# Unified Approach to Fractional Calculus Images Involving the Pathway Transform of Extendedk-Gamma Function and Applications

Authors: Asifa Tassaddiq; Kwara Nantomah
Journal: Advances in Mathematical Physics
 (2022)

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## Abstract

A path way model is concerned with the rules of swapping among different classes of functions. Such model is significant to fit a parametric class of distributions for new data. Based on a such model, the path way orPδ transform is of binomial type and contains a family of different transforms. Taking inspiration from these facts, present research is concerned with the computation of new fractional calculus images involving the extended k-gamma function. The non-integer kinetic equations containing the extended k-gamma function is solved by using pathway transform as well as validated with the earlier obtained results. Pδ transform of Dirac delta function is obtained which proved useful to achieve the purpose. As customary, the results for the frequently used Laplace transform can be recovered by taking δ⟶1 in the definition of Pδ transform. Important new identities involving the Fox-Wright function are obtained and used to simplify the results. It is remarkable that the several new and novel results involving the classical gamma function became possible by using this approach.


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## Body

## 1. Introduction

Research and studies reveal that a large number of integral transforms are found in the literature. Each one of them is suitable for different problems despite a simple mathematical relation existing between them. This role is vital to understanding and applying the modern and classical theories. More recently, fractional type integral transform calledPδ transform or pathway transform is introduced in [1] based on pathway model [2, 3]. This model is significant to study important forms of statistics [4, 5] due to their applications in astrophysics [6–8] and applied analysis [9–11]. By varying the pathway parameter, one can obtain three useful forms. For a>0,δ>0,γ>0,η>0, and the normalizing constants c1,c2,c3; the pathway model is defined as follows:
(1)fx=c1xγ1−a1−αxδη/1−α,1−a1−αxδ>0,α<1c2xγ1−a1−αxδ−η/1−α,−∞<x<∞,α>1c3xγe−aηxδ,−∞<x<∞,α⟶1.For a complex valued integrable functionft,Pδ transform is defined as follows [1]:
(2)Pδft;s=Fs=∫0∞1+δ−1s−t/δ−1ftdt,δ>1;s∈ℂ,t∈ℝ,which is convergent for Rln1+δ−1s/δ−1>0. Involved variable t is transformed from ln1+δ−1s−t/δ−1 to e−st similar to as the paths are transformed from binomial to the exponential under pathway transformations [1].Agarwal et al. [12] recently used such transforms to solve the non-integer order differential and integral equations. Srivastava et al. [13, 14] used this transform to find certain results involving different special functions. For more details, see [1–15] and references therein. Some properties of the pathway transform are given in [1, 12]. For example, the following result is detailed in [1]:
(3)Pδzχ−1;s=Γχδ−1ln1+δ−1sχ,tχ−1Γχ=Pδ−1δ−1ln1+δ−1sχ.The pathway orPδ transform of Riemann–Liouville fractional derivatives of function ft of order ν is computed by [1]:
(4)PδI0+νΩt;s=δ−1ln1+δ−1sνΩs.The pathway orPδ transform of Caputo derivatives of function ft of non-integer order ν and n−1<ν≤n is given by [12]:
(5)PδD0ctνft;s=ln1+α−1sα−1νFs−∑k=0n−1ln1+α−1sα−1ν−k−1fk0.If Fs=Pδft;s and Gs=Pδgt;s, then the convolution theorem is stated in [1] as follows:
(6)FsGs=Pδft;sPδgt;s.ThePδ transform (2) diminishes to Laplace transform (Sneddon [16]) in the limiting case as δ⟶1:
(7)Lft;s=∫0∞e−stftdt;Rs>0,which is a classical tool to solve nontrivial problems [17–19] of diverse nature. It is remarkable that the following relation between two transforms exist [13–15]:
(8)Lft;s=Pδft;es−1δ−1δ−1,Pδft;s=Lft;ln1+δ−1sδ−1.Hence, by using (2), (7), and (8), the pathway or Pδ transform of delta function is given by
(9)Lδrz;ξ=ξrandPαδrz;ξ=ln1+δ−1ξδ−1r.A nontrivial generalization of the classical gamma function is investigated by Chaudhry and Zubair [20]:
(10)Γbz=∫0∞tz−1e−t−b/tdt,σ=Rz>0,b≥0,which attracted the attention of many researchers due to its applications, and therefore further extensions were focused on by them [21]. One such generalization is named as the “extendedk-gamma function” given by [21]:
(11)Γb,kz=∫0∞tz−1exp−tk−bk/tkkdt,Rz>0,b≥0,k>0.Whenk is unity, it diminishes to (10), and when b is zero, it becomes k-gamma function defined by [22]:
(12)Γkz=∫0∞tz−1e−tk/kdtkϵℝ,and whenk=1, the classical gamma function Γ1z=Γz is obtained, while for k=2, it is significant to investigate the important Gaussian distributions [23, 24]. k-gamma function has diverse applications for example, in physics and chemistry [24, 25]; in non-integer order calculus [26, 27]; and in statistical analysis [28]. The interested reader is directed to [29–33] and associated references therein for a more extensive and exhaustive review of related work. In recent years, Tassaddiq [34] has investigated a distributional representation of the extended k-gamma function in terms of complex delta function as follows:
(13)Γb,kz=2π∑n,r=0∞−1/kn+rbkrn!r!δz+nk−kr,(14)Γb,kz=2π∑n,r,p=0∞−1/kn+rbkrkpn−rpn!r!p!δpz.Hence, its action on a suitably chosen function over a specific domain is easily obtainable by means of the standard techniques applied to delta function, and it is found that [34]
(15)LΓb,kz;ξ=2π∑n,r,p=0∞−1/kn+rbkrkpn−rpn!r!p!ξp=2πexp−ekξk−bke−kξk.For further information on such representations concerning other special functions, and for properties of delta functions, one may refer to [34–45].


