the unified Mittag-Leffler function will become(14)LMα,β,γ,δ,μ,νλ,ρ,θ,k,nt;a¯,b¯,c¯,p=LQα,β,γ,δ,μ,νλ,ρ,θ,k,nt;b¯,c¯.Proof.
From Theorem1, we have(15)LMα,β,γ,δ,μ,νλ,ρ,θ,k,nt;a¯,b¯,c¯,p=s−1∑l=0∞∏i=1nBpbi,aiλρlθkl∏i=1nBci,aiγδlμνl1ls−lΓαl+β.
Forp=0, Bpx,y=Bx,y.
Therefore the above expression becomes(16)LMα,β,γ,δ,μ,νλ,ρ,θ,k,nt;a¯,b¯,c¯,p=s−1∑l=0∞∏i=1nBbi,aiλρlθkl∏i=1nBci,aiγδlμνl1ls−lΓαl+β.
Puttingai=l and ℜρ>0, we get the following:(17)LMα,β,γ,δ,μ,νλ,ρ,θ,k,nt;a¯,b¯,c¯,p=s−1∑l=0∞∏i=1nBbi,lλρlθkl∏i=1nBci,lγδlμνll!s−lΓαl+β.
Hence we get(18)LMα,β,γ,δ,μ,νλ,ρ,θ,k,nt;a¯,b¯,c¯,p=LQα,β,γ,δ,μ,νλ,ρ,θ,k,nt;b¯,c¯.
Similarly forn = 1, b1 = c1 + lk, a1 = θ − λ, c1 = λ, ρ = ν = 0 and δ > 0 one can have(19)LMα,β,γ,δ,μ,νλ,ρ,θ,k,nt;a¯,b¯,c¯,p=LEα,β,γλ,δ,k,θt;p.


#### 2.1.2. Euler Beta Transform

The Euler beta transformation of the unified Mittag-Leffler function is given in the next theorem.Theorem 2.
Fora¯=a1,a2,…,an,b¯=b1,b2,…,bn,c¯=c1,c2,…,cn,ai,bi,ci∈ℂ;i=1,…,n such that ℜai,ℜbi,ℜci>0,∀i. Also let α,β,γ,δ,μ,ν,λ,ρ, θ,t∈ℂ, minℜα,ℜβ,ℜγ,ℜδ,ℜλ,ℜθ>0, and k∈0,1∪ℕ with k+ℜρ<ℜδ+ν+α, Imρ=Imδ+ν+α, and the Euler beta transform of the unified Mittag-Leffler function is given as follows:(20)BMα,β,γ,δ,μ,νλ,ρ,θ,k,nt;a¯,b¯,c¯,p;m,n=∑l=0∞∏i=1nBpbi,aiλρlθkl∏i=1nBci,aiγδlμνlBl+m,nΓαl+β.Proof.
By the definition of beta transform of an integrable function, we have the following:(21)BMα,β,γ,δ,μ,νλ,ρ,θ,k,nt;a¯,b¯,c¯,p;m,n=∫01tm−11−tn−1∑l=0∞∏i=1nBpbi,aiλρlθkl∏i=1nBci,aiγδlμνltlΓαl+βdt=∑l=0∞Πi=1nBpbi,aiλρlθklΠi=1nBci,aiγδlμνl1Γαl+β∫01tl+m−11−tn−1dt=∑l=0∞∏i=1nBpbi,aiλρlθkl∏i=1nBci,aiγδlμνlBl+m,nΓαl+β.Corollary 2.
Forai=l,p=0, and ℜρ>0, the Euler beta transform of unified Mittag-Leffler function will become(22)BMα,β,γ,δ,μ,νλ,ρ,θ,k,nt;a¯,b¯,c¯,p=BQα,β,γ,δ,μ,νλ,ρ,θ,k,nt;b¯,c¯.Similarly forn = 1, b1 = c1 + lk, a1 = θ − λ, c1 = λ, ρ = ν = 0 and δ > 0 one can have(23)BMα,β,γ,δ,μ,νλ,ρ,θ,k,nt;a¯,b¯,c¯,p=BEα,β,γλ,δ,k,θt;p.


