# On Modifications of the Gamma Function by Using Mittag-Leffler Function

Authors: Asifa Tassaddiq; Abdulrahman Alruban
Journal: Journal of Mathematics
 (2021)

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## Abstract

Mittag-Leffler function is a natural generalization of the exponential function. Recent applications of Mittag-Leffler function have reshaped the scientific literature due to its fractional effects that cannot be obtained by using exponential function. Present motivation is to define a new special function by modification in the original gamma function with Mittag-Leffler function. Properties of this modified function are discussed by investigating a new series representation involving delta function. Hence, the results are also validated with the earlier obtained results for gamma function as special cases. Furthermore, the new function is used to generate a probability density function, and its statistical properties are explored. Similar properties of existing distributions can be deduced.


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## Body

## 1. Introduction and Preliminaries

### 1.1. Motivation

Use of fractional operators has become a remarkable tool to study new technology and researches [1–17]. One attractive aspect of this approach is to model complex systems by taking into account the effects of filtration and memory. How the fractional operators are defined? It might be summarized as follows: “each fractional operator is the action of a particular special function on a general function, which usually represents the solution of a well-defined problem by using a suitable integral transform.” See for example [18–22]. Hence, a basic motivation and a significant contribution of this research are to study a special function containing the Mittag-Leffler function (also known as “queen of fractional calculus” [23]), which is represented in terms of complex delta function. Hence, its action on a suitably chosen function over a specific domain is easily obtainable in view of the standard properties of the delta function. This gives insights for the emergence of a new fractional operator as one of the natural consequences of this research.The review of literature [1–23] demonstrates that many of the new researches depend on the involvement of Mittag-Leffler function in place of exponential function. More recently, Mainardi and Masina [24] have presented modified exponential integral by using Mittag-Leffler function. Mainardi et al. [25] have explored a new model which depends on ν∈0,1. This model represents the Maxwell body at one end point, namely, ν=0, and the Becker body at other end point, namely ν=1. Referring to these phenomena, creep rule is generalized in the sense that Becker model involves Mittag-Leffler function of order ν in place of exponential function. Following them, Paris [26] has discussed asymptotic behavior of the modified exponential integral and studied its more general form by using two-parameter Mittag-Leffler functions. Taking motivation from these researches [24–26], the aim of the present paper is to study the modified gamma function by using Mittag-Leffler function. Before going on to our main results, we describe necessary background and preliminaries in the subsequent paragraphs.


### 1.2. Preliminaries

Mittag-Leffler [27] proposed a function that appears a natural substitute of exponential function in the case of fractional order. It is known as Mittag-Leffler function defined by(1)Eαs=∑r=0∞srΓαr+1,α∈ℂ,ℜα>0.Forα=1, one-parameter Mittag-Leffler function reduces to the exponential function E1s=Σr=0∞sr/r!=es. There are many generalizations of Mittag-Leffler function. For example, two- and three-parameter Mittag-Leffler functions are defined by(2)Eα,βs=∑r=0∞srΓαr+β,α,β∈ℂ,ℜα>0,Eα,βγs=∑r=0∞γrsrr!Γαr+β,α,β,γ∈ℂ,ℜα>0,respectively. Here, ℂ denotes the set of complex numbers and ℜ denotes the real part of any complex number. Three-parameter Mittag-Leffler function is also known as Prabhakar function [20]. The decades of this century are mainly devoted to the analysis and applications of this function, see for example [1–24], and related references therein. For our purposes, we would be using the following form of Mittag-Leffler function [28]:(3)Eεtε=∑r=0∞tεrΓεr+1;t∈ℝ;0<ε<1.Here,ℝ denotes the set of real numbers. Similarly, the definition of fractional circular functions is [28, 29](4)sinεt=∑r=0∞−1rtε2r+1Γε2r+1+1,(5)cosεt=∑r=0∞−1rt2rεΓ2εr+1,which are obviously related to the Mittag-Leffler function defined in equation (3). Similarly, the hyperbolic functions can be defined as [28, 29](6)sinhεt=Eεtε−Eε−tε2=∑r=0∞tε2r+1Γε2r+1+1,coshεt=Eεtε+Eε−tε2=∑r=0∞t2rεΓ2rε+1.The focus point of the current investigation is to study a modified (generalized) form of gamma function [30], defined by(7)Γu=∫0∞tu−1e−tdt,u∈ℂ;ℜu>0.It is a basic special function so that Pochhammer symbolsuκ have the following relationship with the gamma function:(8)uκ=Γu+κΓu=1,κ=0,uu+1…u+n−1κ∈ℂ\0;κ=n∈ℕ;u∈ℂ.Here,ℕ denotes the set of natural numbers or positive integers. Further detailed analysis of the gamma function can be found in [31–34] and related bibliography therein. It is important to mention that this contribution remained unachievable without reviewing [35–46]. As we will be using delta function for our results and therefore we briefly describe about the spaces of test functions and distributions in the subsequent paragraphs.A dual space also known as space of generalized functions (distributions) corresponds to a space of test functions. Such phenomenon involves the representation of singular functions and hence the certain functions can be taken into account and various calculus operations can be applied in this way. Let us begin with the following simple identities:(9)δy−σ,uy=uσ,σ∈R,and for a nonvanishing parameter σ,δ−y=δy;δσy=δy/σ.Gelfand and Shilov [47] have provided a comprehensive discussion and explanation of distributions (or generalized functions). According to them, [47], the commonly used test functions that have compact support are the elements of a space of test functions denoted by D and another space denoted by S is of infinitely differentiable as well as fast decaying functions. These spaces D and S have their dual spaces denoted by D′ and S′, respectively. When we talk about the closeness of such spaces under Fourier transform, then these spaces behave differently. Hence, the spaces S and S′ are closed but D and D′ are not closed under Fourier transform. As a result, the elements of D′ have Fourier transforms that form distributions for entire functions spaces Z whose Fourier transforms belong to D [48]. In addition to it, since the entire function is nonzero only for a specific interval m1<y<m2, so the following relation among these spaces is valid(10)Z⊂S⊂S′⊂Z′;Z∩D≡0;D⊂S⊂S′⊂D′.For the benefit of this investigation, spaceZ comprises of entire and analytic functions sustaining the subsequent criteria(11)yηuy≤Cηeqθ;η=0,1,2,…,where q and Cη are numbers that depend on function u. The subsequent equations ([46], Vol 1, p. 169, equation (8); [47], p. 159, equation (4); and [48], p. 201, equation (10)) are vital to follow the steps of the proofs in this research(12)ℱeyt;ξ=2πδξ−iy.Hence,ℱeyt;ξ∈Z′and for each u∈Z′,(13)uy+θ=∑j=0∞gjyθjj!,(14)δy+θ=∑j=0∞δjyθjj!;δjy,uy=−1juj0,(15)δσ1−yδy−σ2=δσ1−σ2.In addition to this, the other functions whose Fourier transform involves delta function aresinθ,cosθ,sinhθ, and coshθ. Further detailed study about similar type of functions is given in [47–50].Plan of the remaining paper is given as follows: Section2.1 deals with the definition of the new function by investigating a series form of modified gamma function in terms of complex delta function. Section 2.2 discusses the conditions of persistence along with possible uses of the new series. Authentication of such results is discussed in Section 2.3. Advance properties of the new function as a distribution are analyzed in Section 2.4. Further examples by using new representation are discussed in Section 2.5. Generalized gamma distribution is discussed in Section 2.6. This work is concluded in Section 3. The results presented here are validated with the earlier obtained results in [35–37].