### 1.1. Novelty and Significance of the Research

The gamma function is a basic and widely studied function, but if we google “Laplace transform of gamma function,” then we cannot find it in the existing literature [20] (see also https://math.stackexchange.com/questions/2938166/what-is-the-laplace-transform-of-gamma-function).More recently, it has been computed by Tassaddiq in [34] by using the distributional representation of the extended k-gamma function. Present motivation is to further explore and extend the results of [34] keeping in view their novelty and significance to crack the problems that remained unsolved for many years. For instance, quite a few recent articles by Kiryakova ([46, 47] and references therein) point out that the special functions are either generalized fractional calculus operators of the basic functions (or the generalized fractional calculus operators are defined as the action of a specific special function on a general class of functions and vice versa). This marriage of special functions and fractional calculus is astounding. As a result, several authors have worked to compute the image formulae of a wide range of special functions using important fractional transforms. Such findings are discussed in a review article [46], and [15, 48, 49] contain an interesting debate between the researchers. The authors are inspired by these fractional operators because they connect numerous frequently used fractional operators [46]. As a result, we use the generalized fractional calculus operators to calculate the novel fractional images of the extended k-gamma function. They are known as generalized fractional integrals (multiple E–K operators) in [46, p. 8, Equation (19)]. In addition, the lately common and highly used fractional operators Riemann–Liouville (R-L), Saigo, and Marichev–Saigo–Maeda (M-S-M) are described with reference to such generalized operators. This approach has been used first time in this research and was not possible to use it for the famous gamma function (and its generalizations) by using its known (old) and classical representations.This study plans as follows: Section2 contains the necessary preliminaries related to the family of generalized fractional integrals (multiple E–K operators) and involved special functions. Section 3.1 includes fractional images that use the gamma function and its extensions. Section 3.2 goes over fractional derivatives. The following Section 3.3 focuses on the formulation and solution of a non-integer order kinetic equation using the pathway transform. Section 3.4 computes integrals involving the products of a class of special functions. Section 4 includes conclusion with a detailed discussion and comparison of the findings with other studies.


## 1.1. Novelty and Significance of the Research

The gamma function is a basic and widely studied function, but if we google “Laplace transform of gamma function,” then we cannot find it in the existing literature [20] (see also https://math.stackexchange.com/questions/2938166/what-is-the-laplace-transform-of-gamma-function).More recently, it has been computed by Tassaddiq in [34] by using the distributional representation of the extended k-gamma function. Present motivation is to further explore and extend the results of [34] keeping in view their novelty and significance to crack the problems that remained unsolved for many years. For instance, quite a few recent articles by Kiryakova ([46, 47] and references therein) point out that the special functions are either generalized fractional calculus operators of the basic functions (or the generalized fractional calculus operators are defined as the action of a specific special function on a general class of functions and vice versa). This marriage of special functions and fractional calculus is astounding. As a result, several authors have worked to compute the image formulae of a wide range of special functions using important fractional transforms. Such findings are discussed in a review article [46], and [15, 48, 49] contain an interesting debate between the researchers. The authors are inspired by these fractional operators because they connect numerous frequently used fractional operators [46]. As a result, we use the generalized fractional calculus operators to calculate the novel fractional images of the extended k-gamma function. They are known as generalized fractional integrals (multiple E–K operators) in [46, p. 8, Equation (19)]. In addition, the lately common and highly used fractional operators Riemann–Liouville (R-L), Saigo, and Marichev–Saigo–Maeda (M-S-M) are described with reference to such generalized operators. This approach has been used first time in this research and was not possible to use it for the famous gamma function (and its generalizations) by using its known (old) and classical representations.This study plans as follows: Section2 contains the necessary preliminaries related to the family of generalized fractional integrals (multiple E–K operators) and involved special functions. Section 3.1 includes fractional images that use the gamma function and its extensions. Section 3.2 goes over fractional derivatives. The following Section 3.3 focuses on the formulation and solution of a non-integer order kinetic equation using the pathway transform. Section 3.4 computes integrals involving the products of a class of special functions. Section 4 includes conclusion with a detailed discussion and comparison of the findings with other studies.