#### 2.1.3. Whittaker Transform

The Whittaker transformation of the unified Mittag-Leffler function is given in the next theorem.Theorem 3.
Fora¯=a1,a2,…,an,b¯=b1,b2,…,bn,c¯=c1,c2,…,cn,ai,bi,ci∈ℂ;i=1,…,n such that ℜai,ℜbi,ℜci>0,∀i. Also let α,β,γ,δ,μ,ν,ζ,ρ, θ,t∈ℂ, minℜα,ℜβ,ℜγ,ℜδ,ℜζ,ℜθ>0, and k∈0,1∪ℕ with k+ℜρ<ℜδ+ν+α, Imρ=Imδ+ν+α, and we have(24)∫0∞e−ϕt/2tξ−1ωλ,ψϕtMα,β,γ,δ,μ,νζ,ρ,θ,k,nωtη;a¯,b¯,c¯,pdt=ϕ−ξf−1!g−1!Mα,β,γ,δ,μ,ν,g,η,1ζ,ρ,θ,k,f,η,nωϕ−η;a¯,b¯,c¯,p,where f=1/2±ψ+ξ and g=1−λ+ξ.Proof.
Consider the following improper integral:(25)∫0∞e−ϕt/2tξ−1ωλ,ψϕtMα,β,γ,δ,μ,νζ,ρ,θ,k,nωtη;a¯,b¯,c¯,pdt.
By substitutingϕt=q in the above integral, we obtain the following:(26)∫0∞e−q/2qϕξ−1ωλ,ψqMα,β,γ,δ,μ,νζ,ρ,θ,k,nωqϕη;a¯,b¯,c¯,pdqϕ=ϕ−ξ∫0∞e−q/2qξ−1ωλ,ψq∑l=0∞∏i=1nBpbi,aiζρlθkl∏i=1nBci,aiγδlμνlωlq/ϕηlΓαl+βdq=ϕ−ξ∑l=0∞∏i=1nBpbi,aiζρlθkl∏i=1nBci,aiγδlμνlωl1/ϕηlΓαl+β∫0∞e−q/2qξ+ηl−1ωλ,ψqdq.
By using the definition of Whittaker transformation, we get(27)=ϕ−ξ∑l=0∞∏i=1nBpbi,aiζρlθkl∏i=1nBci,aiγδlμνlωl1/ϕηlΓαl+βΓ1/2+ψ+ξ+ηlΓ1/2−ψ+ξ+ηlΓ1−λ+ξ+ηl=ϕ−ξ∑l=0∞∏i=1nBpbi,aiζρlθkl∏i=1nBci,aiγδlμνlωl1/ϕηlΓαl+βΓ1/2±ψ+ξ+ηlΓ1−λ+ξ+ηlΓ1+l=ϕ−ξ∑l=0∞∏i=1nBpbi,aiζρlθkl∏i=1nBci,aiγδlμνlωl1/ϕηlΓαl+βΓgΓf+ηlΓfΓgΓg+ηlΓ1+lΓf=ϕ−ξ∑l=0∞∏i=1nBpbi,aiζρlθkl∏i=1nBci,aiγδlμνlωl1/ϕηlΓαl+βf−1!fηlg−1!gηl1l=ϕ−ξf−1!g−1!Mα,β,γ,δ,μ,ν,g,η,1ζ,ρ,θ,k,f,η,nωϕ−η;a¯,b¯,c¯,p,where f:=1/2±ψ+ξ and g:=1−λ+ξ, and the required result is obtained.Corollary 3.
Forai=l,p=0 and ℜρ>0, we have(28)∫0∞e−ϕt/2tξ−1ωλ,ψϕtMα,β,γ,δ,μ,νζ,ρ,θ,k,nωtη;a¯,b¯,c¯,pdt=∫0∞e−ϕt/2tξ−1ωλ,ψϕtQα,β,γ,δ,μ,νζ,ρ,θ,k,nωtη;b¯,c¯dt.Similarly forn = 1, b1 = c1 + lk, a1 = θ − λ, c1 = λ, ρ = ν = 0 and δ > 0 one can have(29)∫0∞e−ϕt/2tξ−1ωλ,ψϕtMα,β,γ,δ,μ,νζ,ρ,θ,k,nωtη;a¯,b¯,c¯,pdt∫0∞e−ϕt/2tξ−1ωλ,ψϕtEα,β,γζ,δ,k,nωtηpdt.


## 2.1. Integral Transforms of Unified Mittag-Leffler Function

Now we give the integral transforms of the unified Mittag-Leffler function. These transformations include the Laplace transform, Euler beta transform, and Whittaker transform.