## 1.1. Motivation

Use of fractional operators has become a remarkable tool to study new technology and researches [1–17]. One attractive aspect of this approach is to model complex systems by taking into account the effects of filtration and memory. How the fractional operators are defined? It might be summarized as follows: “each fractional operator is the action of a particular special function on a general function, which usually represents the solution of a well-defined problem by using a suitable integral transform.” See for example [18–22]. Hence, a basic motivation and a significant contribution of this research are to study a special function containing the Mittag-Leffler function (also known as “queen of fractional calculus” [23]), which is represented in terms of complex delta function. Hence, its action on a suitably chosen function over a specific domain is easily obtainable in view of the standard properties of the delta function. This gives insights for the emergence of a new fractional operator as one of the natural consequences of this research.The review of literature [1–23] demonstrates that many of the new researches depend on the involvement of Mittag-Leffler function in place of exponential function. More recently, Mainardi and Masina [24] have presented modified exponential integral by using Mittag-Leffler function. Mainardi et al. [25] have explored a new model which depends on ν∈0,1. This model represents the Maxwell body at one end point, namely, ν=0, and the Becker body at other end point, namely ν=1. Referring to these phenomena, creep rule is generalized in the sense that Becker model involves Mittag-Leffler function of order ν in place of exponential function. Following them, Paris [26] has discussed asymptotic behavior of the modified exponential integral and studied its more general form by using two-parameter Mittag-Leffler functions. Taking motivation from these researches [24–26], the aim of the present paper is to study the modified gamma function by using Mittag-Leffler function. Before going on to our main results, we describe necessary background and preliminaries in the subsequent paragraphs.


## 1.2. Preliminaries

Mittag-Leffler [27] proposed a function that appears a natural substitute of exponential function in the case of fractional order. It is known as Mittag-Leffler function defined by(1)Eαs=∑r=0∞srΓαr+1,α∈ℂ,ℜα>0.Forα=1, one-parameter Mittag-Leffler function reduces to the exponential function E1s=Σr=0∞sr/r!=es. There are many generalizations of Mittag-Leffler function. For example, two- and three-parameter Mittag-Leffler functions are defined by(2)Eα,βs=∑r=0∞srΓαr+β,α,β∈ℂ,ℜα>0,Eα,βγs=∑r=0∞γrsrr!Γαr+β,α,β,γ∈ℂ,ℜα>0,respectively. Here, ℂ denotes the set of complex numbers and ℜ denotes the real part of any complex number. Three-parameter Mittag-Leffler function is also known as Prabhakar function [20]. The decades of this century are mainly devoted to the analysis and applications of this function, see for example [1–24], and related references therein. For our purposes, we would be using the following form of Mittag-Leffler function [28]:(3)Eεtε=∑r=0∞tεrΓεr+1;t∈ℝ;0<ε<1.Here,ℝ denotes the set of real numbers. Similarly, the definition of fractional circular functions is [28, 29](4)sinεt=∑r=0∞−1rtε2r+1Γε2r+1+1,(5)cosεt=∑r=0∞−1rt2rεΓ2εr+1,which are obviously related to the Mittag-Leffler function defined in equation (3). Similarly, the hyperbolic functions can be defined as [28, 29](6)sinhεt=Eεtε−Eε−tε2=∑r=0∞tε2r+1Γε2r+1+1,coshεt=Eεtε+Eε−tε2=∑r=0∞t2rεΓ2rε+1.The focus point of the current investigation is to study a modified (generalized) form of gamma function [30], defined by(7)Γu=∫0∞tu−1e−tdt,u∈ℂ;ℜu>0.It is a basic special function so that Pochhammer symbolsuκ have the following relationship with the gamma function:(8)uκ=Γu+κΓu=1,κ=0,uu+1…u+n−1κ∈ℂ\0;κ=n∈ℕ;u∈ℂ.Here,ℕ denotes the set of natural numbers or positive integers. Further detailed analysis of the gamma function can be found in [31–34] and related bibliography therein. It is important to mention that this contribution remained unachievable without reviewing [35–46]. As we will be using delta function for our results and therefore we briefly describe about the spaces of test functions and distributions in the subsequent paragraphs.A dual space also known as space of generalized functions (distributions) corresponds to a space of test functions. Such phenomenon involves the representation of singular functions and hence the certain functions can be taken into account and various calculus operations can be applied in this way. Let us begin with the following simple identities:(9)δy−σ,uy=uσ,σ∈R,and for a nonvanishing parameter σ,δ−y=δy;δσy=δy/σ.Gelfand and Shilov [47] have provided a comprehensive discussion and explanation of distributions (or generalized functions). According to them, [47], the commonly used test functions that have compact support are the elements of a space of test functions denoted by D and another space denoted by S is of infinitely differentiable as well as fast decaying functions. These spaces D and S have their dual spaces denoted by D′ and S′, respectively. When we talk about the closeness of such spaces under Fourier transform, then these spaces behave differently. Hence, the spaces S and S′ are closed but D and D′ are not closed under Fourier transform. As a result, the elements of D′ have Fourier transforms that form distributions for entire functions spaces Z whose Fourier transforms belong to D [48]. In addition to it, since the entire function is nonzero only for a specific interval m1<y<m2, so the following relation among these spaces is valid(10)Z⊂S⊂S′⊂Z′;Z∩D≡0;D⊂S⊂S′⊂D′.For the benefit of this investigation, spaceZ comprises of entire and analytic functions sustaining the subsequent criteria(11)yηuy≤Cηeqθ;η=0,1,2,…,where q and Cη are numbers that depend on function u. The subsequent equations ([46], Vol 1, p. 169, equation (8); [47], p. 159, equation (4); and [48], p. 201, equation (10)) are vital to follow the steps of the proofs in this research(12)ℱeyt;ξ=2πδξ−iy.Hence,ℱeyt;ξ∈Z′and for each u∈Z′,(13)uy+θ=∑j=0∞gjyθjj!,(14)δy+θ=∑j=0∞δjyθjj!;δjy,uy=−1juj0,(15)δσ1−yδy−σ2=δσ1−σ2.In addition to this, the other functions whose Fourier transform involves delta function aresinθ,cosθ,sinhθ, and coshθ. Further detailed study about similar type of functions is given in [47–50].Plan of the remaining paper is given as follows: Section2.1 deals with the definition of the new function by investigating a series form of modified gamma function in terms of complex delta function. Section 2.2 discusses the conditions of persistence along with possible uses of the new series. Authentication of such results is discussed in Section 2.3. Advance properties of the new function as a distribution are analyzed in Section 2.4. Further examples by using new representation are discussed in Section 2.5. Generalized gamma distribution is discussed in Section 2.6. This work is concluded in Section 3. The results presented here are validated with the earlier obtained results in [35–37].


