# The New Mittag-Leffler Function and Its Applications

Authors: U. Ayub; S. Mubeen; T. Abdeljawad; G. Rahman; Kottakkaran Sooppy Nisar
Journal: Journal of Mathematics
 (2020)

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## Abstract

In this paper, we investigate some properties of the Pochhammerp,s,k-symbol ξn,k,sp and gamma p,s,k-function Γs,kpξ. We then prove several identities for newly defined symbol ξn,k,sp and the function Γs,kpξ. The integral representations for the gamma p,s,k-function and beta p,s,k-function are presented. Also, we define a new Mittag-Leffler p,s,k-function and study its analytic properties and its transforms.


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## Body

## 1. Introduction

The theory of special functions comprises a major part of mathematics. In the last three centuries, the essential of solving the problems taking place in the fields of classical mechanics, hydrodynamics, and control theories motivated the development of the theory of special functions. This field also has wide applications in both pure mathematics and applied mathematics. The interested readers may consult the literature [1–4].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 its wide applications in diverse fields. We suggest the readers to review the literature [5–16] for more details.Throughout this article, letℂ,ℝ+,ℤ−, and ℕ be the sets of complex numbers, positive real numbers, negative integers, and natural numbers, respectively.


## 2. Preliminaries

This section contains some basic definitions and mathematical preliminaries. We begin with the well-known Mittag-Leffler function.In [17], Gosta Mittag-Leffler introduced the following Mittag-Leffler function which is defined by(1)Eθz=∑n=0∞znΓθn+1,z∈ℂ,ℜθ>0.In [18], a generalization of Mittag-Leffler function Eθz (1) is given by(2)Eθ,ϑz=∑n=0∞znΓθn+ϑ,z,ϑ∈ℂ,ℜθ>0.In [6], Prabhakar proposed the following three parameters of Mittag-Leffler function, which is defined by(3)Eθ,ϑϱz=∑n=0∞znϱnΓθn+ϑn!,z,ϑ,ϱ∈ℂ,ℜθ>0,where ϱn is the well-known Pochhammer’s symbol defined as follows (see [19]). For ϱ∈ℂ,(4)ϱn=ϱϱ+1ϱ+2⋯ϱ+n−1,forn∈ℕ=1,forn=0.Definition 1.
In [20], Gehlot introduced the two parameters Pochhammer’s symbol and two parameters of gamma p,k-function.
Letk,p>0, ξ∈ℂ with ℜξ>0, and n∈ℕ. Then, the Pochhammer p,k-symbol ξn,kp is defined as(5)ξn,kp=ξpkξpk+p⋯ξpk+n−1pand the gamma p,k-function is defined by(6)Γkpξ=1klimn⟶∞n!pn+1npξ/kpξn+1,k.Definition 2.
The gammap,k-function is also represented by the following forms:(7)Γkpξ=∫0∞e−tk/ptx−1dt,(8)Γkpξ=pkξ/kΓkξ=pξ/kkΓξk.
Also, the relationship between the Pochhammerp,k-symbol, Pochhammer k-symbol, and the classical Pochhammer’s symbol is represented by [4](9)ξn,kp=pknξn,k=pnξkn.
The gammap,k-function Γkpξ satisfies the following relation:(10)ξn,kp=pΓkξ+nkpΓkξ,where(11)Γkpξ+k=ξpkΓkξ.Definition 3.
In [21], the Mittag-Leffler k-function is defined by(12)Ek,θ,ϑϱz=∑n=0∞ϱn,kznΓkθn+ϑn!,where k∈ℝ, z, θ, ϑ, ϱ∈ℂ, ℜθ, and ℜϑ>0.
Recently, Cerutti et al. [22] introduced the following Mittag-Leffler p,k-function is defined as follows.Definition 4.
Forp,k∈ℝ,z,θ,ϑ,ϱ∈ℂ,ℜθ>0,andℜϑ>0,(13)Ek,θ,ϑϱpz=∑n=0∞pϱn,kznpΓkθn+ϑn!,where ϱn,kp is the Pochhammer p,k-symbol given by (5) and Γkpξ is gamma p,k-function given by (6).
In [23], Pochhammer p,s,k-symbol ξn,k,sp is defined as follows.Definition 5.
Forp,k,ξ∈ℝ, 0<s<1, and n∈ℕ,(14)ξn,k,sp=ξpksξpk+ps⋯ξpk+n−1ps=∏i=0n−1ξpk+ips,where(15)ξs=1−sξ1−s,∀ξ∈ℝ.
The following identities are satisfied:(1)
ξ+ys=ξs+sξys∀ξ,y∈ℝ(2)
1s=1(3)
ξys=ξsyys∀ξ,y∈ℝ(4)
0s=0Definition 6.
In [23], the gamma p,s,k-function in term of s-series is given by(16)Γs,kpkn=pkssp;spn−11−sn−1.
The relationship betweenξn,k,sp and Γs,kpξ is given by [23](17)ξn,k,sp=pΓs,kξ+nkpΓs,kξ.
The betap,k-function is defined by(18)Bkpξ,y=pΓkξpΓkypΓkξ+y,ℜξ>0,ℜy>0.Definition 7.
The betap,k-function is represented by the following integral:(19)Bkpξ,y=1k∫01tξ/k−11−ty/k−1e−p/tdt.Definition 8.
The well-known Laplace transform of piecewise continuous functionf:ℝ⟶ℝ is defined by(20)ℒfθ=∫0∞e−sθfθdθ,ℜs>0.