### 2.1.1. Laplace Transform

First we give the Laplace transform of the unified Mittag-Leffler function.Theorem 1.
Fora¯=a1,a2,…,an,b¯=b1,b2,…,bn,c¯=c1,c2,…,cn,ai,bi,ci∈ℂ;i=1,…,n such that ℜai,ℜbi,ℜci>0,∀i. Also let α,β,γ,δ,μ,ν,λ,ρ, θ,t∈ℂ, minℜα,ℜβ,ℜγ,ℜδ,ℜλ,ℜθ>0, and k∈0,1∪ℕ with k+ℜρ<ℜδ+ν+α, Imρ=Imδ+ν+α, and the Laplace transform of unified Mittag-Leffler function is given as follows:(11)LMα,β,γ,δ,μ,νλ,ρ,θ,k,nt;a¯,b¯,c¯,p=s−1Mα,β,γ,δ,μ,νλ,ρ,θ,k,1,ns−1;a¯,b¯,c¯,p.Proof.
By the definition of the Laplace transform of a function, we have(12)LMα,β,γ,δ,μ,νλ,ρ,θ,k,nt;a¯,b¯,c¯,p=∫0∞e−st∑l=0∞∏i=1nBpbi,aiλρlθkl∏i=1nBci,aiγδlμνltlΓαl+βdt=∑l=0∞∏i=1nBpbi,aiλρlθkl∏i=1nBci,aiγδlμνl1Γαl+βl!sl+1=1s∑l=0∞∏i=1nBpbi,aiλρlθkl∏i=1nBci,aiγδlμνl1Γαl+βs−l1l.
Hence the Laplace transform of the unified Mittag-Leffler function can be given as follows:(13)LMα,β,γ,δ,μ,νλ,ρ,θ,k,nt;a¯,b¯,c¯,p=s−1Mα,β,γ,δ,μ,νλ,ρ,θ,k,1,ns−1;a¯,b¯,c¯,p.Corollary 1.
Forai=l,p=0, and ℜρ>0, the Laplace transform of the unified Mittag-Leffler function will become(14)LMα,β,γ,δ,μ,νλ,ρ,θ,k,nt;a¯,b¯,c¯,p=LQα,β,γ,δ,μ,νλ,ρ,θ,k,nt;b¯,c¯.Proof.
From Theorem1, we have(15)LMα,β,γ,δ,μ,νλ,ρ,θ,k,nt;a¯,b¯,c¯,p=s−1∑l=0∞∏i=1nBpbi,aiλρlθkl∏i=1nBci,aiγδlμνl1ls−lΓαl+β.
Forp=0, Bpx,y=Bx,y.
Therefore the above expression becomes(16)LMα,β,γ,δ,μ,νλ,ρ,θ,k,nt;a¯,b¯,c¯,p=s−1∑l=0∞∏i=1nBbi,aiλρlθkl∏i=1nBci,aiγδlμνl1ls−lΓαl+β.
Puttingai=l and ℜρ>0, we get the following:(17)LMα,β,γ,δ,μ,νλ,ρ,θ,k,nt;a¯,b¯,c¯,p=s−1∑l=0∞∏i=1nBbi,lλρlθkl∏i=1nBci,lγδlμνll!s−lΓαl+β.
Hence we get(18)LMα,β,γ,δ,μ,νλ,ρ,θ,k,nt;a¯,b¯,c¯,p=LQα,β,γ,δ,μ,νλ,ρ,θ,k,nt;b¯,c¯.
Similarly forn = 1, b1 = c1 + lk, a1 = θ − λ, c1 = λ, ρ = ν = 0 and δ > 0 one can have(19)LMα,β,γ,δ,μ,νλ,ρ,θ,k,nt;a¯,b¯,c¯,p=LEα,β,γλ,δ,k,θt;p.


### 2.1.2. Euler Beta Transform

The Euler beta transformation of the unified Mittag-Leffler function is given in the next theorem.Theorem 2.
Fora¯=a1,a2,…,an,b¯=b1,b2,…,bn,c¯=c1,c2,…,cn,ai,bi,ci∈ℂ;i=1,…,n such that ℜai,ℜbi,ℜci>0,∀i. Also let α,β,γ,δ,μ,ν,λ,ρ, θ,t∈ℂ, minℜα,ℜβ,ℜγ,ℜδ,ℜλ,ℜθ>0, and k∈0,1∪ℕ with k+ℜρ<ℜδ+ν+α, Imρ=Imδ+ν+α, and the Euler beta transform of the unified Mittag-Leffler function is given as follows:(20)BMα,β,γ,δ,μ,νλ,ρ,θ,k,nt;a¯,b¯,c¯,p;m,n=∑l=0∞∏i=1nBpbi,aiλρlθkl∏i=1nBci,aiγδlμνlBl+m,nΓαl+β.Proof.
By the definition of beta transform of an integrable function, we have the following:(21)BMα,β,γ,δ,μ,νλ,ρ,θ,k,nt;a¯,b¯,c¯,p;m,n=∫01tm−11−tn−1∑l=0∞∏i=1nBpbi,aiλρlθkl∏i=1nBci,aiγδlμνltlΓαl+βdt=∑l=0∞Πi=1nBpbi,aiλρlθklΠi=1nBci,aiγδlμνl1Γαl+β∫01tl+m−11−tn−1dt=∑l=0∞∏i=1nBpbi,aiλρlθkl∏i=1nBci,aiγδlμνlBl+m,nΓαl+β.Corollary 2.
Forai=l,p=0, and ℜρ>0, the Euler beta transform of unified Mittag-Leffler function will become(22)BMα,β,γ,δ,μ,νλ,ρ,θ,k,nt;a¯,b¯,c¯,p=BQα,β,γ,δ,μ,νλ,ρ,θ,k,nt;b¯,c¯.Similarly forn = 1, b1 = c1 + lk, a1 = θ − λ, c1 = λ, ρ = ν = 0 and δ > 0 one can have(23)BMα,β,γ,δ,μ,νλ,ρ,θ,k,nt;a¯,b¯,c¯,p=BEα,β,γλ,δ,k,θt;p.