## 2. Preliminaries

Note: In this research,R, ℂ, and ℝ are the symbols to denote the real part of any complex number, complex numbers, and real numbers separately. ℝ+ is a set of positive reals, and ℤ0− is a set of negative integers including 0.Definition 1. (see [50]).
Forαϵℂ;Rα>0, Mittag-Leffler function is defined in a series form:
(16)Eαz=∑r=0∞zrΓαr+1.
Γαr+1 denotes the familiar gamma function [20], and, when α=1, it diminishes to the factorial, and hence Mittag-Leffler function diminishes to the exponential function. Similarly, Mittag-Leffler function of parameters 2 and 3 i-e α,β,γϵℂ,Rα>0 is defined as
(17)Eα,βz=∑r=0∞zrΓαr+β;Eα,βγz=∑r=0∞γrzrΓαr+β.Definition 2 (see [46, 51, 52]).
The Fox-Wright function (Ψqp) has a series representation
(18)Ψqpai,Aibj,Bj;z=∑m=0∞∏l=1pΓai+Aim∏l=1qΓbj+Bjmzmm!aiϵℝ+i=1,⋯.p;Bjϵℝ+j=1,⋯.q;1+∑i=1qBi−∑j=1pAj>0.Definition 3 (see [46, 51, 52]).
The foxH-function is defined as follows:
(19)Hp,qm,nz=Hp,qm,nzai,Aibj,Bj=Hp,qm,nza1,A1,⋯,ai,Aib1,B1,⋯,bj,Bj=12πi∫L∏j=1mΓbj+Bjs∏i=1nΓ1−aj−Ajs∏j=m+1qΓ1−bj−Bjs∏i=n+1pΓaj+Ajsz−sds1≤m≤q;0≤n≤p;Ai>0;Bj>0;ai,bj∈ℂ;i=1,⋯,p;j=1,⋯,q,where L denotes a suitable Mellin-Barnes curve to keep secluded the poles of Γbj+Bjsj=1m from that of Γ1−aj−Ajsj=1n.Remark 4 (see [46, 51, 52]).
IfAp=Bq=1, then H-function turns into Meijer G-function:
(20)Hp,qm,nza1,A1,⋯,ai,Aib1,B1,⋯,bj,Bj=∑m=0∞∏j=1mΓbj+Bjm∏i=1nΓ1−ai−Aim∏j=m+1qΓ1−bj−Bjm∏i=n+1pΓai+imzmm!,Hp,qm,nza1,1,⋯,ai,1b1,1,⋯,bj,1=Gp,qm,nza1⋯,aib1,⋯,bj,
and the Fox-Wright function is related with H-function(21)Ψqpai,Aibj,Bj;z=Hp,q+11,p−z1−a1,A1,⋯,1−ai,Ai0,1,1−b1,B1,⋯,1−bj,Bj,
and the hypergeometric functions as(22)Ψqpai,1bj,1;z=Gp,q+11,p−z1−a1,1,⋯,1−ai,10,1−b1,1,⋯,1−bj,1=Fqpaibj;z.Γa1⋯ΓaiΓb1⋯Γbj.ai>0;bj∉ℤ0−.Definition 5 (see [46]).
The generalized fractional integrals namely (multiple) E–K operators as defined in are(23)Iβk,mγk,νkfz=∫01fzσHm,mm,0σγk+νk+1−1βk,1βk1mγk+1−1βk,1βk1mdσ;∑kνk>0z−1∫0zfξHm,mm,0ξzγk+νk+1−1βk,1βk1mγk+1−1βk,1βk1mdξ;∑kνk>0.
Here, the order of integration is represented byνk′s, the additional parameters are represented by βk′s, and weights are symbolized by γk′s. It is important to notice that Iβk,mγk,νkfz=fz;νk=1. Since the function Hm,mm,0 cease to exist for σ>1, therefore the upper limit in (23) can be replaced by ∞.Definition 6. (see [46]).
Similarly, the fractional derivative of orderν=ν1≥0,⋯,νm≥0 in response to (23) are defined as
(24)Dβk,mγk1m,νkfz≔DηIβk,mγk+νk,ηk−νkfz=Dη∫01fzσHm,mm,0σγk+ηk+1−1βk,1βk1mγk+1−1βk,1βk1mdσ,where Dη (polynomial of degree η1+⋯+ηm in variable zd/dz is
(25)Dη=∏r=1m∏j=1ηr1βrzddz+γr+j;ηk=νk+1;νk∉ℤνk;νk∈ℤ.
Furthermore, the conforming non-integer order derivatives of Caputo type are(26)∗Dβk,mγk1m,νkfz=Iβk,mγk+νk,ηk−νkDηfz.Lemma 7 (see [46]).