## 3. Applications and Properties of the Pochhammerp,s,k-Symbol ξn,k,sp and Gamma p,s,k-Function Γs,kpξ

In this section, we define gammap,s,k-function Γs,kpξ in terms of limit function and give its integral representation. Also, we define beta p,s,k-function Bs,kpξ,y and its integral representation. Furthermore, we prove some identities of the Pochhammer p,s,k-symbol ξn,k,sp and the gamma p,s,k-function Γs,kpξ.Definition 9.
Suppose thatk, s, p>0, ξ∈ℂ/kℤ− with ℜξ>0, and n∈ℕ. Then, the gamma p,s,k-function Γs,kpξ is given as(21)Γs,kpξ=sklimn⟶∞n!pn+1snpξ/k−1pξn,k.Theorem 1.
Suppose thatk,p,s,r,q>0, ξ∈ℂ/kℤ− with ℜξ>0, and n∈ℕ. Then, the following identities hold:(22)1.ξn,r,sp=pqξrn,q,s(23)2.ξn,r,sp=prnqξrn,q,sr(24)3.ξn,k,sp=ξn,k,ps(25)4.Γs,kpξ=qkΓs,qpqξk(26)5.Γs,qrξ=kqrpξ/sΓs,kpkξs(27)6.Γk,qrξ=rpξ/qΓk,qpξProof.
The properties (22), (23), and (24), respectively, follow from definition (11) and equations (2.8), (2.9), and (2.10) of [16]. The properties (25), (26), and (27), respectively, follow by using equation (21) and (2.11), (2.12), and (2.13) of [16].Theorem 2.
Letk,p,s,r>0, ξ∈ℂ/kℤ− with ℜξ>0, and n∈ℕ. Then, the following integral representation of gamma p,s,k-function is defined by(28)Γs,kpξ=∫0∞e−tk/sptξ−1dt.Proof.
Consider the right hand side of (28). By applying ([8], page 2) Tannery theorem and using (21), we have(29)∫0∞e−tk/sptξ−1dt=limn⟶∞∫0snp1/k1−tknspntξ−1dt.
LetCn,iξ,i=0,1,…,n, be given by(30)Cn,iξ=limn⟶∞∫0snp1/k1−tknspitξ−1dt.
After integrating by parts, we obtain(31)Cn,iξ=kinspxCn,i−1ξ+k,∀i=1,2,…,n.
Also,(32)Cn,0ξ=snpξ/kξ.
Therefore,(33)Cn,nξ=skn!pn+1snpξ/k−1ξn,kp1+ξ/kn,Γs,kpξ=limn⟶∞Cn,nξ=sklimn⟶∞n!pn+1snpξ/k−1ξn,kp.Definition 10.
Letξ,y∈ℂ/kℤ−, k,p,s,r∈ℝ+−0, and ℜξ>0,n∈ℕ. Then, the integral representation of Bs,kpξ,y is given by(34)Bs,kpξ,y=1k∫01tξ/k−11−ty/k−1e−p/t−s/1−tdt,where ℜp>0,ℜs>0,ℜξ>0,andℜy>0.Theorem 3.
The relation between three parameters, two parameters, and the classical Pochhammer’s symbol is given by(35)ξn,k,sp=snξn,kp=spknξn,k=spnξkn.Proof.
Using (14) and (9), we get the desired result.Theorem 4.
The relation between gammap,s,k-function, gamma p,k-function, gamma k-function, and classic gamma function is given by(36)Γs,kpξ=sξ/kΓkpξ=spkξ/kΓkξ=spξ/kkΓξk.Proof.
Using (21) and (8), we get the desired result.Theorem 5.
Givenξ∈ℂkZ−,k,s,p∈ℝ+−0,ℜξ>0, and n∈ℕ, the recurrence relation for Pochhammer p,s,k-symbol is given by(37)ξn+j,k,sp=ξj,k,sp×ξ+jkn,k,sp,nsξn−1,k,sp=ξn,k,sp−ξ−kn,k,sp.Proof.
Using the definition of Pochhammerp,s,k-symbol, we get the desired result.


