# An Extension of the Mittag-Leffler Function and Its Associated Properties

Authors: Gauhar Rahman; Abdus Saboor; Zunaira Anjum; Kottakkaran Sooppy Nisar; Thabet Abdeljawad
Journal: Advances in Mathematical Physics
 (2020)

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## Abstract

Inspired by certain fascinating ongoing extensions of the special functions such as an extension of the Pochhammer symbol and generalized hypergeometric function, we present a new extension of the generalized Mittag-Leffler (ML) functionεa,b;p,vκz1 in terms of the generalized Pochhammer symbol. We then deliberately find certain various properties and integral transformations of the said function εa,b;p,vκz1. Some particular cases and outcomes of the main results are also established.


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## Body

## 1. Introduction

The well-known ML (Mittag-Leffler) function with one parameter is defined by(1)εaz1=∑l=0∞z1lΓal+1a∈ℂ;Ra>0,z1∈ℂ.The generalization of (1) with two parameters defined by
(2)εa,bz1=∑l=0∞z1lΓal+ba,b∈ℂ;Ra>0,Rb>0was presented and contemplated by Mittag-Leffler [1–7] and other researchers. In [8], the generalization of (1) was given by
(3)εa,bκz1=∑l=0∞κlΓal+bz1ll!a,b,κ∈ℂ;Ra>0,Rb>0.Shukla and Prajapati [9] defined the following generalization of the ML function by
(4)εa,bκ,qz1=∑l=0∞κlqΓal+bz1ll!a,b,κ∈ℂ;Ra>0,Rb>0,q>0.Rahman et al. [10] defined the following extension of generalized ML function by
(5)εa,b,cκ,q,cz1=∑l=0∞Bκ+lq;c−κclqBκ,c−κΓal+bz1ll!,where a,b,κ,c∈ℂ;Rc>0,Ra>0,Rb>0,Rp>0,q>0, and Bp(x,y) is the extension of beta function (see [11]).Moreover, the generalization of ML function (3) was presented by [12] as follows:
(6)εa,b,pκz1=∑l=0∞κ;plΓal+bz1ll!p≥0,a,b,κ∈ℂ;Ra>0,Rb>0,Rp>0,where κ;pl is the Pochhammer symbol which is defined as
(7)λ;σμ=Γσλ+μΓλ,Rσ,λ,μ∈ℂ,λμ,σ=0,λ,μ∈ℂ\0.The researchers studied these extensions (6) and (7) and investigated their further extensions and associated properties and applications. (The readers may consult [13–16].) Recently, Srivastava et al. [17] have presented and concentrated in a fairly productive way the following extension of the generalized hypergeometric function:
(8)Fstδ1;ρ,v,⋯,δs;ζ1,⋯,ζt;z1=∑l=0∞δ1;ρ,vl⋯δslζ1l⋯ζtlz1ll!,where δj∈ℂ for j=1,2,⋯,s, ζk∈ℂ for k=1,2,⋯,t, and ζk≠0,−1,−2,⋯, and where μ;ω,aη is the extension of the generalized Pochhammer symbol defined by [23]:
(9)μ;ω,aη=Γaμ+η;ωΓμ,Rω,Ra>0,μ,η∈ℂ,μ;ωη,a=0,μ,η∈ℂ\0.The integral representation ofμ;ω,aη is explained by
(10)μ;ω,aη=2ωπ1Γμ∫0∞sμ+η−3/2e−sKa+1/2ωsds,where Ka· is the modified Bessel function of order a. Clearly, when a=0 in (10), at that point, by utilizing the way that K1/2t=π/2te−t, it will lead to formula [(31)]:
(11)μ;ω,0η=μ;ωη=1Γμ∫0∞sμ+η−1e−sds.Specifically, the relating extensions of the confluent hypergeometric function1F1 and the Gauss hypergeometric function 2F1 are given by
(12)F21δ1;ρ,v,b;κ;z1=∑l=0∞δ1;ρ,vlblκlz1ll!,(13)F11δ1;ρ,v;κ;z1=Φδ1;ρ,v;κ;z1=∑l=0∞δ1;ρ,vlκlz1ll!.The extension of generalized hypergeometric functionrFs of r numerator and s denominator parameters was investigated by [18]. Recently, the researchers defined various extensions of special functions and their associated properties and applications in the diverse field. (The interested readers may consult [19–22].) In [23–25], the authors introduced an extension of fractional derivative operators based on the extended beta functions.Next, motivated by the above such extensions of special functions, we define an extension of ML function (6) in terms of the generalized Pochhammer symbol (9) and investigate its certain variations.