### 2.1.3. Whittaker Transform

The Whittaker transformation of the unified Mittag-Leffler function is given in the next theorem.Theorem 3.
Fora¯=a1,a2,…,an,b¯=b1,b2,…,bn,c¯=c1,c2,…,cn,ai,bi,ci∈ℂ;i=1,…,n such that ℜai,ℜbi,ℜci>0,∀i. Also let α,β,γ,δ,μ,ν,ζ,ρ, θ,t∈ℂ, minℜα,ℜβ,ℜγ,ℜδ,ℜζ,ℜθ>0, and k∈0,1∪ℕ with k+ℜρ<ℜδ+ν+α, Imρ=Imδ+ν+α, and we have(24)∫0∞e−ϕt/2tξ−1ωλ,ψϕtMα,β,γ,δ,μ,νζ,ρ,θ,k,nωtη;a¯,b¯,c¯,pdt=ϕ−ξf−1!g−1!Mα,β,γ,δ,μ,ν,g,η,1ζ,ρ,θ,k,f,η,nωϕ−η;a¯,b¯,c¯,p,where f=1/2±ψ+ξ and g=1−λ+ξ.Proof.
Consider the following improper integral:(25)∫0∞e−ϕt/2tξ−1ωλ,ψϕtMα,β,γ,δ,μ,νζ,ρ,θ,k,nωtη;a¯,b¯,c¯,pdt.
By substitutingϕt=q in the above integral, we obtain the following:(26)∫0∞e−q/2qϕξ−1ωλ,ψqMα,β,γ,δ,μ,νζ,ρ,θ,k,nωqϕη;a¯,b¯,c¯,pdqϕ=ϕ−ξ∫0∞e−q/2qξ−1ωλ,ψq∑l=0∞∏i=1nBpbi,aiζρlθkl∏i=1nBci,aiγδlμνlωlq/ϕηlΓαl+βdq=ϕ−ξ∑l=0∞∏i=1nBpbi,aiζρlθkl∏i=1nBci,aiγδlμνlωl1/ϕηlΓαl+β∫0∞e−q/2qξ+ηl−1ωλ,ψqdq.
By using the definition of Whittaker transformation, we get(27)=ϕ−ξ∑l=0∞∏i=1nBpbi,aiζρlθkl∏i=1nBci,aiγδlμνlωl1/ϕηlΓαl+βΓ1/2+ψ+ξ+ηlΓ1/2−ψ+ξ+ηlΓ1−λ+ξ+ηl=ϕ−ξ∑l=0∞∏i=1nBpbi,aiζρlθkl∏i=1nBci,aiγδlμνlωl1/ϕηlΓαl+βΓ1/2±ψ+ξ+ηlΓ1−λ+ξ+ηlΓ1+l=ϕ−ξ∑l=0∞∏i=1nBpbi,aiζρlθkl∏i=1nBci,aiγδlμνlωl1/ϕηlΓαl+βΓgΓf+ηlΓfΓgΓg+ηlΓ1+lΓf=ϕ−ξ∑l=0∞∏i=1nBpbi,aiζρlθkl∏i=1nBci,aiγδlμνlωl1/ϕηlΓαl+βf−1!fηlg−1!gηl1l=ϕ−ξf−1!g−1!Mα,β,γ,δ,μ,ν,g,η,1ζ,ρ,θ,k,f,η,nωϕ−η;a¯,b¯,c¯,p,where f:=1/2±ψ+ξ and g:=1−λ+ξ, and the required result is obtained.Corollary 3.
Forai=l,p=0 and ℜρ>0, we have(28)∫0∞e−ϕt/2tξ−1ωλ,ψϕtMα,β,γ,δ,μ,νζ,ρ,θ,k,nωtη;a¯,b¯,c¯,pdt=∫0∞e−ϕt/2tξ−1ωλ,ψϕtQα,β,γ,δ,μ,νζ,ρ,θ,k,nωtη;b¯,c¯dt.Similarly forn = 1, b1 = c1 + lk, a1 = θ − λ, c1 = λ, ρ = ν = 0 and δ > 0 one can have(29)∫0∞e−ϕt/2tξ−1ωλ,ψϕtMα,β,γ,δ,μ,νζ,ρ,θ,k,nωtη;a¯,b¯,c¯,pdt∫0∞e−ϕt/2tξ−1ωλ,ψϕtEα,β,γζ,δ,k,nωtηpdt.