## 4. Definition and Convergence Condition of the Mittag-Lefflerp,s,k-Function Ek,θ,ϑϱ,spz

In this section, we define a new generalization of the Mittag-Lefflerp,s,k-function. Also, we check the convergence of the Mittag-Leffler p,s,k-function.Definition 11.
Suppose thatp,k∈ℝ,θ,ϑ,ϱ∈ℂ,ℜθ>0,ℜϑ,andℜϱ>0. Then, Mittag-Leffler p,s,k-function is defined by(38)Ek,θ,ϑϱ,spz=∑n=0∞ϱn,k,spznΓs,kpθn+ϑn!,where ϱn,k,sp is Pochhammer p,s,k-symbol defined in (14), and Γs,kpξ is defined in (21).The recurrence relation of gamma p,s,k-function Γs,kpξ given in [23] is(39)Γs,kpx+k=ξpksΓs,kpξ.
Now, some characteristics of the Mittag-Lefflerp,s,k-function are presented. We show that the M-L p,s,k-function is an entire function. Also, its order and type are given.Theorem 6.
The Mittag-Lefflerp,s,k-function, defined in (38), is an entire function of order ρ and type σ given by(40)ρ=kReθ,σ=ρpeReθ/klnθ/kρ−1.Proof.
LetR denotes the radius of convergence of the Mittag-Leffler p,s,k-function. By considering the properties (5) and (8) and using the asymptotic expansions for the gamma function [1] and the asymptotic Stirling’s formula, we have(41)Γz=2π1/2zz−1/2ez1+Oz−1argz<π;z⟶∞.
In particular,(42)n!=2πn1/2nne−n1+On−1n∈N;n⟶∞,and the following quotient expansion of two gamma functions at infinity is given as(43)Γz+aΓz+b=za−b1+Oz−1argz+a<π;z⟶∞.
Series (38) can be written in the following forms:(44)Ek,θ,ϑϱ,spz=∑n=0∞ϱn,k,spznΓs,kpθn+ϑn!=∑n=0∞cnzn,since(45)R=limn⟶∞supcncn+1.
In view of the properties (35) and (36) and using III.9 of Theorem (1) in [22], we get(46)cncn+1=ϱn,k,spΓs,kpθn+1+ϑn+1!ϱn+1,k,spΓs,kpθn+ϑn!≃snϱn,kpsθn+θ+ϑ/kΓkpθn+1+ϑn+1!sn+1ϱn+1,kpspθn+ϑ/kΓkθn+ϑn!≃n+1sϑ/kpθ/k−1θnkθ/k⟶∞.
Thus, the Mittag-Lefflerp,s,k-function is an entire function.
To obtain the orderρ and the type σ, we apply the following definitions of ρ and σ, respectively:(47)ρ=limn⟶∞supnlnnln1/cn,(48)eρσ=limn⟶∞supncnρ/n.
Consider(49)1cn=Γs,kpϱΓs,kpθn+ϑn!Γs,kpϱ+nk=sθn−1+ϑ/kΓkpϱΓkpθn+ϑn!pΓkϱ+nk.
By using Theorem (1) equationIII.12 ([22]) and definition of ρ (47), we get(50)ρ=kReθ.
Similarly, by putting the value ofcn in the definition of σ (48) and simplify as the same in Theorem (1) in equation III.14 ([22]), we obtain(51)σ=ρpeℜθ/klnθ/kρ−1.