## 2. Extension of ML Function

We present an extension of the generalized ML function in (6) regarding the extended Pochhammer symbol in (9) as follows:
(14)εa,b;p,vκz1=∑l=0∞κ;p,vlΓal+bz1ll!a,b,κ∈ℂ;Ra>0,Rb>0,Rp>0,p≥0,v≥0,given that the series on the right hand side converges.Clearly, it diminishes to the extended generalized ML function (6) for ν=0. The special case for a=1 in (14) can be communicated regarding extended confluent hypergeometric function (13) as follows:
(15)ε1,b;p,νκz1=1Γb1F1κ1;p,ν;b;z1=1ΓbΦκ1;p,ν;b;z1.


## 3. Basic Properties ofεκa,b;p,νz1

In this section, we present certain basic properties and integral representations of the extended generalized ML functionεa,b;p,νκz1 in (14).Theorem 1.
For the functionεa,b;p,νκz1 in (14), the following relation holds true:(16)εa,b;p,νκz1=bεa,b+1;p,νκz1+az1ddzεa,b+1;p,νκz1a,b,κ∈ℂ;Ra>0,Rb>0,Rp>0,p≥0,ν≥0.
Specifically, we have(17)Ea,b;pκz1=bEa,b+1;pκz1+az1ddz1Ea,b+1;pκz1.Proof.
From (14), we have
(18)bεa,b+1,p,νκz1+az1ddz1εa,b+1,p,νκz1=b∑l=0∞κ,p,νlΓal+b+1z1ll!+az1ddz1∑l=0∞κ,p,νlΓal+b+1z1ll!=b∑l=0∞κ,p,νlΓal+b+1z1ll!+az1ddz1∑l=0∞κ,p,νlΓal+b+1z1ll!=b∑l=0∞κ,p,νlΓal+b+1z1ll!+∑l=0∞az1κ,p,νlΓal+b+1lzl−1l!=b∑l=0∞κ,p,νlΓal+b+1z1ll!+∑l=0∞κ,p,νlalΓal+b+1z1ll!=∑l=0∞κ,p,νlΓal+b+1z1ll!al+b=∑l=0∞κ,p,νlal+bal+bΓal+bz1ll!,usingΓz1+1=z1Γz1,=∑l=0∞κ,p,νlΓal+bz1ll!=εa,b,p,νκz1.
Equation (17) can be obtained from (16) when we put ν=0.Theorem 2.
For the functionεκa,b;p,νz1 in (14), the following higher order differentiation formulas hold true:
(19)ddz1mεa,b;p,νκz1=κmεa,b+m;p,νκ+mz1,(20)ddz1mz1b−1εa,b;p,νκωz1a=z1b−m−1εa,b−m;p,νκωz1aRb−m>0,m∈ℕ,(21)ddz1mz1b−1Φκ;p,ν;b;ωz1=ΓbΓb−mz1b−m−1Φκ;p,ν;b−m;ωz1,where a,b,κ∈ℂ;Ra>0,Rb>0,Rp>0,p≥0,ν≥0.
Specifically, we have(22)ddz1mEa,b,pκz1=κmEa,b+m,pκ+mz1,(23)ddz1mz1b−1Ea,b,pκωz1a=z1b−m−1Ea,b−m,pκωz1a,(24)ddz1mz1b−1Φκ,p;b;ωz1=ΓbΓb−mz1b−m−1Φκ,p;b−m;ωz1.Proof.
Operating term wise differentiationm times on (14), we get
(25)ddz1mεa,b;p,νκz1=ddz1m∑l=0∞κ;p,νlΓal+bz1ll!=∑l=m∞κ;p,νlΓal+bl!zl−ml−m!l!=∑l=0∞κ;p,νl+mΓal+b+amz1ll!,replacinglbyl+m,=∑l=0∞κmκ+m;p,νlΓal+b+amz1ll!,usingλ;σ,vμ+p=λμλ+μ;σ,vp,=κm∑l=0∞κ+m;p,νnΓal+b+amz1ll!=κmεa,b+m;p,νκ+mz1.
In a similar manner from (20), we get
(26)ddz1mzb−1εa,b;p,νκωz1a=ddz1mz1b−1∑l=0∞κ;p,νlΓal+bωz1all!=ddz1m∑l=0∞κ;p,νlΓal+bz1b−1+all!ωl=∑l=0∞κ;p,νlΓal+bωll!al+b−1!al+b−m−1!z1al+b−m−1,differentiatingmtimes,=∑l=0∞κ;p,νlΓal+bΓal+bΓal+b−mωlz1al+b−1−ml!,usingl−1!=Γl,=z1b−m−1∑l=0∞κ;p,νlΓal+b−mωz1all!=z1b−m−1εa,b−m;p,νκωz1a.
Moreover, puttinga=1 in (20) gives (21). For the special case of (19), (20), and (21), when we put v=0, we get (22), (23), and (24), respectively.Corollary 3.
The following integral representations for ML functionεκa,b;p,νz1 (14) hold true:
(27)∫0z1tb−1εa,b;p,νκωtadt=z1bεa,b+1;p,νκωz1a,∫0z1tb−1Φκ,p,ν;b;ωtdt=1ΓbzbΦκ;p,ν;b+1;ωz1,where a,b,κ,ω∈ℂ;Ra>0,Rb>0,Rp>0,p≥0,ν≥0.
Specifically, we have(28)∫0z1tb−1Ea,b;pκωtadt=z1bEa,b+1;pκωz1a,∫0z1tb−1Φκ,p;b;ωtdt=1ΓbzbΦκ;p;b+1;ωz1.