For−βk1+γk<p;νk≥0;k=1,⋯,m, we have
(27)Iβk,mγk,νkzp=∏i=1mΓγi+1+p/βi/Γγi+νi+1+p/βizp.Definition 8 (see [52–56]).
For complex parametersγ1,γ1′,γ2,γ2′, Rν>0, the M-S-M fractional integral operators are defined by
(28)I0+γ1,γ1′,γ2,γ2′,νfx=x−γ1Γν∫0xx−tν−1t−γ1′F3γ1,γ1′,γ2,γ2′,ν;1−tx;1−xtftdt,I0−γ1,γ1′,γ2,γ2′,νfx=t−γ1′Γν∫x∞x−tν−1t−γ1F3γ1,γ1′,γ2,γ2′,ν;1−xt;1−txftdt.where F3 represents Appel Function (Horn function) which is defined as [52]
(29)F3γ1,γ1′,γ2,γ2′,ν;u;v=∑k,l=0∞γ1kγ1′lγ2kγ2′lνl+mukk!vll!,maxu,v<1.Lemma 9 (see [52–56]).
Letγ1,γ1′,γ2,γ2′∈ℂ,ω>0∧Rχ>max0,Rγ1+γ1′+γ2–ν,Rγ1′−γ2′,Rν>0,then
(30)I0+γ1,γ1′,γ2,γ2′,νωχ−1=ΓχΓχ+ν−γ1−γ1′−γ2Γχ+γ2′−γ1′Γχ+γ2′Γχ+ν−γ1−γ1′Γχ+ν−γ1′−γ2ων+χ−γ1−γ1′−1.
Similarly, letγ1,γ1′,γ2,γ2′∈ℂ,ω>0,andifRδ>0,Rχ<1+minR−γ2,Rγ1+γ1′−δ,Rγ1+γ2′−ν, then following image formula holds true:
(31)I0−γ1,γ1′,γ2,γ2′,νωχ−1=Γ1−χ−ν+γ1+γ1′Γ1−χ+γ1+γ2′−νΓ1−χ−γ1Γ1−χΓ1−χ+γ1+γ1′+γ2+γ2′−νΓ1−χ+γ1−γ2ων+χ−γ1−γ1′−1.Definition 10. (see [55]).
Forγ1,γ2,ν∈ℂ with x;Rν>0 by Saigo fractional integral operators are defined by
(32)I0+γ1,γ2,ν=x−ν−γ1Γν∫0xx−tν−1F21ν+γ2,−γ1;ν;1−txftdt,I−γ1,γ2,νfx=1Γν∫x∞t−xν−1t−ν−γ1F21ν+γ2,−γ1;ν;1−xtftdt,where F21 represents the Gauss hypergeometric function given by (see [56])
(33)F21γ1,γ2,γ3;u=∑k=0∞γ1kγ2kγ3kukk!,u<1;u=1u≠1,Rγ3−γ1−γ2>0.Lemma 11 (see [53–56]).
Forγ1,γ2,ν∈ℂ;Rν>0,Rχ>max0,Rγ1−γ2, we have
(34)I0+γ1,γ2,νωχ−1=ΓχΓχ+γ2−γ1Γχ−γ2Γχ+ν+γ2ωχ−γ1−1,
and forγ1,γ2,ν∈ℂ∧Rν>0∧Rχ<1+minRγ1,Rγ2:
(35)I−γ1,γ2,νωχ−1=Γγ1−χ+1Γγ2−χ+1Γ1−χΓγ1+γ2+ν−χ+1ωχ−γ1−1.Definition 12 (see [53–56]).
For complexγ,ν∈ℂ,Rν>0, Erdélyi–Kober integrals are defined by
(36)I0+0,γ,νfx=I0+γ,νfx=x−ν−γΓχ∫0xx−tν−1tγftdtx>0,I0−0,γ,νfx=I0−γ,νfx=xγΓχ∫x∞t−xν−1t−ν−γftdtx>0.Definition 13 (see [57, 58]).
Forν∈ℂ∧Rν>0, the R-L fractional integrals are defined as
(37)I0+νfx=1Γν∫0xx−tν−1ftdtx>0,I−νfx=1Γν∫x∞x−tν−1ftdtx>0,respectively. These are also related to Weyl transform [57, 58].Remark 14.
All of the above fractional operators are related and can be obtained as special cases of (17) by varying and specifying different parameter values. This is described below in the following Table 1.Table 1
Important cases of (multiple) E–K operators.
Cases of (23)Relation between kernels of the above fractional operators [46]m=3;β1=β2=β3=β=1Marichev–Saigo–Maeda (M-S-M)H3,33,0tx=G3,33,0txγ1′+γ2′,ν–γ1,ν–γ2γ1′,γ2′,ν–γ1−γ2=x−γ1Γνx−tδ−1t−γ1′F3γ1,γ1′,γ2,γ2′,ν;1−tx;1−xtm=2;β1=β2=β>0;σ=tx∧σ=xtSaigoH2,22,0σγ1+ν1+1−1β,1β,γ2+ν2+1−1β,1βγ1+1−1β,1β,γ2+1−1β,1β=G2,22,0σβγ1+ν1,γ2+ν2γ1,γ2=βσβγ21–σβν1+ν2−1Γν1+ν2F21γ2+ν2−γ1,ν1;ν1+ν2;1−σβ(m=1)Erdélyi–Kober (E-K)H1,01,1σγ+ν,1βγ,1β=βσβ−1G1,01,1σβγ+νγ=βσβγ+β−11–σβν−1Γνm=1;β=1;σ=tx∧σ=xtRiemann–Liouville (R-L)H1,01,1σγ+ν,1γ,1=G1,01,1txγ+νγ=x−tν−1tγΓν