## 2. Results

### 2.1. Modified Form of the Gamma Function and Its Series Representation

By following the methodology of [24], we introduce(16)Γεu=∫0∞tu−1Eε−tεdt,u∈ℂ;0<ℜu<1;0<ε<1,and for ε=1, we get the original gamma function (6). According to the particular asymptotic behavior of Mittag-Leffler function ([2], p. 286, equations (2.8-2.9)], this integral converges for 0<ℜu<1 when 0<ε<1. Next, by following Lebedeve [30], we rewrite (16) as(17)∫0∞tu−1Eε−tεdt=∫01tu−1Eε−tεdt+∫1∞tu−1Eε−tεdt=Pεu+Qεu.Here, first integralPεu is an analytic function of u and second integral Qεu is an entire function. Now, we replace Eε−tε by its power series such as(18)Pεu=∑n=0∞−1nΓnε+1∫01tεntu−1dt=∑n=0∞−1nΓnε+1u+nε,u∈ℂ;ℜu>0.Uniform convergence of the involved integral allows one to reverse the order of integration and summation in equation (18). It shows that Γεu has singularities at u=−nε.Continuing in the same way, we investigate a new series representation ofΓεu in the form of the following theorem.Theorem 1.
The modified gamma functionΓεu has the following representation:(19)Γεu=2π∑n=0∞−1nΓnε+1δθ–iν+nε.Proof.
Let us first replacet=ex and u=ν+iθ in equation (16), then we get(20)Γεu=∫−∞∞exν+iθEε−exεdx.
Next, we replace Mittag-Leffler function by its series form as given in (1),(21)Eε−exε=∑n=0∞−exεnΓnε+1,and then by rearranging the involved exponential terms(22)Γεu=∫−∞∞exν+iθ∑n=0∞−exεnΓnε+1dx,we get(23)Γεu=∑n=0∞−1nΓnε+1∫−∞∞eixθeν+nεxdx.
Next, the uniform convergence of the involved integral allows us to interchange the operations of integration and summation. Hence, by using equation (12), we can write(24)∫−∞∞eiθxeν+nεxdx=ℱeν+nεx;θ=2πδθ–iν+nε.
The above expressions (23) and (24) lead to the result (19) as mentioned above.Corollary 1.
The modified gamma functionΓεu has the subsequent demonstration(25)Γεu=2π∑n,r=0∞−1n−iν+nεrΓnε+1r!δrθ.Proof.
This can be obtained by using equations (14) and (19) as follows:(26)δθ–iν+nϵ=∑r=0∞−iν+nϵrr!δrθ.
Next, by using this relation in (19), we obtain the desired result.Corollary 2.
The modified gamma functionΓεu has the subsequent demonstration(27)Γεu=2π∑n=0∞−1nΓnε+1δu+nε.Proof.
This result can be obtained by a small alteration in equation (19) as follows:(28)∫−∞∞eixθeν+nεxdx=ℱeν+nεx;θ=2πδθ−iν+nε=2πδ1iiθ+ν+nε=2πiδν+iθ+nε=2πδu+nε,which leads to the required result in view of (19).Corollary 3.
The modified Gamma functionΓεu has the subsequent demonstration:(29)Γεu=2π∑n,r=0∞−1nnεrΓnε+1r!δru.Proof.
This can be deduced in view of the relation (14) as follows:(30)δu+nε=∑r=0∞nεrr!δru.
Next, by making use of this result in (27), we obtain the required form.Corollary 4.
Gamma function has the subsequent representation ([36], p. 2092, equation (2.7))(31)Γu=Γ1u=2π∑n=0∞−1nn!θ−iν+n.Proof.
This can be deduced by fixingε=1 in equation (19).Corollary 5.
Gamma function has the subsequent series representation [37](32)Γu=2π∑n,r=0∞−1n−iν+nrn!r!δrθ.Proof.
This can be deduced by fixingε=1 in equation (25).Corollary 6.
Gamma function has the following representation [37]:(33)Γu=2π∑n=0∞−1nn!δu+n.Proof.
This can be deduced by fixingε=1 in equation (27).Corollary 7.
Gamma function has the subsequent representation [37]:(34)Γu=2π∑n,r=0∞−1nnrn!r!δru.Proof.
This can be deduced by fixingε=1 in equation (29).Remark 1.
It is to be remarked that the newly obtained series representation is only meaningful if defined as a distribution acting over a space of test functions. We can observe it by simply multiplying the both sides of equation (27) with 1/Γεu,(35)1=2π∑n=0∞−1n/n!δu+nεΓεu,which produces the following:(36)1=2π∑n=0∞−1nn!Γε−nε.
As proved above thatΓεu has poles at u=−nε, here, the right hand side sum will disappear because Γε−nε is in the denominator(37)⇒1=0,which is obviously false. At the same time, if we consider the following inner product,(38)Γεu,1Γεu=2π∑n=0∞−1nn!δu+nε,1Γεu,we obtain(39)∫uϵℂ1du=2π∑n=0∞−1nn!Γε−nε.
Now, the right-hand side will vanish due to the fact thatΓεu has poles at u=−nε, hence(40)∫uϵℂ1du=0,⇒∞=0,which is false. This discussion demonstrates that we have to be very careful and rigorous in selecting the set of functions for a meaningful definition of this new representation. This is a part of Section 2.2.


### 2.2. Existence and Applications of New Representation

This exemplification of the generalized gamma functionΓεu is an infinite summation over delta function, which is certain only if defined in the way as distributions (generalized functions). Therefore, it is curious to show that the obtained series of delta function is a distribution (generalized function) on a particular space of test functions as proved in the subsequent theorem.Theorem 2.
Prove that the modified gamma functionΓεu acts as a distribution over the space of slowly increasing complex test functions denoted by Z.Proof.
Let us first take the following expression for anyf∗u,f∗∗u∈Z, and c∗,c∗∗∈ℂ:(41)Γεu,c∗f∗u+c∗∗f∗∗u=2π∑n=0∞−1nΓnε+1δu+εn,c∗f∗u+c∗∗f∗∗u,⇒Γεu,c∗f∗u+c∗∗f∗∗u=c∗Γεu,f∗u+c∗∗Γεu,f∗∗u.
It shows thatΓεu is a linear function over Z. Now, we take an arbitrary sequence Z⊃fμμ=1∞⟶0 and use the fact that δu acts as continuous function over Z,(42)⇒Γεu,fμuμ=1∞=2π∑n=0∞−1nΓnε+1δu+nε,fμuμ=1∞.
Then, we can show thatδu+εn,fμuμ=1∞⟶0 (tends to zero). Hence, the modified gamma function Γεu is a distribution (continuous linear functional) over Z because of the convergence of (27) given by(43)Γϵu,fu=2π∑n=0∞−1nΓnϵ+1δu+nϵ,fu∀fuϵZ,(44)=2π∑n=0∞−1nΓnϵ+1f−nϵ.
In the above equation, we have applied the following property for delta function,(45)δu+nε,fu=f−nε.
In this way, the attained functionsf−nε∈Z are test functions of slow growth (bounded by a polynomial); also, we can observe that summation of the coefficients in (44) is(46)sum over the coefficients=∑n=0∞−1nΓnε+1=Eε−1.
Hence, we get a finite and definite answer, which shows thatΓϵu,fu is convergent, for ∀fuϵΖ. Here, we have used the well-known fact that “the product of a rapid decay and slow growth functions is convergent.” However, it is also obvious from the famous Abel’s theorem. Hence, the same is true for its special cases which are given in (25)–(29).
Let us move to a larger class of functions and considerfu=τuξ. In view of equation (27) and shifting characteristic of delta functions, the product fu,Γεu is computed as follows:(47)∫uεℂτuξΓεudu=2π∑n=0∞−1nΓnε+1τ−nεξ=2π∑n=0∞−τ−εξnΓnε+1=2πEε−τ−εξ.
By puttingε=1 in (47), we can get the following known result for gamma function:(48)∫uεℂτuξΓudu=2π∑n=0∞−1nn!τ−nξ=2π∑n=0∞−τ−ξnn!=2πexp−τ−ξ.Remark 2.
Particular engineering problems involve sum of delta functions. It is remarked that when one considers the product of2πEε−1 with sequence of delta functions δu+nεn=0∞, then it yields the modified form Γεu. One can validate this result for gamma function Γu at ε=1.
The above discussion explores the existence of further results. To illustrate, we takeτ=e−1 in equation (47), which then leads to Laplace transform of Γεu. Hence, we consider the validation of such outcomes in the subsequent part.