## 4. Representation ofεa,b;p,νκz1 in terms of Generalized Hypergeometric Function

Here, we establish the representation ofεa,b;p,νκz1 (14) in terms of generalized hypergeometric function as follows.Theorem 4.
The functionεa,b;p,νκz1 defined in (14) fora∈N can be represented in the form of generalized hypergeometric function as given by
(29)εa,b,p,νκz1=1Γb1Fqκ,p,ν;bq,b+1q,⋯,b+k−1q;z1qq=1Γb1Fqκ,p,ν;Δq;b;z1qq,where q∈ℕ and Δq;b is an array pfq parameters b/q,b+1/q,⋯,b+q−1/q.Proof.
Takinga=q∈ℕ in (14) and utilizing the well-known multiplication formula for the gamma function, we have
(30)εq,b,p,νκz1=∑l=0∞κ,p,νlΓqn+bz1ll!=∑l=0∞κ,p,νlΓb/Γb×Γqn+bz1ll!=1Γb∑l=0∞κ,p,νlΓqn+b/Γbz1ll!=1Γb∑l=0∞κ,p,νlbnqz1ll!=1Γb∑l=0∞κ,p,νl∏j=1qb+j−1/qnz1qql1l!,sincebqn=∏j=1qb+j−1qn,=1Γb∑l=0∞κ,p,νlb/ql,b+1/ql,⋯,b+q−1/qlz1qql1l!=1Γb1Fqκ,p,ν;bq,b+1q,⋯,b+q−1q;z1qq=1Γb1Fqκ,p,ν;Δq;b;z1qq.


## 5. Integral Transformation ofεa,b,p,νκz1

Here, we present various integral representations of the functionεa,b;p,νκz1 in (14) such as the Mellin, the Euler-beta, and the Laplace transformations.