## 2.1.1. Laplace Transform

First we give the Laplace transform of the unified Mittag-Leffler function.Theorem 1.
Fora¯=a1,a2,…,an,b¯=b1,b2,…,bn,c¯=c1,c2,…,cn,ai,bi,ci∈ℂ;i=1,…,n such that ℜai,ℜbi,ℜci>0,∀i. Also let α,β,γ,δ,μ,ν,λ,ρ, θ,t∈ℂ, minℜα,ℜβ,ℜγ,ℜδ,ℜλ,ℜθ>0, and k∈0,1∪ℕ with k+ℜρ<ℜδ+ν+α, Imρ=Imδ+ν+α, and the Laplace transform of unified Mittag-Leffler function is given as follows:(11)LMα,β,γ,δ,μ,νλ,ρ,θ,k,nt;a¯,b¯,c¯,p=s−1Mα,β,γ,δ,μ,νλ,ρ,θ,k,1,ns−1;a¯,b¯,c¯,p.Proof.
By the definition of the Laplace transform of a function, we have(12)LMα,β,γ,δ,μ,νλ,ρ,θ,k,nt;a¯,b¯,c¯,p=∫0∞e−st∑l=0∞∏i=1nBpbi,aiλρlθkl∏i=1nBci,aiγδlμνltlΓαl+βdt=∑l=0∞∏i=1nBpbi,aiλρlθkl∏i=1nBci,aiγδlμνl1Γαl+βl!sl+1=1s∑l=0∞∏i=1nBpbi,aiλρlθkl∏i=1nBci,aiγδlμνl1Γαl+βs−l1l.
Hence the Laplace transform of the unified Mittag-Leffler function can be given as follows:(13)LMα,β,γ,δ,μ,νλ,ρ,θ,k,nt;a¯,b¯,c¯,p=s−1Mα,β,γ,δ,μ,νλ,ρ,θ,k,1,ns−1;a¯,b¯,c¯,p.Corollary 1.
Forai=l,p=0, and ℜρ>0, the Laplace transform of the unified Mittag-Leffler function will become(14)LMα,β,γ,δ,μ,νλ,ρ,θ,k,nt;a¯,b¯,c¯,p=LQα,β,γ,δ,μ,νλ,ρ,θ,k,nt;b¯,c¯.Proof.
From Theorem1, we have(15)LMα,β,γ,δ,μ,νλ,ρ,θ,k,nt;a¯,b¯,c¯,p=s−1∑l=0∞∏i=1nBpbi,aiλρlθkl∏i=1nBci,aiγδlμνl1ls−lΓαl+β.
Forp=0, Bpx,y=Bx,y.
Therefore the above expression becomes(16)LMα,β,γ,δ,μ,νλ,ρ,θ,k,nt;a¯,b¯,c¯,p=s−1∑l=0∞∏i=1nBbi,aiλρlθkl∏i=1nBci,aiγδlμνl1ls−lΓαl+β.
Puttingai=l and ℜρ>0, we get the following:(17)LMα,β,γ,δ,μ,νλ,ρ,θ,k,nt;a¯,b¯,c¯,p=s−1∑l=0∞∏i=1nBbi,lλρlθkl∏i=1nBci,lγδlμνll!s−lΓαl+β.
Hence we get(18)LMα,β,γ,δ,μ,νλ,ρ,θ,k,nt;a¯,b¯,c¯,p=LQα,β,γ,δ,μ,νλ,ρ,θ,k,nt;b¯,c¯.
Similarly forn = 1, b1 = c1 + lk, a1 = θ − λ, c1 = λ, ρ = ν = 0 and δ > 0 one can have(19)LMα,β,γ,δ,μ,νλ,ρ,θ,k,nt;a¯,b¯,c¯,p=LEα,β,γλ,δ,k,θt;p.


## 2.1.2. Euler Beta Transform

The Euler beta transformation of the unified Mittag-Leffler function is given in the next theorem.Theorem 2.
Fora¯=a1,a2,…,an,b¯=b1,b2,…,bn,c¯=c1,c2,…,cn,ai,bi,ci∈ℂ;i=1,…,n such that ℜai,ℜbi,ℜci>0,∀i. Also let α,β,γ,δ,μ,ν,λ,ρ, θ,t∈ℂ, minℜα,ℜβ,ℜγ,ℜδ,ℜλ,ℜθ>0, and k∈0,1∪ℕ with k+ℜρ<ℜδ+ν+α, Imρ=Imδ+ν+α, and the Euler beta transform of the unified Mittag-Leffler function is given as follows:(20)BMα,β,γ,δ,μ,νλ,ρ,θ,k,nt;a¯,b¯,c¯,p;m,n=∑l=0∞∏i=1nBpbi,aiλρlθkl∏i=1nBci,aiγδlμνlBl+m,nΓαl+β.Proof.
By the definition of beta transform of an integrable function, we have the following:(21)BMα,β,γ,δ,μ,νλ,ρ,θ,k,nt;a¯,b¯,c¯,p;m,n=∫01tm−11−tn−1∑l=0∞∏i=1nBpbi,aiλρlθkl∏i=1nBci,aiγδlμνltlΓαl+βdt=∑l=0∞Πi=1nBpbi,aiλρlθklΠi=1nBci,aiγδlμνl1Γαl+β∫01tl+m−11−tn−1dt=∑l=0∞∏i=1nBpbi,aiλρlθkl∏i=1nBci,aiγδlμνlBl+m,nΓαl+β.Corollary 2.
Forai=l,p=0, and ℜρ>0, the Euler beta transform of unified Mittag-Leffler function will become(22)BMα,β,γ,δ,μ,νλ,ρ,θ,k,nt;a¯,b¯,c¯,p=BQα,β,γ,δ,μ,νλ,ρ,θ,k,nt;b¯,c¯.Similarly forn = 1, b1 = c1 + lk, a1 = θ − λ, c1 = λ, ρ = ν = 0 and δ > 0 one can have(23)BMα,β,γ,δ,μ,νλ,ρ,θ,k,nt;a¯,b¯,c¯,p=BEα,β,γλ,δ,k,θt;p.