### 2.3. Validation of the Results Obtained by New Representation

In this section, we will testify our results with the existing results. Let us taket=ex and u=ν+iξ in (16), then the modified gamma function Γεu is expressible as(49)Γεν+iθ=2πℱeνxEε−exε;ξ,and putting ε=1, in the above equation, we obtain(50)Γν+iθ=2πℱeνxexp−ex;θ.We know that Fourier transform satisfies the following dual property for any functionut,(51)ℱ2πℱut;θ;ξ=2πu−ξ.Hence, by applying this property for equations (49)–(50), we get the following result:(52)ℱΓεuν+iθ;ξ=ℱ2πℱeνxEε−exε;ξ=f−ξ=2πe−νξEε−e−εξ,or(53)∫−∞+∞eiθξΓεuν+iθdθ=2πe−νξEε−e−εξ.This can also be accessible as a particular case of the main result (47) when one puts τ=e;u=ν+iθ. So, the results obtained by using the new series representation are accurate when compared with well-known results. Furthermore, when one takes ξ=0 in (49), then the following identity can be obtained:(54)∫−∞+∞Γεuν+iθdθ=2πEε−1,which is also a special case of (53). Hence, it is testified that the new representation of Γεu creates new results that are unachievable in view of known representations but the special cases of these novel results are reliable with the existing methods. Continuing in this way, by taking ε=1 in (53) and (54), we obtain(55)∫−∞+∞eiθξΓ1ν+iθdθ=∫−∞+∞eiθξΓν+iθdθ=2πe−νξexp−e−ξ,and ξ=0 in (55) yields the following well-known result:(56)∫−∞+∞Γν+iθdθ=2πe.


### 2.4. Advanced Distributional Properties of Modified Gamma Function

Here, by taking motivation from ([48], Chapter 7), we state and prove a list of basic properties of the modified gamma function.Theorem 3.
For an arbitrary test functionfuϵZ, Γεu do satisfy the subsequent properties as a distribution over the space of test functions:(1)
Γεu,f∗u+f∗∗u=Γεu,f∗u+Γεu,f∗∗u.(2)
c∗Γεu,fu=Γεu,c∗fu.(3)
Γεu−γ,fu=Γεu,fu+γ.(4)
Γεc∗u,fu=Γεu,1/c∗fu/c∗.(5)
Γεc∗u−γ,fu=Γεu,1/c∗fu/c∗+γ.(6)
For any regular distributionψu, ψuΓεuϵZ is a distribution over Z.(7)
Γεu+ε=uΓεuifffu−ε=ufu,wheref∈Z.(8)
Γεmu,fu=∑n=0∞−1n/Γnε+1−1mfm−nε.(9)
Γεc∗−uΓεu−c∗∗=2πEε−12δc∗−c∗∗,∀fu∈Z.(10)
ℱΓεu,fu=Γεu,ℱfu.(11)
ℱΓεu,ℱfu=2πΓεt,f−t,t=ℜu.(12)
ℱΓεu¯,ℱfu=2πΓεt,fτt where f−t¯=fτt.(13)
ℱΓεu,ℱfu¯=2πΓεt,fτt.(14)
ℱΓεu¯,ℱfu¯=2πℱΓεu,ℱfu.(15)
ℱΓεmu=−itmℱΓεu.(16)
Γεu+c∗=∑n=0∞c∗n/n!Γεnu,where c∗,c∗∗,andγ are any complex or real numbers.Proof.
First six results (1)–(6) can be proved by using the basic properties of delta function and by following the procedure of Theorem 2. Here, we will prove the next result (7). Hence, by taking mth derivative of representation (27) w.r.t u and using (14), one can obtain(57)Γεmu,fu=∑n=0∞−1nΓnε+1−1mfm−εn,which is a well-defined and finite product. Then, one can prove the next result as follows:(58)Γεu+ε,fu=Γεu,fu−ε,⟺uΓεu,fu=Γεu,fu−ε,⟺Γεu,fu=Γεu,fu−ε.
Again, by using the property of delta function, one can get the next result (9),(59)Γεω1−uΓεu−ω2,fu=2π∑n=0∞−1nΓnε+12δω1−ω2,fu=2πEε−12δω1−ω2,fu.
Similarly, the results (10)–(15) are obvious to hold in view of Fourier transform and the properties of complex delta function. Let us start proving the next identity (10) from the above list,(60)ℱΓεu,fu=2π∑n=0∞−1nΓnε+1ℱδu+εn,fu,=2π∑n=0∞−1nΓnε+1δu+εn,ℱfu=Γεu,ℱfu.
Next, we prove identity (11) from the above list,(61)ℱΓεu,ℱfu=2πℱΓεu,ℱfu=2πΓεt,f−t,and identity (12),(62)ℱΓεu¯,ℱfu=2πℱΓεu,ℱfu¯=2πΓεt,f−t¯=2πΓεt,fτt,where fτ represents transpose of the involved test function f. Hence, the above identity (13) can be observed by the following:(63)ℱΓεu,ℱfu¯=2πΓεt,f−t¯=2πΓεt,fτt,and by applying Parseval’s identity of Fourier transform, the proof of identity (14) is(64)ℱΓεu¯,ℱfu¯=2πℱΓεu,ℱfu¯¯=2πℱΓεu,ℱfu.
Next, the identity (15) follows from following details:(65)ℱΓε1u,fu=ℱΓεu,f1u=Γεu,ℱf1u,⟹ℱΓε1u,fu=Γεu,−itfue−iut,⟹ℱΓε1u,fu=−itℱΓεu,fu,and so on, we get(66)ℱΓεmu,fu=−itmℱΓεu,fu.
Continuing in this way, one can prove the last identity (16) by making use of the result ([48], p. 201),(67)fu+α=∑n=0∞αnn!fnu.
Therefore, we have(68)Γεu+c∗,fu=Γεu,fu−c∗=limν⟶∞Γεu,∑n=0ν−c∗nn!fnu=limν⟶∞∑n=0νc∗nn!Γεnu,fu,as required.Remark 3.
According to the fact stated and proved in ([48], p. 203), “Fourier transform maps D′ onto Z′ and the inverse Fourier transform maps Z′ onto D′.” It is remarked that eνxEε−tε is an element of D′, and ℱΓεu is also in D′ in view of (49) and (52), respectively.