### 5.1. Mellin Transform

The well-known Millen transform [26] of integrable function fz1 with index s is defined by
(31)Mfz1:z1→s=∫0∞z1s−1fz1dz1,if the improper integral in (31) exists.Theorem 5.
For the functionεa,b;p,νκz1 in (14), the following Mellin transform exists:
(32)Mεa,b,p,νκz1:ν→s,(33)2s−1πΓs−v2Γs+v+12κsEa,bκ+sz1Rs−v>0,Rκ+s>−1.Proof.
By (32), we get from (14)
(34)Mεa,b;p,νκz1:ν→s=∫0∞νs−1εa,b;p,νκz1dν=∫0∞νs−1∑l=0∞κ;p,νlΓal+bz1ll!dν=∫0∞νs−1∑l=0∞Γνκ+l;pΓκΓal+bz1ll!dν.
Interchanging the order of summation and integration, we get(35)Mεa,b;p,νκz1:ν→s=∑l=0∞1Γal+bz1ll!1Γκ∫0∞νs−1Γνκ+l;pdν.
Now, by utilizing the well-known result ([11], Equation 4.105),
(36)∫0∞νs−1Γνκ+l;pdν=2s−1πΓs−v2Γs+v+12Γκ+s+lRκ+s>−1,Rs−v>0.
Therefore, we have(37)Mεa,b;p,νκz1:ν→s=2s−1/πΓs−v/2Γs+v+1/2Γκ∑l=0∞Γκ+s+lΓal+bz1ll!=2s−1πΓs−v2Γs+v+12κs∑l=0∞κ+slΓal+bz1ll!=2s−1πΓs−v2Γs+v+12κsEa,bκ+sz1.Corollary 6.
The below integral transform exists:(38)Mεa,b;p,νκz1:ν→s=2s−1/πΓs−v/2Γs+v+1/2Γκ1Ψ1κ+s,1;b,a;z,where pΨq is the Wright hypergeometric function (see [27, 28]).


### 5.2. Euler-Beta Transform

In [26], the well-known Euler-beta transform of the function fz1 is defined by
(39)Bfz1;a1,b1=∫01z1a1−11−z1b1−1fz1dz1.Theorem 7.
For the functionεκa,b;p,νz1in (14), the following Euler-beta transform holds:
(40)Bεa,b;p,νκxz1σ:a1,b1=Γb1Γa1ΓbΓa1+b1σ+1Fa+σκ,p,ν,Δσ;a1;Δa;b,Δσ;a1+b1;xaaa,σ∈ℕ;Rp>0,Ra1>0,Rb1>0,Rb>0,Rκ>0,p≥0,ν≥0.Proof.
By using (40) and (14), we obtain
(41)Bεa,b;p,νκxz1σ:a1,b1=∫01z1a1−11−z1b1−1εa,b;p,νκxz1σdz1=∫01z1a1−11−z1b1−1∑l=0∞κ;p,νlΓal+bxlz1σll!dz1=∑l=0∞κ;p,νlΓal+bxll!∫01z1a1+σl−11−z1b1−1dz1=∑l=0∞κ;p,νlΓal+bxll!Γσl+a1Γb1Γσl+a1+b1=Γb1Γa1Γa1+b1∑l=0∞κ;p,νla1σlΓal+b×a1+b1σlxll!=Γb1Γa1Γa1+b1Γb∑l=0∞κ;p,νla1σlbala1+b1σlxll!=Γb1Γa1Γa1+b1Γb∑l=0∞κ;p,νl∏j=1σa1+j−1/σlσσl∏j=1ab+j−1/alaal∏j=1σa1+b1+j−1/σlσσlxll!=Γb1Γa1Γa1+b1Γb∑l=0∞κ;p,νl∏j=1σa1+j−1/σl∏j=1ab+j−1/alaal∏j=1σa1+b1+j−1/σlxll!=Γb1Γa1ΓbΓa1+b1σ+1Fa+σκ,p,ν,Δσ;a1;Δa;b,Δσ;a1+b1;xaa.Corollary 8.
By puttinga1=bandσ=a∈ℂ in (33) and then using (14), we have
(42)∫01z1b−11−z1b1−1εa,b;p,νκxzadz1=Γb1εa,b+b1;p,νκx.
Similarly, we have(43)∫01z1a1−11−z1b−1εa,b;p,νκx1−z1adz1=Γa1εa,b+a1;p,νκx.
In general, we get(44)∫01x−z1a1−1z1−tb−1εa,b;p,νκwz1−tadz1=Γa1x−tb+a1−1εa,b+a1;p,νκwz−ta.