## 2.1.3. Whittaker Transform

The Whittaker transformation of the unified Mittag-Leffler function is given in the next theorem.Theorem 3.
Fora¯=a1,a2,…,an,b¯=b1,b2,…,bn,c¯=c1,c2,…,cn,ai,bi,ci∈ℂ;i=1,…,n such that ℜai,ℜbi,ℜci>0,∀i. Also let α,β,γ,δ,μ,ν,ζ,ρ, θ,t∈ℂ, minℜα,ℜβ,ℜγ,ℜδ,ℜζ,ℜθ>0, and k∈0,1∪ℕ with k+ℜρ<ℜδ+ν+α, Imρ=Imδ+ν+α, and we have(24)∫0∞e−ϕt/2tξ−1ωλ,ψϕtMα,β,γ,δ,μ,νζ,ρ,θ,k,nωtη;a¯,b¯,c¯,pdt=ϕ−ξf−1!g−1!Mα,β,γ,δ,μ,ν,g,η,1ζ,ρ,θ,k,f,η,nωϕ−η;a¯,b¯,c¯,p,where f=1/2±ψ+ξ and g=1−λ+ξ.Proof.
Consider the following improper integral:(25)∫0∞e−ϕt/2tξ−1ωλ,ψϕtMα,β,γ,δ,μ,νζ,ρ,θ,k,nωtη;a¯,b¯,c¯,pdt.
By substitutingϕt=q in the above integral, we obtain the following:(26)∫0∞e−q/2qϕξ−1ωλ,ψqMα,β,γ,δ,μ,νζ,ρ,θ,k,nωqϕη;a¯,b¯,c¯,pdqϕ=ϕ−ξ∫0∞e−q/2qξ−1ωλ,ψq∑l=0∞∏i=1nBpbi,aiζρlθkl∏i=1nBci,aiγδlμνlωlq/ϕηlΓαl+βdq=ϕ−ξ∑l=0∞∏i=1nBpbi,aiζρlθkl∏i=1nBci,aiγδlμνlωl1/ϕηlΓαl+β∫0∞e−q/2qξ+ηl−1ωλ,ψqdq.
By using the definition of Whittaker transformation, we get(27)=ϕ−ξ∑l=0∞∏i=1nBpbi,aiζρlθkl∏i=1nBci,aiγδlμνlωl1/ϕηlΓαl+βΓ1/2+ψ+ξ+ηlΓ1/2−ψ+ξ+ηlΓ1−λ+ξ+ηl=ϕ−ξ∑l=0∞∏i=1nBpbi,aiζρlθkl∏i=1nBci,aiγδlμνlωl1/ϕηlΓαl+βΓ1/2±ψ+ξ+ηlΓ1−λ+ξ+ηlΓ1+l=ϕ−ξ∑l=0∞∏i=1nBpbi,aiζρlθkl∏i=1nBci,aiγδlμνlωl1/ϕηlΓαl+βΓgΓf+ηlΓfΓgΓg+ηlΓ1+lΓf=ϕ−ξ∑l=0∞∏i=1nBpbi,aiζρlθkl∏i=1nBci,aiγδlμνlωl1/ϕηlΓαl+βf−1!fηlg−1!gηl1l=ϕ−ξf−1!g−1!Mα,β,γ,δ,μ,ν,g,η,1ζ,ρ,θ,k,f,η,nωϕ−η;a¯,b¯,c¯,p,where f:=1/2±ψ+ξ and g:=1−λ+ξ, and the required result is obtained.Corollary 3.
Forai=l,p=0 and ℜρ>0, we have(28)∫0∞e−ϕt/2tξ−1ωλ,ψϕtMα,β,γ,δ,μ,νζ,ρ,θ,k,nωtη;a¯,b¯,c¯,pdt=∫0∞e−ϕt/2tξ−1ωλ,ψϕtQα,β,γ,δ,μ,νζ,ρ,θ,k,nωtη;b¯,c¯dt.Similarly forn = 1, b1 = c1 + lk, a1 = θ − λ, c1 = λ, ρ = ν = 0 and δ > 0 one can have(29)∫0∞e−ϕt/2tξ−1ωλ,ψϕtMα,β,γ,δ,μ,νζ,ρ,θ,k,nωtη;a¯,b¯,c¯,pdt∫0∞e−ϕt/2tξ−1ωλ,ψϕtEα,β,γζ,δ,k,nωtηpdt.