## 5. Applications and Properties of the Mittag-Lefflerp,s,k-Function Ek,θ,ϑϱ,spz

Some basic properties of Mittag-Lefflerp,s,k-function Ek,θ,ϑϱ,spz are presented in this section.Theorem 7.
Suppose thatp,k,s∈ℕ,θ,ϑ,ϱ∈ℂ with ℜθ>0,ℜϑ and ℜϱ>0, then(1)(52)ddzmEk,θ,ϑϱ,spz=ϱm,k,spEk,θ,θm+ϑϱ+mk,spz.(2)(53)ddzmzϑ/kEk,θ,ϑϱ,spzθ/k=zθn/k−m+ϑ/k−1ϱm,k,spEk,θ,θm+ϑϱ+mk,spzθ/k.Proof.
(1)
Taking L.H.S of (42),(54)ddzmEk,θ,ϑϱ,spz=ddzm∑n=0∞ϱn,k,spznΓs,kpθn+ϑn!=∑n=m∞ϱn,k,spzn−mΓs,kpθn+ϑn−m!=∑n=0∞ϱn+m,k,spznΓs,kpθn+m+ϑn!.By using(55)ξn+j,k,sp=ξj,k,sp×ξ+jkn,k,sp,in the above equation, we get the desired result (52).(2)
Taking L.H.S of (43),(56)ddzmzpϑ/kEk,θ,ϑϱ,szθ/k=ddzm∑n=0∞ϱn,k,spzθn/k+ϑ/k−1Γs,kpθn+ϑn!=∑n=m∞ϱn,k,spzθn/k+ϑ/k−m−1Γs,kpθn+ϑn−m!=∑n=0∞ϱn+m,k,spzθ/kn+m+ϑ/k−m−1Γq,kpθn−m+ϑn!.By using(57)ξn+j,k,sp=ξj,k,sp×ξ+jkn,k,sp,in the above equation, we get the desired result (53).Theorem 8.
Suppose thatk>0 and θ,ϑ and ϱ∈ℂ with ℜθ>0,ℜϑ>0 and ℜϱ>0, then(58)∫0ztϑ/k−1Ek,θ,ϑϱ,spatθ/kdt=sθn+ϑ/kpzϑ/kEk,θ,ϑ+kϱ,spazθ/k.Proof.
Consider the L.H.S.(59)∫0ztϑ/k−1Ek,θ,ϑϱ,spatθ/kdt=∫0ztϑ/k−1∑n=0∞ϱn,k,spatθ/knΓs,kpθn+ϑn!.
By changing integration and summation orders in the above equation, we have(60)∫0ztϑ/k−1Ek,θ,ϑϱ,spatθ/kdt=∑n=0∞ϱn,k,spanΓs,kpθn+ϑn!∫0ztθn/k+ϑ/k−1dt=∑n=0∞ϱn,k,spanzθn/k+ϑ/kθn+ϑ/kΓs,kpθn+ϑn!.
By using the recurrence relation ofΓs,kpξ and Γkpξ and equation III.31 in [22], we get(61)θn+ϑkΓs,kpθn+ϑ=sθn+ϑ/kpΓs,kpθn+ϑ+k.
By using the above result in equation (60), we have(62)∫0ztϑ/k−1Ek,θ,ϑϱ,spatθ/kdt=sθn+ϑ/kpzpϑ/kEk,θ,ϑ+kϱ,sazθ/k.


## 6. The Euler-Beta Transform of the Mittag-Lefflerp,s,k-Function Ek,θ,ϑϱ,spz

The well-known Euler-beta transform is defined by(63)Bfθ;a;b=∫01θa−11−θb−1fθdθ,where a,b∈ℝ and minℜθ,ℜϑ>0.Next, we define the beta transform of newly defined M-L function.Theorem 9.
Suppose thatp,s,k>0 and θ,ϑ,ϱ,a,b∈ℂ with ℜθ>0,ℜϑ>0, ℜϱ>0,ℜa>0, and ℜb>0, then(64)BEk,θ,ϑϱ,spλzθ/k;ϑk;bk=kΓs,kpbEk,θ,ϑ+bϱ,spλ.Proof.
(65)BEk,θ,ϑϱ,spλzθ/k;ϑk;bk=∫01zϑ/k−11−zb/k−1∑n=0∞ϱn,k,spλnzθn/kΓs,kpθn+ϑn!dz.
After interchanging the integration and summation orders of the above equation and using the relation between the gammap,s,k-function and the classical gamma function given by (3.4), we have(66)BEk,θ,ϑϱ,spλzθ/k;ϑk;bk=∑n=0∞ϱn,k,spλnΓs,kpθn+ϑn!∫01zθn/k+ϑ/k−11−zb/k−1dz=∑n=0∞ϱn,k,spλnΓs,kpθn+ϑn!Γθn+ϑ/kΓb/kΓθn+ϑ+b/k=∑n=0∞ϱn,k,spλnsθn+ϑ/kΓθn+ϑ/kn!Γθn+ϑ/kΓb/kΓθn+ϑ+b/k.
After simplification, we obtain the desired result:(67)BEk,θ,ϑϱ,spzθ/k;ϑk;bk=kΓs,kpbEk,θ,ϑ+bϱ,spλ.Remark 1.
(i)
Ifs=1, then equation (64) coincides with the result of [22], sec IV.2(ii)
Ifs=1andp=k, then equation (64) coincides with the formula 11,37 in [21](iii)
Ifs=p=k=1, then equation (64) coincides with the formula 2.2.14 in [24]