## 3. Main Results

### 3.1. New Fractional Image Formulae Containing the Extendedk-Gamma Function

Lemma 15.
Assumingb,k>0, prove that the subsequent relation for the Fox-Wright function holds true:
(38)∑n,r=0∞−1/kn+rbkrn!r!Ψ00−−kn−rξ=Ψ00−−−ekξk−bke−kξk.Proof.
Let us consider (14), then
(39)LΓb,kz;ξ=2π∑n,r=0∞−1/kn+rbkrn!r!Ψ00−−kn−rξ,(40)LΓb,kz;ξ=2πexp−ekξk−bke−kξk=Ψ00−−−ekξk−bke−kξk.
Hence, from both of the above Equations (39)–(40), the required result is proved.Remark 16.
It is to be remarked that a general result is obvious from (38) as follows:
(41)∑n,r=0∞−1/kn+rbkrn!r!Ψpqai,Aibj,Bjkn−rξ=Ψpqai,Aibj,Bj−ekξk−bke−kξk.
Similar results will also hold for Mittag-Leffler function and other related functions.Remark 17.
A similar result can be proved by using the path way transform. It can be deduced from (38) by using the following replacement on LHS:
(42)ξ⟶ln1+δ−1ξδ−1.
Henceforth,(43)PδΓb,kz;ξ=2π∑n,r,p=0∞−1/kn+rbkrkpn−rpn!r!p!ln1+δ−1ξδ−1p=2πexp−ekln1+δ−1ξ/δ−1k−bke−kln1+δ−1ξ/δ−1k=2πΨ00−−−ekln1+δ−1ξ/δ−1k−bke−kln1+δ−1ξ/δ−1k.
Or equivalently, in equation (43), 2\pi\psi should not be bold face(44)PδΓb,kz;ξ=2π∑n,r=0∞−1/kn+rbkrn!r!Ψ00−−kn−rln1+δ−1ξδ−1.Theorem 18.
The E-K fractional transform (of multiplicity m) involving the extendedk-gamma function is computed as
(45)Iβk,mγk,νkln1+δ−1sδ−1χ−1PδΓb,kz;s=2πln1+δ−1sδ−1χ−1Ψmmγi+1+χ−1βi,1βi1mγi+δi+1+χ−1βi,1βi1m−ekln1+δ−1ξ/δ−1k−bke−kln1+δ−1ξ/δ−1k−βk1+γk<p;δk≥0;k=1,⋯,m.Proof.
Let us first consider(46)Iβk,mγk,νkln1+δ−1sδ−1χ−1PδΓb,kz;sx=Iβk,mγk,δkln1+δ−1sδ−1χ−12π∑n,r,p=0∞−1/kn+rbkrkpn−rpn!r!p!ln1+δ−1sδ−1px,
then exchanging the summation and integration(47)Iβk,mγk,νkln1+δ−1sδ−1χ−1PδΓb,kz;sx=2π∑n,r,p=0∞−1/kn+rbkrkpn−rpn!r!p!Iβk,mγk,νkln1+δ−1sδ−1χ−1ln1+δ−1sδ−1px.
Next, to solve the RHS integralIβk,mγk,νkln1+δ−1s/δ−1χ−1ln1+δ−1s/δ−1p, we make substitution u=ln1+δ−1s/δ−1; then, using (27) with back substitution ln1+δ−1s/δ−1=u executes the following:
(48)Iβk,mγk,νkln1+δ−1sδ−1χ−1PδΓb,kz;sx=2π∑n,r,p=0∞−1/kn+rbkrkpn−rpn!r!p!∏i=1mΓγi+1+χ+p−1/βiΓγi+δi+1+χ+p−1/βiln1+δ−1sδ−1p+χ−1,which after modifications by using (18) leads to the following:
(49)Iβk,mγk,νkln1+δ−1sδ−1χ−1PδΓb,kz;s=2πln1+δ−1sδ−1χ−1∑n,r=0∞−1/kn+rbkrn!r!Ψmmγi+1+χ−1βi,1βi1mγi+δi+1+χ−1βi,1βi1mkn−rln1+δ−1sδ−1−βk1+γk<p;δk≥0;k=1,⋯,m.
Henceafter, by making use of Lemma15 leads to the required result.Corollary 19.
The E-K fractional transform (of multiplicity m) involving the extended gamma function is computed as(50)Iβk,mγk,νkln1+δ−1sδ−1χ−1PδΓbz;sx=2πln1+δ−1sδ−1χ−1Ψmmγi+1+χ−1βi,1βi1mγi+δi+1+χ−1βi,1βi1m−eln1+δ−1ξ/δ−1−be−ln1+δ−1ξ/δ−1.Proof.
It follows by consideringk=1 in (45).Corollary 20.
The E-K fractional transform (of multiplicity m) involving thek-gamma function is computed as
(51)Iβk,mγk,νkln1+δ−1sδ−1χ−1PδΓkz;sx=2πln1+δ−1sδ−1χ−1Ψmmγi+1+χ−1βi,1βi1mγi+δi+1+χ−1βi,1βi1m−ekln1+δ−1ξ/δ−1k.Proof.
It follows by consideringb=0 in (45).Corollary 21.
The E-K fractional transform (of multiplicity m) involving the classical gamma function is computed as(52)Iβk,mγk,νkln1+δ−1sδ−1χ−1PδΓz;sx=2πln1+δ−1sδ−1χ−1Ψmmγi+1+χ−1βi,1βi1mγi+δi+1+χ−1βi,1βi1m−eln1+δ−1ξ/δ−1.Proof.
It can be proved by consideringk=1;b=0 in (45).Corollary 22.
The E-K fractional transform of multiplicitym=3 or the Marichev–Saigo–Maeda fractional integral operator of the extended k-gamma function is given by