### 2.5. Further Examples by Using New Representation

Since delta function is a linear mapping which can map each function to its value at zero. By taking into account this fact, we consider the following examples. Let us consider (29) and restrict the variable, u=t, to real numbers, then we have(69)Γεt=2π∑n,r=0∞−1nnεrΓnε+1r!δrt.Hence,(70)Γεt,ft=2π∑n,r=0∞−1nnεrΓnε+1r!δrt,ft=2π∑n,r=0∞−1nnεrΓnε+1r!−1rfr0.Next, we consider the following examples.Example 1.
Letft=ect, then fr0=cr,(71)Γεt,ect=2π∑n,r=0∞−1nnεrΓnε+1r!−1rcr=2π∑n=0∞−e−cεnΓnε+1=2πEε−e−cε.Example 2.
Letft=sinct, then f2r+10=−1rc2r+1;fr0=0;r=0,2,4,…,(72)Γεt,ft=2π∑n,r=0∞−1nnε2r+1Γnε+12r+1!−12r+1−1rc2r+1=2π∑n=0∞−1nΓnε+1sin−cεn=IMG2π∑n=0∞−1ne−icεnΓnε+1=IMG2π∑n=0∞−e−icεnΓnε+1=IMG2πEε−e−icε.
Similarly,ft=cosct, then(73)fr0=−1rc2r;Λr0=0;r=1,3,5,…,Γεt,ft=Re2πEε−e−icε.Example 3.
Letft=1/1−t, then fr0=r!,(74)Γεt,11−t=2π∑n,r=0∞−1nnεrΓnε+1r!−1rr!=2π∑n=0∞−1nΓnε+11+nε=2π∑n=0∞−1nΓnε+2=2πEε,2−1.Example 4.
Letft=ln1+t, then fr0=−1r+1r−1!,(75)Γεt,ln1+t=2π∑n,r=0∞−1nnεrΓnε+1r!−1r−1r+1r−1!=2π∑n=0∞−1nln1−nεΓnε+1.
Here, we restrict the value ofε very small such that 1−nε>0. These examples show that our new representation of the modified gamma function is meaningful for all those functions that have derivatives of all orders at point 0. Similar results hold for its special case gamma function given by equation (34).
Next, we consider Laplace transform of delta function that yields the Laplace transform of the modified gamma and the original gamma function itself. We know that(76)Lδru;s=sr.
Therefore, we get(77)LΓεu;s=L2π∑n,r=0∞−1nnεrΓnε+1r!δru;s=2π∑n,r=0∞−1nnεrΓnε+1r!Lδru;s=2π∑n,r=0∞−1nnεrΓnε+1r!sr=2π∑n=0∞−1nΓnε+1eεsn=2πEε−eεs.
Puttingε=1 in the above equation gives(78)LΓu;s=2πexp−es,which yields further(79)LΓεu−c;s=2πe−scEε−e−εsLΓu−c;s=2πe−scexp−es.
A special case takings=0 in (78) leads to the well-known result given above in (56). Hence, it validates that the results obtained by new representation are consistent with the known results.Remark 4.
It is to be remarked that the success of the application of this new representation lies in the fact that the sum over the coefficients in the new representation is finite and well-defined as given in (46).


### 2.6. Generalized Gamma Distribution

Using a probability density function, Dirac delta function is frequently used to represent certain probability distributions. For example, considering the probability density function u(x) of a discrete distribution with pointsx=x1,…,xi,… and corresponding probabilities pi, it can be written as(80)ux=∑i=0∞piδx−xi.Moreover, gamma distribution and its generalizations are of interest for a large audience (see [51] and the cited bibliography therein). Let us consider the following new probability density function of a statistical distribution for a random variable X as follows:(81)ΓεX=Xu−1Eε−XεΓεu;u>0,0,elsewhere,where Γεu is defined in (16) and it is suitably assumed that for the involved parameters,(82)ΓεX>0.It can also be verified that(83)∫0∞ΓεXdX=∫0∞tu−1Eε−tεΓεudt=1.Next, we discuss the usual properties of a random variableX which is distributed according to the probability density function ΓεX given in (81).The general moment of ordern for the random variable X is obtained as(84)EXn=∫0∞tn+u−1Eε−tεΓεudt=Γεu+nΓεu,where E denotes the mathematical expectation.Mean and expected value for the probability distribution function is the first moment of the variableX and obtained as a special case of the above for n=1,(85)μX=EX=∫0∞tuEε−tεΓεudt=Γεu+1Γεu,and for n=2,(86)EX2=∫0∞tu+1Eε−tεΓεudt=Γεu+2Γεu.Hence, the variance of the random variableX is computed as(87)σX2=EX2−EX2=Γεu+2Γεu−Γεu+1Γεu2.By using (84), the characteristic function of the random variable X is given as follows:(88)EeitX=∑n=0∞itnn!EXn=∑n=0∞itnn!Γεu+nΓεu.Similarly, the generalized moment generating function of the random variableX by using Mittag-Leffler function instead of exponential function is given as(89)EEεtX=E∑n=0∞tXnΓnε+1=∑n=0∞tnΓnε+1EXn=∑n=0∞tnΓnε+1Γεu+nΓεu,and the usual moment generating function of the random variable X is obtained as a special case of (89),(90)ℳXt=EetX=∑n=0∞tnn!Γεu+nΓεu.The cumulative distribution function of the random variableX for X>0 is computed as(91)ℱεX=∫0Xtu−1Eε−tεΓεudt=1Γεu∫0Xtu−1Eε−tεdt=1Γεu∑n=0∞−1nΓnε+1∫0Xtεntu−1dt=1Γεu∑n=0∞−1nXu+nεΓnε+1u+nε,u∈ℂ;ℜu>0.The Lorenz curve as introduced by Lorenz [52] is commonly used to represent the disparity of money and earnings in economics. For our new probability function, it is computed for a random variable X as follows:(92)LX=ℱεXμX=1Γεu+1∑n=0∞−1nXu+nεΓnε+1u+nε,u∈ℂ;ℜu>0.Hence, it is to be remarked that the results for the other probability functions, for example,(93)fX=Xu−1e−tΓu;u>0,0,elsewhere,can be obtained as special case of the above statistical measures by taking ε=1.