### 5.3. Laplace Transform

The well-known Laplace transform [26] of fz1 is defined by
(45)Lfz1=∫0∞e−sz1fz1dz1.Theorem 9.
For the functionεκa,b;p,νz1 in (14), the following Laplace transform holds:
(46)Lz1a1−1εa,b;p,νκxz1σ=Γa1Γbsa1σ+1Faκ,p,ν,Δσ;a1;Δa;b;xaasσσa,σ∈ℕ;Rp>0,Ra1>0,Rs>0,Rb>0,Rκ>0,p≥0,ν≥0.Proof.
By using (45) and from (14), we have
(47)Lz1a1−1εa,b;p,νκxz1σ=∫0∞z1a1−1e−sz1εa,b;p,νκxz1σdz1=∫0∞z1a1−1e−sz1∑l=0∞κ;p,νlΓal+bxlz1σll!dz1=∑l=0∞κ;p,νlΓal+bxll!∫0∞z1a1+σl−1e−sz1dz1=∑l=0∞κ;p,νlΓal+bxll!Γa1+σlsa1+σl=∑l=0∞κ;p,νlΓal+bxll!Γa1+σlsa1+σl=Γa1Γbsa1∑l=0∞κ;p,νla1σlbalxaalsσσσl1l!=Γa1Γbsa1σ+1Faκ,p,ν,Δσ;a1;Δa;b;xaasσσ.Corollary 10.
By settinga1=bandσ=a in (46), we obtain
(48)∫0∞zb−1e−szεa,b;p,νκxzadz=1sb1F0κ,p,ν;−;xsa.


### 5.4. Whittaker Transformation

To determine the Whittaker transforms, we use the following formula:(49)∫0∞tv−1e−1/2tWη,ξtdt=Γ1/2±ξ+vΓ1−η+vRv±ξ>−12.Theorem 11.
For the functionεκa,b;p,νz1in (14), the following Whittaker transform holds:
(50)∫0∞tρ−1e−1/2stWη,ξstεa,b;p,νκωzδdt=Γ1/2±ξ+ρΓ1−η+ρΓbsρ2ϑ+1Fa+ϑκ,p,ν,Δϑ;12±ξ+ρ;Δa,b,Δϑ;1−η+ρ;ωaasϑϑa,ϑ∈ℕ;p≥0;Rp>0,Rρ>0,Rb>0,Rκ>0,ν≥0.Proof.
By the definition of Whittaker transform, we have from (14)
(51)∫0∞tρ−1e−1/2stWη,ξstεa,b;p,νκωtϑdt=∫0∞tρ−1e−1/2stWη,ξst∑l=0∞κ;p,νlΓal+bωltϑnl!dt=∑l=0∞κ;p,νlΓal+bωll!∫0∞tρ+ϑl−1e−1/2stWη,ξstdt.By putting v=st and then using the definition of Whittaker transforms, we get
(52)∫0∞tρ−1e−1/2tWη,ξstεa,b;p,νκωzϑdt=∑l=0∞κ;p,νlΓal+bωll!Γ1/2±ξ+ρ+ϑlΓ1−η+ρ+ϑls−ρ−ϑl=1sρ∑l=0∞κ;p,νlΓ1/2±ξ+ρ+ϑl×Γ1/2±ξ+ρ/Γ1/2±ξ+ρΓal+b×Γ1−η+ρ+ϑl×Γ1−η+ρ/Γ1−η+ρ×1l!ωsϑl=Γ1/2±ξ+ρΓ1−η+ρΓbsρ∑l=0∞κ;p,νl1/2±ξ+ρϑlbal1−η+ρϑl1l!ωsϑl=Γ1/2±ξ+ρΓ1−η+ρΓbsρ2ϑ+1Fa+ϑκ,p,ν,Δϑ;12±ξ+ρ;Δa,b,Δϑ;1−η+ρ;ωaasϑϑ.