(53)I0+γ1,γ1′,γ2,γ2′,νln1+δ−1sδ−1χ−1PδΓb,kz;s=2πln1+δ−1sδ−1δ+χ−γ1−γ1′−1Ψ33χ,1χ+ν−γ1−γ1′−γ2,1χ+γ2′−γ1′,1χ+γ2′,1χ+ν−γ1−γ1′,1χ+ν−γ1′−γ2−ekln1+δ−1ξ/δ−1k−bke−kln1+δ−1ξ/δ−1k.Proof.
It can be proved by using the casem=3 of Table 1 along with (29) in the main result (45).Corollary 23.
The E-K fractional transform of multiplicitym=3 or the Marichev–Saigo–Maeda fractional integral operator of the extended gamma function is given by
(54)I0+γ1,γ1′,γ2,γ2′,νln1+δ−1sδ−1χ−1PδΓbz;sx=2πln1+δ−1sδ−1δ+χ−γ1−γ1′−1Ψ33χ,1χ+ν−γ1−γ1′−γ2,1χ+γ2′−γ1′,1χ+γ2′,1χ+ν−γ1−γ1′,1χ+ν−γ1′−γ2−eln1+δ−1ξ/δ−1−be−ln1+δ−1ξ/δ−1.Proof.
It follows by consideringk=1 in (53).Corollary 24.
The E-K fractional transform of multiplicitym=3 or the Marichev–Saigo–Maeda fractional integral operator of the k-gamma function is given by
(55)I0+γ1,γ1′,γ2,γ2′,νln1+δ−1sδ−1χ−1PδΓkz;sx=2πln1+δ−1sδ−1δ+χ−γ1−γ1′−1Ψ33χ,1χ+ν−γ1−γ1′−γ2,1χ+γ2′−γ1′,1χ+γ2′,1χ+ν−γ1−γ1′,1χ+ν−γ1′−γ2−ekln1+δ−1ξ/δ−1k.Proof.
It follows by consideringb=0 in (53).Corollary 25.
The E-K fractional transform of multiplicitym=3 or the Marichev–Saigo–Maeda fractional integral operator of gammafunction is given by
(56)I0+γ1,γ1′,γ2,γ2′,νln1+δ−1sδ−1χ−1PδΓz;sx=2πln1+δ−1sδ−1δ+χ−γ1−γ1′−1Ψ33χ,1χ+ν−γ1−γ1′−γ2,1χ+γ2′−γ1′,1χ+γ2′,1χ+ν−γ1−γ1′,1χ+ν−γ1′−γ2−ekln1+δ−1ξ/δ−1k.Proof.
It follows by consideringk=b=0 in (53).Corollary 26.
The subsequent new result is valid containing the Laplace transform of extendedk-gamma function:
(57)I0−γ1,γ1′,γ2,γ2′,νln1+δ−1sδ−1χ−1PδΓb,kz;s=2πln1+δ−1sδ−1δ+χ−γ1−γ1′−1Ψ331−χ−ν+γ1+γ1′,11−χ+γ1+γ2′−ν,11−χ−γ11−χ,11−χ+γ1+γ1′+γ2+γ2′−ν,11−χ+γ1−γ2−ekln1+δ−1ξ/δ−1k−bke−kln1+δ−1ξ/δ−1k.Proof.
It can be proved by using casem=3 of Table 1 along with (31) in the main result (45).Corollary 27.
The subsequent new result is valid containing the Laplace transform of extended gamma function:(58)I0−γ1,γ1′,γ2,γ2′,νln1+δ−1sδ−1χ−1PδΓbz;sx=2πln1+δ−1sδ−1δ+χ−γ1−γ1′−1Ψ331−χ−ν+γ1+γ1′,11−χ+γ1+γ2′−ν,11−χ−γ11−χ,11−χ+γ1+γ1′+γ2+γ2′−ν,11−χ+γ1−γ2−eln1+δ−1ξ/δ−1−be−ln1+δ−1ξ/δ−1.Proof.
It follows by consideringk=1 in (57).Corollary 28.
The subsequent new result is valid containing the Laplace transform ofk-gamma function:
(59)I0−γ1,γ1′,γ2,γ2′,νln1+δ−1sδ−1χ−1PδΓkz;sx=2πln1+δ−1sδ−1δ+χ−γ1−γ1′−1,Ψ331−χ−ν+γ1+γ1′,11−χ+γ1+γ2′−ν,11−χ−γ11−χ,11−χ+γ1+γ1′+γ2+γ2′−ν,11−χ+γ1−γ2−ekln1+δ−1ξ/δ−1k.Proof.
It follows by consideringb=1 in (57).Corollary 29.
The subsequent new result is valid containing the Laplace transform of gamma function:(60)I0−γ1,γ1′,γ2,γ2′,νln1+δ−1sδ−1χ−1PδΓz;sx=2πln1+δ−1sδ−1δ+χ−γ1−γ1′−1Ψ331−χ−ν+γ1+γ1′,11−χ+γ1+γ2′−ν,11−χ−γ11−χ,11−χ+γ1+γ1′+γ2+γ2′−ν,11−χ+γ1−γ2−eln1+δ−1ξ/δ−1.Proof.
It follows by consideringk=1 in (57).For the benefit of the users, further deductions from the main result (45) and in the light of the discussion presented in Section 2.1 are given as follows:Ifγ1,γ2,ν∈ℂ∧Rν>0∧Rχ>max0,Rγ1−γ2, then the E-K fractional transform of multiplicity m=2 or the left handed Saigo integral operator containing the extended k-gamma function and its specific cases are given by