## 2.1. Modified Form of the Gamma Function and Its Series Representation

By following the methodology of [24], we introduce(16)Γεu=∫0∞tu−1Eε−tεdt,u∈ℂ;0<ℜu<1;0<ε<1,and for ε=1, we get the original gamma function (6). According to the particular asymptotic behavior of Mittag-Leffler function ([2], p. 286, equations (2.8-2.9)], this integral converges for 0<ℜu<1 when 0<ε<1. Next, by following Lebedeve [30], we rewrite (16) as(17)∫0∞tu−1Eε−tεdt=∫01tu−1Eε−tεdt+∫1∞tu−1Eε−tεdt=Pεu+Qεu.Here, first integralPεu is an analytic function of u and second integral Qεu is an entire function. Now, we replace Eε−tε by its power series such as(18)Pεu=∑n=0∞−1nΓnε+1∫01tεntu−1dt=∑n=0∞−1nΓnε+1u+nε,u∈ℂ;ℜu>0.Uniform convergence of the involved integral allows one to reverse the order of integration and summation in equation (18). It shows that Γεu has singularities at u=−nε.Continuing in the same way, we investigate a new series representation ofΓεu in the form of the following theorem.Theorem 1.
The modified gamma functionΓεu has the following representation:(19)Γεu=2π∑n=0∞−1nΓnε+1δθ–iν+nε.Proof.
Let us first replacet=ex and u=ν+iθ in equation (16), then we get(20)Γεu=∫−∞∞exν+iθEε−exεdx.
Next, we replace Mittag-Leffler function by its series form as given in (1),(21)Eε−exε=∑n=0∞−exεnΓnε+1,and then by rearranging the involved exponential terms(22)Γεu=∫−∞∞exν+iθ∑n=0∞−exεnΓnε+1dx,we get(23)Γεu=∑n=0∞−1nΓnε+1∫−∞∞eixθeν+nεxdx.
Next, the uniform convergence of the involved integral allows us to interchange the operations of integration and summation. Hence, by using equation (12), we can write(24)∫−∞∞eiθxeν+nεxdx=ℱeν+nεx;θ=2πδθ–iν+nε.
The above expressions (23) and (24) lead to the result (19) as mentioned above.Corollary 1.
The modified gamma functionΓεu has the subsequent demonstration(25)Γεu=2π∑n,r=0∞−1n−iν+nεrΓnε+1r!δrθ.Proof.
This can be obtained by using equations (14) and (19) as follows:(26)δθ–iν+nϵ=∑r=0∞−iν+nϵrr!δrθ.
Next, by using this relation in (19), we obtain the desired result.Corollary 2.
The modified gamma functionΓεu has the subsequent demonstration(27)Γεu=2π∑n=0∞−1nΓnε+1δu+nε.Proof.
This result can be obtained by a small alteration in equation (19) as follows:(28)∫−∞∞eixθeν+nεxdx=ℱeν+nεx;θ=2πδθ−iν+nε=2πδ1iiθ+ν+nε=2πiδν+iθ+nε=2πδu+nε,which leads to the required result in view of (19).Corollary 3.
The modified Gamma functionΓεu has the subsequent demonstration:(29)Γεu=2π∑n,r=0∞−1nnεrΓnε+1r!δru.Proof.
This can be deduced in view of the relation (14) as follows:(30)δu+nε=∑r=0∞nεrr!δru.
Next, by making use of this result in (27), we obtain the required form.Corollary 4.
Gamma function has the subsequent representation ([36], p. 2092, equation (2.7))(31)Γu=Γ1u=2π∑n=0∞−1nn!θ−iν+n.Proof.
This can be deduced by fixingε=1 in equation (19).Corollary 5.
Gamma function has the subsequent series representation [37](32)Γu=2π∑n,r=0∞−1n−iν+nrn!r!δrθ.Proof.
This can be deduced by fixingε=1 in equation (25).Corollary 6.
Gamma function has the following representation [37]:(33)Γu=2π∑n=0∞−1nn!δu+n.Proof.
This can be deduced by fixingε=1 in equation (27).Corollary 7.
Gamma function has the subsequent representation [37]:(34)Γu=2π∑n,r=0∞−1nnrn!r!δru.Proof.
This can be deduced by fixingε=1 in equation (29).Remark 1.
It is to be remarked that the newly obtained series representation is only meaningful if defined as a distribution acting over a space of test functions. We can observe it by simply multiplying the both sides of equation (27) with 1/Γεu,(35)1=2π∑n=0∞−1n/n!δu+nεΓεu,which produces the following:(36)1=2π∑n=0∞−1nn!Γε−nε.
As proved above thatΓεu has poles at u=−nε, here, the right hand side sum will disappear because Γε−nε is in the denominator(37)⇒1=0,which is obviously false. At the same time, if we consider the following inner product,(38)Γεu,1Γεu=2π∑n=0∞−1nn!δu+nε,1Γεu,we obtain(39)∫uϵℂ1du=2π∑n=0∞−1nn!Γε−nε.
Now, the right-hand side will vanish due to the fact thatΓεu has poles at u=−nε, hence(40)∫uϵℂ1du=0,⇒∞=0,which is false. This discussion demonstrates that we have to be very careful and rigorous in selecting the set of functions for a meaningful definition of this new representation. This is a part of Section 2.2.


## 2.2. Existence and Applications of New Representation

This exemplification of the generalized gamma functionΓεu is an infinite summation over delta function, which is certain only if defined in the way as distributions (generalized functions). Therefore, it is curious to show that the obtained series of delta function is a distribution (generalized function) on a particular space of test functions as proved in the subsequent theorem.Theorem 2.
Prove that the modified gamma functionΓεu acts as a distribution over the space of slowly increasing complex test functions denoted by Z.Proof.
Let us first take the following expression for anyf∗u,f∗∗u∈Z, and c∗,c∗∗∈ℂ:(41)Γεu,c∗f∗u+c∗∗f∗∗u=2π∑n=0∞−1nΓnε+1δu+εn,c∗f∗u+c∗∗f∗∗u,⇒Γεu,c∗f∗u+c∗∗f∗∗u=c∗Γεu,f∗u+c∗∗Γεu,f∗∗u.
It shows thatΓεu is a linear function over Z. Now, we take an arbitrary sequence Z⊃fμμ=1∞⟶0 and use the fact that δu acts as continuous function over Z,(42)⇒Γεu,fμuμ=1∞=2π∑n=0∞−1nΓnε+1δu+nε,fμuμ=1∞.
Then, we can show thatδu+εn,fμuμ=1∞⟶0 (tends to zero). Hence, the modified gamma function Γεu is a distribution (continuous linear functional) over Z because of the convergence of (27) given by(43)Γϵu,fu=2π∑n=0∞−1nΓnϵ+1δu+nϵ,fu∀fuϵZ,(44)=2π∑n=0∞−1nΓnϵ+1f−nϵ.
In the above equation, we have applied the following property for delta function,(45)δu+nε,fu=f−nε.
In this way, the attained functionsf−nε∈Z are test functions of slow growth (bounded by a polynomial); also, we can observe that summation of the coefficients in (44) is(46)sum over the coefficients=∑n=0∞−1nΓnε+1=Eε−1.
Hence, we get a finite and definite answer, which shows thatΓϵu,fu is convergent, for ∀fuϵΖ. Here, we have used the well-known fact that “the product of a rapid decay and slow growth functions is convergent.” However, it is also obvious from the famous Abel’s theorem. Hence, the same is true for its special cases which are given in (25)–(29).
Let us move to a larger class of functions and considerfu=τuξ. In view of equation (27) and shifting characteristic of delta functions, the product fu,Γεu is computed as follows:(47)∫uεℂτuξΓεudu=2π∑n=0∞−1nΓnε+1τ−nεξ=2π∑n=0∞−τ−εξnΓnε+1=2πEε−τ−εξ.
By puttingε=1 in (47), we can get the following known result for gamma function:(48)∫uεℂτuξΓudu=2π∑n=0∞−1nn!τ−nξ=2π∑n=0∞−τ−ξnn!=2πexp−τ−ξ.Remark 2.
Particular engineering problems involve sum of delta functions. It is remarked that when one considers the product of2πEε−1 with sequence of delta functions δu+nεn=0∞, then it yields the modified form Γεu. One can validate this result for gamma function Γu at ε=1.
The above discussion explores the existence of further results. To illustrate, we takeτ=e−1 in equation (47), which then leads to Laplace transform of Γεu. Hence, we consider the validation of such outcomes in the subsequent part.