## 7. The Laplace Transform of the Mittag-Lefflerp,s,k-Function Ek,θ,ϑϱ,spz

Theorem 10.
Letp,s,k,c>0 and θ,ϑ,ϱ∈ℂ with ℜθ>0,ℜϑ>0, and ℜϱ>0. Then, the Laplace transform of Ek,θ,ϑϱ,sp is given by(68)ℒzϑ/k−1Ek,θ,ϑϱ,sp±czθ/k=ksps−ϑ/k1∓pcsksθ/k−ϱ/k.Proof.
Applying the Laplace transform on the left hand side of (68), the Laplace transform of the potential function (see [1], equation (1.4.58)) and the generalized binomial formula is given by(69)1−kw−ϱ/k=∑k=0∞ϱn,kwnn!.
We have(70)ℒzϑ/k−1Ek,θ,ϑϱ,sp±czθ/k=∫0∞e−szzϑ/k−1∑n=0∞ϱn,k,spΓs,kpθn+ϑn!±1nczθn/kdz=∑n=0∞ϱn,k,sp±1ncθn/kΓs,kpθn+ϑn!∫0∞e−szzθn/k+ϑ/k−1dz=kspsϑ/k∑n=0∞ϱn,k,spn!c/spsθn/k.
Now, using the relation betweenϱn,k,sp and ϱn,kp, we have(71)ℒzϑ/k−1Ek,θ,ϑϱ,sp±czθ/k=kspsϑ/k∑n=0∞spnϱn,kn!cspsθn/k=kspsϑ/k∑n=0∞snp/knϱn,kn!cspsθn/k=kspsϑ/k∑n=0∞ϱn,kn!cspsθ/kspkn.
Writing the series in compact form, we have(72)ℒzϑ/k−1Ek,θ,ϑϱ,sp±czθ/k=kspsϑ/k1∓pscksθ/k−ϱ/k.Remark 2.
(a)
Ifs=1 in the above theorem, then the result coincides with the formula of [22](b)
Ifp=s=k=c=1 in Theorem 10, then(73)ℒzϑ−1E1,θ,ϑϱ,11zθ=sϑ1−s−θ−ϱ,which coincides with formula (11.13) of [25].


## 8. Conclusion

In this paper, we established a new extension of the Mittag-Leffler function and investigated some of its properties. We concluded as follows:(1)
Ifs=1, then we get the results of the Mittag-Leffler p,s,k-function defined in [22](2)
Ifp=1ands=1, then we get the results of the Mittag-Leffler k-function defined in [21](3)
Ifk=1,p=1,ands=1, then we get the results of the Mittag-Leffler defined in [6](4)
Ifp=1,k=1,s=1,andϱ=1, then we get the results of the Mittag-Leffler defined in [18](5)
Ifp=1,k=1,s=1,ϑ=1,andϱ=1, then we get the results of the Mittag-Leffler k-function defined in [17](6)
Ifp=1,k=1,s=1,θ=1,ϑ=1,andϱ=1, then we get the exponential functionRecently, the Mittag-Leffler function is used to construct the fractional operators with nonsingular kernels [26, 27]. In [28, 29], the authors introduced the generalized fractional integrals and differential operators, which contain the Mittag-Leffler k-function in the kernels, and proved their various properties. Recently, Samraiz et al. [30] introduced the Hilfer Prabhakar k;s-fractional derivative by using the Mittag-Leffler k-function. They discussed its various properties and the generalized Laplace transform of the said operator. They also discussed the applications of Hilfer Prabhakar’s k;s-derivative in mathematical physics. In this study, we defined further generalization of the Mittag-Leffler and proved its various basic properties. Hence, it would be of great interest that the Mittag-Leffler function studied in this article will be utilized to generalize such classes of fractional and differential operators.



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