(61)I0+γ1,γ2,νln1+δ−1sδ−1χ−1PδΓb,kz;sx=2πln1+δ−1sδ−1χ−γ1−1Ψ22χ,1χ+γ2−γ1,1χ−γ2,1χ+ν+γ2−ekln1+δ−1ξ/δ−1k−bke−kln1+δ−1ξ/δ−1k,I0+γ1,γ2,νln1+δ−1sδ−1χ−1PδΓbz;sx=2πln1+δ−1sδ−1χ−γ1−1Ψ22χ,1χ+γ2−γ1,1χ−γ2,1χ+ν+γ2−eln1+δ−1ξ/δ−1−be−ln1+δ−1ξ/δ−1,I0+γ1,γ2,νln1+δ−1sδ−1χ−1PδΓkz;sx=2πln1+δ−1sδ−1χ−γ1−1Ψ22χ,1χ+γ2−γ1,1χ−γ2,1χ+ν+γ2−ekln1+δ−1ξ/δ−1k,I0+γ1,γ2,νln1+δ−1sδ−1χ−1PδΓz;sx=2πln1+δ−1sδ−1χ−γ1−1Ψ22χ,1χ+γ2−γ1,1χ−γ2,1χ+ν+γ2−eln1+δ−1ξ/δ−1.Ifγ1,γ2,ν∈ℂ∧Rν>0∧Rχ<1+minRγ1,Rγ2, then the E-K fractional transform of multiplicity m=2 or the right handed Saigo integral operator containing the extended k-gamma function, and its special cases are given by
(62)I−γ1,γ2,νln1+δ−1sδ−1χ−1PδΓb,kz;sx=2πln1+δ−1sδ−1χ−γ1−1Ψ22γ1−χ+1,1γ2−χ+1,11−χ,1γ1+γ2+ν−χ+1,1−ekln1+δ−1ξ/δ−1k−bke−kln1+δ−1ξ/δ−1k,I−γ1,γ2,νln1+δ−1sδ−1χ−1PδΓbz;sx=2πln1+δ−1sδ−1χ−γ1−1Ψ22γ1−χ+1,1γ2−χ+1,11−χ,1γ1+γ2+ν−χ+1,1−eln1+δ−1ξ/δ−1−be−ln1+δ−1ξ/δ−1,I−γ1,γ2,νln1+δ−1sδ−1χ−1PδΓkz;sx=2πln1+δ−1sδ−1χ−γ1−1Ψ22γ1−χ+1,1γ2−χ+1,11−χ,1γ1+γ2+ν−χ+1,1−ekln1+δ−1ξ/δ−1k,I−γ1,γ2,νln1+δ−1sδ−1χ−1PδΓz;sx=2πln1+δ−1sδ−1χ−γ1−1Ψ22γ1−χ+1,1γ2−χ+1,11−χ,1γ1+γ2+ν−χ+1,1−eln1+δ−1ξ/δ−1.Ifγ,ν∈ℂ,Rν>0,Rν+p>−Rγ, then the E-K fractional transform of multiplicity m=1 or the left handed Erdélyi–Kober integral operator containing the extended k-gamma function, and its special cases are given by
(63)I0+γ,νln1+δ−1sδ−1χ−1PδΓb,kz;sx=2πln1+δ−1sδ−1χ−1Ψ11χ+γ,1χ+γ+ν,1−ekln1+δ−1ξ/δ−1k−bke−kln1+δ−1ξ/δ−1k,I0+γ,νln1+δ−1sδ−1χ−1PδΓbz;sx=2πln1+δ−1sδ−1χ−1Ψ22χ+γ,1χ+γ+ν,1−eln1+δ−1ξ/δ−1−be−ln1+δ−1ξ/δ−1,I0+γ,νln1+δ−1sδ−1χ−1PδΓkz;sx=2πln1+δ−1sδ−1χ−1Ψ11χ+γ,1χ+γ+ν,1−ekln1+δ−1ξ/δ−1k,I0+γ,νln1+δ−1sδ−1χ−1PδΓz;sx=2πln1+δ−1sδ−1χ−1Ψ11χ+γ,1χ+γ+ν,1−eln1+δ−1ξ/δ−1.Ifγ,ν∈ℂ;Rχ+p<1+Rγ, then the E-K fractional transform of multiplicity m=1 or the right handed Erdélyi–Kober integral operator containing the extended k-gamma function, and its special cases are given by
(64)I0−γ,νln1+δ−1sδ−1χ−1PδΓb,kz;sx=2πln1+δ−1sδ−1χ−1Ψ11γ−χ+1,−1γ+ν−χ+1,−1−ekln1+δ−1ξ/δ−1k−bke−kln1+δ−1ξ/δ−1k,I0−γ,νln1+δ−1sδ−1χ−1PδΓbz;sx=2πln1+δ−1sδ−1χ−1Ψ11γ−χ+1,−1γ+ν−χ+1,−1−eln1+δ−1ξ/δ−1−be−ln1+δ−1ξ/δ−1,I0−γ,νln1+δ−1sδ−1χ−1PδΓkz;sx=2πln1+δ−1sδ−1χ−1γ−χ+1,−1γ+ν−χ+1,−1−ekln1+δ−1ξ/δ−1k,I0−γ,νln1+δ−1sδ−1χ−1PδΓz;sx=2πln1+δ−1sδ−1χ−1Ψ11γ−χ+1,−1γ+ν−χ+1,−1−eln1+δ−1ξ/δ−1.Ifν,χ∈ℂ,Rν>0,Rχ>0, then the E-K fractional transform of multiplicity m=1 or the left handed Riemann-Liouville (R-L) integral operator containing the extended k-gamma function, and its special cases are given by
(65)I0+νln1+δ−1sδ−1χ−1PδΓb,kz;sx=2πln1+δ−1sδ−1χ+δ−1Ψ11χ,1ν+χ,1−ekln1+δ−1ξ/δ−1k−bke−kln1+δ−1ξ/δ−1k,I0+νln1+δ−1sδ−1χ−1PδΓbz;sx=2πln1+δ−1sδ−1χ+δ−1Ψ11χ,1ν+χ,1−eln1+δ−1ξ/δ−1−be−ln1+δ−1ξ/δ−1,I0+νln1+δ−1sδ−1χ−1PδΓkz;sx=2πln1+δ−1sδ−1χ+δ−1Ψ11χ,1ν+χ,1−ekln1+δ−1ξ/δ−1k,I0+νln1+δ−1sδ−1χ−1PδΓz;sx=2πln1+δ−1sδ−1χ+δ−1Ψ11χ,1ν+χ,1−eln1+δ−1ξ/δ−1.Ifδ,χ∈ℂ,0<Rδ<1−Rχ+p, then the E-K fractional transform of multiplicity m=1 or the right handed Riemann-Liouville (R-L) integral operator containing the extended k-gamma function, and its special cases are given by
(66)I−νln1+δ−1sδ−1χ−1PδΓb,kz;sx=2πln1+δ−1sδ−1χ+δ−1Ψ111−ν−χ,−11−χ,−1−ekln1+δ−1ξ/δ−1k−bke−kln1+δ−1ξ/δ−1k,I−νln1+δ−1sδ−1χ−1PδΓbz;sx=2πln1+δ−1sδ−1χ+δ−1Ψ111−ν−χ,−11−χ,−1−eln1+δ−1ξ/δ−1−be−ln1+δ−1ξ/δ−1,I−νln1+δ−1sδ−1χ−1PδΓkz;sx=2πln1+δ−1sδ−1χ+δ−1Ψ111−ν−χ,−11−χ,−1−ekln1+δ−1ξ/δ−1k,I−νln1+δ−1sδ−1χ−1PδΓz;sx=2πln1+δ−1sδ−1χ+δ−1Ψ111−ν−χ,−11−χ,−1−eln1+δ−1ξ/δ−1.It is to be remarked that the results obtained for this case i-em=1 can be expressed in terms of Mittag-Leffler function by using the relation
(67)Ψ11χ,1ν+χ,1kn−rln1+δ−1sδ−1=ΓχE1,ν+χχkn−rln1+δ−1sδ−1.