## 5.1. Mellin Transform

The well-known Millen transform [26] of integrable function fz1 with index s is defined by
(31)Mfz1:z1→s=∫0∞z1s−1fz1dz1,if the improper integral in (31) exists.Theorem 5.
For the functionεa,b;p,νκz1 in (14), the following Mellin transform exists:
(32)Mεa,b,p,νκz1:ν→s,(33)2s−1πΓs−v2Γs+v+12κsEa,bκ+sz1Rs−v>0,Rκ+s>−1.Proof.
By (32), we get from (14)
(34)Mεa,b;p,νκz1:ν→s=∫0∞νs−1εa,b;p,νκz1dν=∫0∞νs−1∑l=0∞κ;p,νlΓal+bz1ll!dν=∫0∞νs−1∑l=0∞Γνκ+l;pΓκΓal+bz1ll!dν.
Interchanging the order of summation and integration, we get(35)Mεa,b;p,νκz1:ν→s=∑l=0∞1Γal+bz1ll!1Γκ∫0∞νs−1Γνκ+l;pdν.
Now, by utilizing the well-known result ([11], Equation 4.105),
(36)∫0∞νs−1Γνκ+l;pdν=2s−1πΓs−v2Γs+v+12Γκ+s+lRκ+s>−1,Rs−v>0.
Therefore, we have(37)Mεa,b;p,νκz1:ν→s=2s−1/πΓs−v/2Γs+v+1/2Γκ∑l=0∞Γκ+s+lΓal+bz1ll!=2s−1πΓs−v2Γs+v+12κs∑l=0∞κ+slΓal+bz1ll!=2s−1πΓs−v2Γs+v+12κsEa,bκ+sz1.Corollary 6.
The below integral transform exists:(38)Mεa,b;p,νκz1:ν→s=2s−1/πΓs−v/2Γs+v+1/2Γκ1Ψ1κ+s,1;b,a;z,where pΨq is the Wright hypergeometric function (see [27, 28]).


## 5.2. Euler-Beta Transform

In [26], the well-known Euler-beta transform of the function fz1 is defined by
(39)Bfz1;a1,b1=∫01z1a1−11−z1b1−1fz1dz1.Theorem 7.
For the functionεκa,b;p,νz1in (14), the following Euler-beta transform holds:
(40)Bεa,b;p,νκxz1σ:a1,b1=Γb1Γa1ΓbΓa1+b1σ+1Fa+σκ,p,ν,Δσ;a1;Δa;b,Δσ;a1+b1;xaaa,σ∈ℕ;Rp>0,Ra1>0,Rb1>0,Rb>0,Rκ>0,p≥0,ν≥0.Proof.
By using (40) and (14), we obtain
(41)Bεa,b;p,νκxz1σ:a1,b1=∫01z1a1−11−z1b1−1εa,b;p,νκxz1σdz1=∫01z1a1−11−z1b1−1∑l=0∞κ;p,νlΓal+bxlz1σll!dz1=∑l=0∞κ;p,νlΓal+bxll!∫01z1a1+σl−11−z1b1−1dz1=∑l=0∞κ;p,νlΓal+bxll!Γσl+a1Γb1Γσl+a1+b1=Γb1Γa1Γa1+b1∑l=0∞κ;p,νla1σlΓal+b×a1+b1σlxll!=Γb1Γa1Γa1+b1Γb∑l=0∞κ;p,νla1σlbala1+b1σlxll!=Γb1Γa1Γa1+b1Γb∑l=0∞κ;p,νl∏j=1σa1+j−1/σlσσl∏j=1ab+j−1/alaal∏j=1σa1+b1+j−1/σlσσlxll!=Γb1Γa1Γa1+b1Γb∑l=0∞κ;p,νl∏j=1σa1+j−1/σl∏j=1ab+j−1/alaal∏j=1σa1+b1+j−1/σlxll!=Γb1Γa1ΓbΓa1+b1σ+1Fa+σκ,p,ν,Δσ;a1;Δa;b,Δσ;a1+b1;xaa.Corollary 8.
By puttinga1=bandσ=a∈ℂ in (33) and then using (14), we have
(42)∫01z1b−11−z1b1−1εa,b;p,νκxzadz1=Γb1εa,b+b1;p,νκx.
Similarly, we have(43)∫01z1a1−11−z1b−1εa,b;p,νκx1−z1adz1=Γa1εa,b+a1;p,νκx.
In general, we get(44)∫01x−z1a1−1z1−tb−1εa,b;p,νκwz1−tadz1=Γa1x−tb+a1−1εa,b+a1;p,νκwz−ta.