## 3. Convergence of Unified Mittag-Leffler Function

Before stating the theorem for the convergence of the unified Mittag-Leffler function, we give an important formula that will be used in the proof of our theorem.Definition 9.
The asymptotic formula for the gamma function is given in [5] the following:(30)Γa+zΓb+z=za−b1+a−ba+b−12z+O1z2,z⟶∞,argz<π.Theorem 4.
The unified Mittag-Leffler functionMα,β,γ,δ,μ,νλ,ρ,θ,k,n converges absolutely for all values of t∈ℂ if k+ℜρ<ℜδ+ν+α with Imρ=Imδ+ν+α.Proof.
By the definition of the unified Mittag-Leffler function, we have the following series:(31)Mα,β,γ,δ,μ,νλ,ρ,θ,k,nt;a¯,b¯,c¯,p=∑l=0∞∏i=1nBpbi,aiλρlθkl∏i=1nBci,aiγδlμνltlΓαl+β=∑l=0∞altl,where(32)al=∏i=1nBpbi,aiλρlθkl∏i=1nBci,aiγδlμνl1Γαl+βalal+1=λρlλρl+ρ⋅θklθkl+k⋅γδl+δγδl⋅μνl+νμνl⋅Γαl+α+βΓαl+β.
Applying limit on both sides, we get the following:(33)liml⟶∞alal+1=liml⟶∞λρlλρl+ρ⋅θklθkl+k⋅γδl+δγδl⋅μνl+νμνl⋅Γαl+α+βΓαl+β.
Using (30), the fractions involving Pochhammer symbols and gamma function in (33) become the following:(34)λρlλρl+ρ=ρl−ρ1−2λ+ρ−12l+O1ρl2,(35)θklθkl+k=kl−k1−2θ+k−12l+O1kl2,(36)γδl+δγδl=δlδ1+2γ+δ−12l+O1δl2,(37)μνl+νμνl=νlν1+2μ+ν−12l+O1νl2,(38)Γαl+α+βΓαl+β=αlα1+2β+α−12l+O1αl2.
Using (34)–(38) in (33), we get the following:(39)liml⟶∞alal+1≈liml⟶∞|ρl−ρ1−2λ+ρ−12l+O1ρl2×kl−k1−2θ+k−12l+O1kl2δlδ1+2γ+δ−12l+O1δl2×νlν1+2μ+ν−12l+O1νl2αlα1+2β+α−12l+O1αl2|,liml⟶∞alal+1≈liml⟶∞δδννααρρkk⋅lδ+ν+α−ρ+k.
The formula for the radius of convergence of a series is(40)liml⟶∞alal+1=R.
Therefore, the functionMα,β,γ,δ,μ,νλ,ρ,θ,k,n converges absolutely for all values of t if k+ℜρ<ℜδ+ν+α with Imρ=Imδ+ν+α. □Next, we give recurrence relations of unified Mittag-Leffler function.Theorem 5.
Leta¯=a1,a2,…,an,b¯=b1,b2,…,bn,c¯=c1,c2,…,cn,ai,bi,ci∈ℂ;i=1,…,n such that ℜai,ℜbi,ℜci>0,∀i. Also let α,β,γ,δ,μ,ν,λ,ρ, θ,t∈ℂ, minℜα,ℜβ,ℜγ,ℜδ,ℜλ,ℜθ>0, and k∈0,1∪ℕ with k+ℜρ<ℜδ+ν+α, Imρ=Imδ+ν+α, and then the difference of two consecutive unified Mittag-Leffler functions is given as follows:(41)Mα,β,γ,δ,μ,νλ,ρ,θ,k,nt;a¯,b¯,c¯,p−Mα,β,γ−1,δ,μ,νλ,ρ,θ,k,nt;a¯,b¯,c¯,p=tδ1−γddtMα,β,γ,δ,μ,νλ,ρ,θ,k,nt;a¯,b¯,c¯,p,with ℜγ>1.Proof.
By the definition of the unified Mittag-Leffler function, we have(42)Mα,β,γ,δ,μ,νλ,ρ,θ,k,nt;a¯,b¯,c¯,p−Mα,β,γ−1,δ,μ,νλ,ρ,θ,k,nt;a¯,b¯,c¯,p=∑l=0∞∏i=1nBpbi,aiλρlθkl∏i=1nBci,aiμνltlΓαl+β1γδl−1γ−1δl=∑l=0∞∏i=1nBpbi,aiλρlθkl∏i=1nBci,aiμνltlΓαl+βΓγΓγ+δlδl1−γ=δt1−γ∑l=0∞∏i=1nBpbi,aiλρlθkl∏i=1nBci,aiμνlγδlltl−1Γαl+β=tδ1−γddtMα,β,γ,δ,μ,νλ,ρ,θ,k,nt;a¯,b¯,c¯,p.Theorem 6.
Form∈ℤ+, a¯=a1,a2,…,an,b¯=b1,b2,…,bn,c¯=c1,c2,…,cn,whereai,bi,ci∈ℂ;i=1,…,n such that ℜai,ℜbi,ℜci>0,∀i. Also let α,β,γ,δ,μ,ν,λ,ρ, θ,t∈ℂ, minℜα,ℜβ,ℜγ,ℜδ,ℜλ,ℜθ>0 and k∈0,1∪ℕ with k+ℜρ<ℜδ+ν+α, Imρ=Imδ+ν+α, and mth derivative of unified Mittag-Leffler function is given by(43)ddtmMα,β,γ,δ,μ,νλ,ρ,θ,k,nt;a¯,b¯,c¯,p=λmρθmkγmδμmν∑l=0∞∏i=1nBpbi,aiλ+ρmρlθ+kmkl∏i=1nBci,aiγ+δmδlμ+νmνl⋅1+lmtlΓαl+m+β.Proof.
Differentiating the unified Mittag-Leffler functionm times, we get(44)ddtmMα,β,γ,δ,μ,νλ,ρ,θ,k,nt;a¯,b¯,c¯,p=∑l=m∞∏i=1nBpbi,aiλρlθkl∏i=1nBci,aiγδlμνlll−1,…,l−m−1tl−mΓαl+β=∑l=0∞∏i=1nBpbi,aiλρl+mθkl+m∏i=1nBci,aiγδl+mμνl+ml+ml+m−1,…,l+1tlΓαl+β.