## 2.3. Validation of the Results Obtained by New Representation

In this section, we will testify our results with the existing results. Let us taket=ex and u=ν+iξ in (16), then the modified gamma function Γεu is expressible as(49)Γεν+iθ=2πℱeνxEε−exε;ξ,and putting ε=1, in the above equation, we obtain(50)Γν+iθ=2πℱeνxexp−ex;θ.We know that Fourier transform satisfies the following dual property for any functionut,(51)ℱ2πℱut;θ;ξ=2πu−ξ.Hence, by applying this property for equations (49)–(50), we get the following result:(52)ℱΓεuν+iθ;ξ=ℱ2πℱeνxEε−exε;ξ=f−ξ=2πe−νξEε−e−εξ,or(53)∫−∞+∞eiθξΓεuν+iθdθ=2πe−νξEε−e−εξ.This can also be accessible as a particular case of the main result (47) when one puts τ=e;u=ν+iθ. So, the results obtained by using the new series representation are accurate when compared with well-known results. Furthermore, when one takes ξ=0 in (49), then the following identity can be obtained:(54)∫−∞+∞Γεuν+iθdθ=2πEε−1,which is also a special case of (53). Hence, it is testified that the new representation of Γεu creates new results that are unachievable in view of known representations but the special cases of these novel results are reliable with the existing methods. Continuing in this way, by taking ε=1 in (53) and (54), we obtain(55)∫−∞+∞eiθξΓ1ν+iθdθ=∫−∞+∞eiθξΓν+iθdθ=2πe−νξexp−e−ξ,and ξ=0 in (55) yields the following well-known result:(56)∫−∞+∞Γν+iθdθ=2πe.


## 2.4. Advanced Distributional Properties of Modified Gamma Function

Here, by taking motivation from ([48], Chapter 7), we state and prove a list of basic properties of the modified gamma function.Theorem 3.
For an arbitrary test functionfuϵZ, Γεu do satisfy the subsequent properties as a distribution over the space of test functions:(1)
Γεu,f∗u+f∗∗u=Γεu,f∗u+Γεu,f∗∗u.(2)
c∗Γεu,fu=Γεu,c∗fu.(3)
Γεu−γ,fu=Γεu,fu+γ.(4)
Γεc∗u,fu=Γεu,1/c∗fu/c∗.(5)
Γεc∗u−γ,fu=Γεu,1/c∗fu/c∗+γ.(6)
For any regular distributionψu, ψuΓεuϵZ is a distribution over Z.(7)
Γεu+ε=uΓεuifffu−ε=ufu,wheref∈Z.(8)
Γεmu,fu=∑n=0∞−1n/Γnε+1−1mfm−nε.(9)
Γεc∗−uΓεu−c∗∗=2πEε−12δc∗−c∗∗,∀fu∈Z.(10)
ℱΓεu,fu=Γεu,ℱfu.(11)
ℱΓεu,ℱfu=2πΓεt,f−t,t=ℜu.(12)
ℱΓεu¯,ℱfu=2πΓεt,fτt where f−t¯=fτt.(13)
ℱΓεu,ℱfu¯=2πΓεt,fτt.(14)
ℱΓεu¯,ℱfu¯=2πℱΓεu,ℱfu.(15)
ℱΓεmu=−itmℱΓεu.(16)
Γεu+c∗=∑n=0∞c∗n/n!Γεnu,where c∗,c∗∗,andγ are any complex or real numbers.Proof.
First six results (1)–(6) can be proved by using the basic properties of delta function and by following the procedure of Theorem 2. Here, we will prove the next result (7). Hence, by taking mth derivative of representation (27) w.r.t u and using (14), one can obtain(57)Γεmu,fu=∑n=0∞−1nΓnε+1−1mfm−εn,which is a well-defined and finite product. Then, one can prove the next result as follows:(58)Γεu+ε,fu=Γεu,fu−ε,⟺uΓεu,fu=Γεu,fu−ε,⟺Γεu,fu=Γεu,fu−ε.
Again, by using the property of delta function, one can get the next result (9),(59)Γεω1−uΓεu−ω2,fu=2π∑n=0∞−1nΓnε+12δω1−ω2,fu=2πEε−12δω1−ω2,fu.
Similarly, the results (10)–(15) are obvious to hold in view of Fourier transform and the properties of complex delta function. Let us start proving the next identity (10) from the above list,(60)ℱΓεu,fu=2π∑n=0∞−1nΓnε+1ℱδu+εn,fu,=2π∑n=0∞−1nΓnε+1δu+εn,ℱfu=Γεu,ℱfu.
Next, we prove identity (11) from the above list,(61)ℱΓεu,ℱfu=2πℱΓεu,ℱfu=2πΓεt,f−t,and identity (12),(62)ℱΓεu¯,ℱfu=2πℱΓεu,ℱfu¯=2πΓεt,f−t¯=2πΓεt,fτt,where fτ represents transpose of the involved test function f. Hence, the above identity (13) can be observed by the following:(63)ℱΓεu,ℱfu¯=2πΓεt,f−t¯=2πΓεt,fτt,and by applying Parseval’s identity of Fourier transform, the proof of identity (14) is(64)ℱΓεu¯,ℱfu¯=2πℱΓεu,ℱfu¯¯=2πℱΓεu,ℱfu.
Next, the identity (15) follows from following details:(65)ℱΓε1u,fu=ℱΓεu,f1u=Γεu,ℱf1u,⟹ℱΓε1u,fu=Γεu,−itfue−iut,⟹ℱΓε1u,fu=−itℱΓεu,fu,and so on, we get(66)ℱΓεmu,fu=−itmℱΓεu,fu.
Continuing in this way, one can prove the last identity (16) by making use of the result ([48], p. 201),(67)fu+α=∑n=0∞αnn!fnu.
Therefore, we have(68)Γεu+c∗,fu=Γεu,fu−c∗=limν⟶∞Γεu,∑n=0ν−c∗nn!fnu=limν⟶∞∑n=0νc∗nn!Γεnu,fu,as required.Remark 3.
According to the fact stated and proved in ([48], p. 203), “Fourier transform maps D′ onto Z′ and the inverse Fourier transform maps Z′ onto D′.” It is remarked that eνxEε−tε is an element of D′, and ℱΓεu is also in D′ in view of (49) and (52), respectively.