## 5.3. Laplace Transform

The well-known Laplace transform [26] of fz1 is defined by
(45)Lfz1=∫0∞e−sz1fz1dz1.Theorem 9.
For the functionεκa,b;p,νz1 in (14), the following Laplace transform holds:
(46)Lz1a1−1εa,b;p,νκxz1σ=Γa1Γbsa1σ+1Faκ,p,ν,Δσ;a1;Δa;b;xaasσσa,σ∈ℕ;Rp>0,Ra1>0,Rs>0,Rb>0,Rκ>0,p≥0,ν≥0.Proof.
By using (45) and from (14), we have
(47)Lz1a1−1εa,b;p,νκxz1σ=∫0∞z1a1−1e−sz1εa,b;p,νκxz1σdz1=∫0∞z1a1−1e−sz1∑l=0∞κ;p,νlΓal+bxlz1σll!dz1=∑l=0∞κ;p,νlΓal+bxll!∫0∞z1a1+σl−1e−sz1dz1=∑l=0∞κ;p,νlΓal+bxll!Γa1+σlsa1+σl=∑l=0∞κ;p,νlΓal+bxll!Γa1+σlsa1+σl=Γa1Γbsa1∑l=0∞κ;p,νla1σlbalxaalsσσσl1l!=Γa1Γbsa1σ+1Faκ,p,ν,Δσ;a1;Δa;b;xaasσσ.Corollary 10.
By settinga1=bandσ=a in (46), we obtain
(48)∫0∞zb−1e−szεa,b;p,νκxzadz=1sb1F0κ,p,ν;−;xsa.


## 5.4. Whittaker Transformation

To determine the Whittaker transforms, we use the following formula:(49)∫0∞tv−1e−1/2tWη,ξtdt=Γ1/2±ξ+vΓ1−η+vRv±ξ>−12.Theorem 11.
For the functionεκa,b;p,νz1in (14), the following Whittaker transform holds:
(50)∫0∞tρ−1e−1/2stWη,ξstεa,b;p,νκωzδdt=Γ1/2±ξ+ρΓ1−η+ρΓbsρ2ϑ+1Fa+ϑκ,p,ν,Δϑ;12±ξ+ρ;Δa,b,Δϑ;1−η+ρ;ωaasϑϑa,ϑ∈ℕ;p≥0;Rp>0,Rρ>0,Rb>0,Rκ>0,ν≥0.Proof.
By the definition of Whittaker transform, we have from (14)
(51)∫0∞tρ−1e−1/2stWη,ξstεa,b;p,νκωtϑdt=∫0∞tρ−1e−1/2stWη,ξst∑l=0∞κ;p,νlΓal+bωltϑnl!dt=∑l=0∞κ;p,νlΓal+bωll!∫0∞tρ+ϑl−1e−1/2stWη,ξstdt.By putting v=st and then using the definition of Whittaker transforms, we get
(52)∫0∞tρ−1e−1/2tWη,ξstεa,b;p,νκωzϑdt=∑l=0∞κ;p,νlΓal+bωll!Γ1/2±ξ+ρ+ϑlΓ1−η+ρ+ϑls−ρ−ϑl=1sρ∑l=0∞κ;p,νlΓ1/2±ξ+ρ+ϑl×Γ1/2±ξ+ρ/Γ1/2±ξ+ρΓal+b×Γ1−η+ρ+ϑl×Γ1−η+ρ/Γ1−η+ρ×1l!ωsϑl=Γ1/2±ξ+ρΓ1−η+ρΓbsρ∑l=0∞κ;p,νl1/2±ξ+ρϑlbal1−η+ρϑl1l!ωsϑl=Γ1/2±ξ+ρΓ1−η+ρΓbsρ2ϑ+1Fa+ϑκ,p,ν,Δϑ;12±ξ+ρ;Δa,b,Δϑ;1−η+ρ;ωaasϑϑ.


## 6. Conclusion

In our current investigation, we presented an extension of the generalized ML functionεκa,b;p,νz1 in (14) by utilizing an extension of Pochhammer symbol μ;ω,aη defined in (9). Further, we have investigated several basic properties of the newly defined function εκa,b;p,νz1. The special cases of the main result for ν=0 can be found in the work of [12]. Thus, the results introduced in this present article are new and an extension of the relating outcomes in the existing literature (see, e.g., [29–31]). The newly defined ML function εκa,b;p,νz1 presented in this article will be applicable in different fields of applied sciences.



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