We know thatθa+b=θ+abθa. Therefore, we obtain(45)ddtmMα,β,γ,δ,μ,νλ,ρ,θ,k,nt;a¯,b¯,c¯,p=λmρθmkγmδμmν∑l=0∞∏i=1nBpbi,aiλ+ρmρlθ+kmkl∏i=1nBci,aiγ+δmδlμ+νmνl⋅1+lmtlΓαl+m+β.Next, we give the definition of fractional integral operator with unified Mittag-LefflerMfunction as the kernel.Definition 10.
Letf∈L1a,b. Then ∀ξ∈a,b, the fractional integral operator with Mfunction as its kernel is defined as follows:(46)Ia+,α,β,γ,δ,μ,νω,λ,ρ,θ,k,nfξ;a¯,b¯,c¯,p=∫aξξ−tβ−1Mα,β,γ,δ,μ,νλ,ρ,θ,k,nωξ−tα;a¯,b¯,c¯,pftdt,Ib−,α,β,γ,δ,μ,νω,λ,ρ,θ,k,nfξ;a¯,b¯,c¯,p=∫ξbt−ξβ−1Mα,β,γ,δ,μ,νλ,ρ,θ,k,nωt−ξα;a¯,b¯,c¯,pftdt,with a¯=a1,a2,…,an,b¯=b1,b2,…,bn,c¯=c1,c2,…,cn,whereai,bi,ci,ω∈ℂ;i=1,…,n such that ℜai,ℜbi,ℜci>0,∀i. Also let α,β,γ,δ,μ,ν,λ,ρ, θ,t∈ℂ, minℜα,ℜβ,ℜγ,ℜδ,ℜλ,ℜθ>0, and k∈0,1∪ℕ with k+ℜρ<ℜδ+ν+α, Imρ=Imδ+ν+α.Remark 1.
Forn = 1, b1 = c1 + lk, a1 = θ − λ, c1 = λ, ρ = ν = 0, δ > 0, we obtain the fractional integral operator containing extended generalized Mittag-Leffler function in its kernel and is given by [5];(47)εa+,α,β,γω,λ,δ,k,θfξ,p=∫aξξ−tβ−1Eα,β,γλ,δ,k,θωξ−tα,pftdt,εb−,α,β,γω,λ,δ,k,θfξ,p=∫ξbt−ξβ−1Eα,β,γλ,δ,k,θωt−ξα,pftdt.Now we give the proof of boundedness of the fractional integral operator defined above.Theorem 7.
Letf∈L1a,b. If a¯=a1,a2,…,an,b¯=b1,b2,…,bn,c¯=c1,c2,…,cn,whereai,bi,ci,ω∈ℂ;i=1,…,n such that ℜai,ℜbi,ℜci>0,∀i. Also let α,β,γ,δ,μ,ν,λ,ρ, θ,t∈ℂ, minℜα,ℜβ,ℜγ,ℜδ,ℜλ,ℜθ>0, and k∈0,1∪ℕ with k+ℜρ<ℜδ+ν+α, Imρ=Imδ+ν+α, and then the fractional integral operator Ia+,α,β,γ,δ,μ,νω,λ,ρ,θ,k,nf is bounded on L1a,b.Proof.
Applying 1-norm to the fractional integral operatorIa+,α,β,γ,δ,μ,νω,λ,ρ,θ,k,nf, we get the following:(48)Ia+,α,β,γ,δ,μ,νω,λ,ρ,θ,k,nf1=∫ab∫aξξ−tβ−1Mα,β,γ,δ,μ,νλ,ρ,θ,k,nωξ−tα;a¯,b¯,c¯,pftdtdξ≤∫abft∫tbξ−tℜβ−1Mα,β,γ,δ,μ,νλ,ρ,θ,k,nωξ−tα;a¯,b¯,c¯,pdξdt.
By substitutingξ−t=s, we obtain the following inequality:(49)Ia+,α,β,γ,δ,μ,νω,λ,ρ,θ,k,nf1≤∫abft∫0b−tsℜβ−1Mα,β,γ,δ,μ,νλ,ρ,θ,k,nωsα;a¯,b¯,c¯,pdsdt≤∫abft∫0b−asℜβ−1Mα,β,γ,δ,μ,νλ,ρ,θ,k,nωsα;a¯,b¯,c¯,pdsdt≤∑l=0∞∏i=1nBpbi,aiλρlθkl∏i=1nBci,aiγδlμνlωlΓαl+β×∫0b−asℜαl+ℜβ−1dsf1≤∑l=0∞∏i=1nBpbi,aiλρlθkl∏i=1nBci,aiγδlμνlωlΓαl+βb−aℜαlℜαl+ℜβ×b−aℜβf1,Ia+,α,β,γ,δ,μ,νω,λ,ρ,θ,k,nf1≤Kf1,where(50)K=∏i=1nBpbi,aiλρlθkl∏i=1nBci,aiγδlμνlωlΓαl+βb−aℜαlℜαl+ℜβb−aℜβ.


## 4. Conclusions

In this paper, we extended the Mittag-Leffler function and generalizedQ function simultaneously. By applying the Laplace, Euler beta, and Whittaker transformations on the unified Mittag-Leffler function, compact formulas are established from which formulas for generalized Q function and extended generalized Mittag-Leffler function are deduced. These formulas also reproduce integral transformations of various deduced Mittag-Leffler functions. Moreover, we proved the convergence of this unified Mittag-Leffler function and constructed the associated fractional integral operator. Our proposed unified Mittag-Leffler function and constructed fractional integral operator will give new directions to the researcher working in this field.



---

*Source: 6043769-2021-10-13.xml*