## 2.5. Further Examples by Using New Representation

Since delta function is a linear mapping which can map each function to its value at zero. By taking into account this fact, we consider the following examples. Let us consider (29) and restrict the variable, u=t, to real numbers, then we have(69)Γεt=2π∑n,r=0∞−1nnεrΓnε+1r!δrt.Hence,(70)Γεt,ft=2π∑n,r=0∞−1nnεrΓnε+1r!δrt,ft=2π∑n,r=0∞−1nnεrΓnε+1r!−1rfr0.Next, we consider the following examples.Example 1.
Letft=ect, then fr0=cr,(71)Γεt,ect=2π∑n,r=0∞−1nnεrΓnε+1r!−1rcr=2π∑n=0∞−e−cεnΓnε+1=2πEε−e−cε.Example 2.
Letft=sinct, then f2r+10=−1rc2r+1;fr0=0;r=0,2,4,…,(72)Γεt,ft=2π∑n,r=0∞−1nnε2r+1Γnε+12r+1!−12r+1−1rc2r+1=2π∑n=0∞−1nΓnε+1sin−cεn=IMG2π∑n=0∞−1ne−icεnΓnε+1=IMG2π∑n=0∞−e−icεnΓnε+1=IMG2πEε−e−icε.
Similarly,ft=cosct, then(73)fr0=−1rc2r;Λr0=0;r=1,3,5,…,Γεt,ft=Re2πEε−e−icε.Example 3.
Letft=1/1−t, then fr0=r!,(74)Γεt,11−t=2π∑n,r=0∞−1nnεrΓnε+1r!−1rr!=2π∑n=0∞−1nΓnε+11+nε=2π∑n=0∞−1nΓnε+2=2πEε,2−1.Example 4.
Letft=ln1+t, then fr0=−1r+1r−1!,(75)Γεt,ln1+t=2π∑n,r=0∞−1nnεrΓnε+1r!−1r−1r+1r−1!=2π∑n=0∞−1nln1−nεΓnε+1.
Here, we restrict the value ofε very small such that 1−nε>0. These examples show that our new representation of the modified gamma function is meaningful for all those functions that have derivatives of all orders at point 0. Similar results hold for its special case gamma function given by equation (34).
Next, we consider Laplace transform of delta function that yields the Laplace transform of the modified gamma and the original gamma function itself. We know that(76)Lδru;s=sr.
Therefore, we get(77)LΓεu;s=L2π∑n,r=0∞−1nnεrΓnε+1r!δru;s=2π∑n,r=0∞−1nnεrΓnε+1r!Lδru;s=2π∑n,r=0∞−1nnεrΓnε+1r!sr=2π∑n=0∞−1nΓnε+1eεsn=2πEε−eεs.
Puttingε=1 in the above equation gives(78)LΓu;s=2πexp−es,which yields further(79)LΓεu−c;s=2πe−scEε−e−εsLΓu−c;s=2πe−scexp−es.
A special case takings=0 in (78) leads to the well-known result given above in (56). Hence, it validates that the results obtained by new representation are consistent with the known results.Remark 4.
It is to be remarked that the success of the application of this new representation lies in the fact that the sum over the coefficients in the new representation is finite and well-defined as given in (46).


## 2.6. Generalized Gamma Distribution

Using a probability density function, Dirac delta function is frequently used to represent certain probability distributions. For example, considering the probability density function u(x) of a discrete distribution with pointsx=x1,…,xi,… and corresponding probabilities pi, it can be written as(80)ux=∑i=0∞piδx−xi.Moreover, gamma distribution and its generalizations are of interest for a large audience (see [51] and the cited bibliography therein). Let us consider the following new probability density function of a statistical distribution for a random variable X as follows:(81)ΓεX=Xu−1Eε−XεΓεu;u>0,0,elsewhere,where Γεu is defined in (16) and it is suitably assumed that for the involved parameters,(82)ΓεX>0.It can also be verified that(83)∫0∞ΓεXdX=∫0∞tu−1Eε−tεΓεudt=1.Next, we discuss the usual properties of a random variableX which is distributed according to the probability density function ΓεX given in (81).The general moment of ordern for the random variable X is obtained as(84)EXn=∫0∞tn+u−1Eε−tεΓεudt=Γεu+nΓεu,where E denotes the mathematical expectation.Mean and expected value for the probability distribution function is the first moment of the variableX and obtained as a special case of the above for n=1,(85)μX=EX=∫0∞tuEε−tεΓεudt=Γεu+1Γεu,and for n=2,(86)EX2=∫0∞tu+1Eε−tεΓεudt=Γεu+2Γεu.Hence, the variance of the random variableX is computed as(87)σX2=EX2−EX2=Γεu+2Γεu−Γεu+1Γεu2.By using (84), the characteristic function of the random variable X is given as follows:(88)EeitX=∑n=0∞itnn!EXn=∑n=0∞itnn!Γεu+nΓεu.Similarly, the generalized moment generating function of the random variableX by using Mittag-Leffler function instead of exponential function is given as(89)EEεtX=E∑n=0∞tXnΓnε+1=∑n=0∞tnΓnε+1EXn=∑n=0∞tnΓnε+1Γεu+nΓεu,and the usual moment generating function of the random variable X is obtained as a special case of (89),(90)ℳXt=EetX=∑n=0∞tnn!Γεu+nΓεu.The cumulative distribution function of the random variableX for X>0 is computed as(91)ℱεX=∫0Xtu−1Eε−tεΓεudt=1Γεu∫0Xtu−1Eε−tεdt=1Γεu∑n=0∞−1nΓnε+1∫0Xtεntu−1dt=1Γεu∑n=0∞−1nXu+nεΓnε+1u+nε,u∈ℂ;ℜu>0.The Lorenz curve as introduced by Lorenz [52] is commonly used to represent the disparity of money and earnings in economics. For our new probability function, it is computed for a random variable X as follows:(92)LX=ℱεXμX=1Γεu+1∑n=0∞−1nXu+nεΓnε+1u+nε,u∈ℂ;ℜu>0.Hence, it is to be remarked that the results for the other probability functions, for example,(93)fX=Xu−1e−tΓu;u>0,0,elsewhere,can be obtained as special case of the above statistical measures by taking ε=1.


## 3. Conclusion

During this study, we explored a new special function as a modified form of gamma function so that the gamma function can be obtained as a special case of it. We discussed some basic properties of this function. The combination of distribution theory with different integral transforms is well explored for the analysis of partial differential equations (PDE). Numerous practical questions are impossible to be answered by applying the known techniques but became possible by using this combination. Hence, a new fractional operator naturally emerges as a consequence of this research which can act on a suitably chosen functionfx as follows:(94)Γεfx=∫sεℂΓεsfsds=2π∑n=0∞−1nΓnε+1f−nε.A new probability density function involving the new function is introduced, and its properties are studied. We end this discussion by proposing a generalized representation of the familiar beta function,(95)Βx;y=ΓxΓyΓx+y,x,y∈ℂ;ℜx;ℜy>0,in terms of the new function as follows:(96)Βεx;y=ΓεxΓεyΓεx+yx,y∈ℂ;ℜx;ℜy>0.It is customary to consider Mittag-Leffler function as given in equation (3) for certain applications. However, it can be noticed that if we consider (1) instead of (3) in (16) for 0 < ℜ (α) < 1, then (18) would be convergent for a greater set of values u ∈ ℂ\{0, −1, −2, …}. Similarly, equation(2) can be used for further extensions.



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