### 3.2. Generalized Fractional Derivatives Containing the Extendedk-Gamma Function

By using the methodology of Theorem18 and new representation of the extended k-gamma function, we can find the multiple fractional derivatives concerning the extended k-gamma function. Here, we obtain them directly by using the general result [46, Theorem 4] also given as
(68)Dβk,mγk1m,νkzcΨpqai,αi1pbj,βj1q;λzμ=zcΨp+mq+mai,αi1p,γk+νk+1+cβk,1βk1mbj,βj1q,γk+1+cβk,1βk1m;λzμ,Generalized fractional derivatives containing the extendedk-gamma function is obtained by applying (68) on (42):
(69)Dβk,mγk1m,νkln1+δ−1sδ−1χ−1PδΓb,kz;s=2πln1+δ−1sδ−1χ−1Ψmmγk+νk+1+cβk,1βk1mγk+1+cβk,1βk1m−ekln1+δ−1ξ/δ−1k−bke−kln1+δ−1ξ/δ−1k.Similarly, one can obtain the corresponding Marichev–Saigo–Maeda fractional derivatives [53–55] as follows:
(70)D0+γ1,γ1′,γ2,γ2′,νln1+δ−1sδ−1χ−1PδΓb,kz;s=2πln1+δ−1sδ−1χ−1Ψ33χ,1χ−γ2+γ1,1χ+γ1+γ1′+γ2′−ν,1χ−γ2,1χ−ν+γ1++γ2′,1χ−ν+γ1′+γ1,1−ekln1+δ−1ξ/δ−1k−bke−kln1+δ−1ξ/δ−1k,D−γ1,γ1′,γ2,γ2′,νln1+δ−1sδ−1χ−1PδΓb,kz;s=2πωχ−1Ψ331−χ+γ2′,11+γ2′−χ−γ2+γ1,11−χ−γ1−γ1′+ν,11−χ,11−χ−γ1′+γ2′,11−χ+ν−γ1′−γ1−γ2,1−ekln1+δ−1ξ/δ−1k−bke−kln1+δ−1ξ/δ−1k.The corresponding Saigo fractional derivatives [53–55] are given as follows:
(71)D0+γ1,γ2,νln1+δ−1sδ−1χ−1PδΓb,kz;s=2πln1+δ−1sδ−1χ−γ1−1Ψ22χ,1χ+ν+γ2+γ1,1χ+γ2,1χ+ν,1−ekln1+δ−1ξ/δ−1k−bke−kln1+δ−1ξ/δ−1k,D−γ1,γ2,νln1+δ−1sδ−1χ−1PδΓb,kz;s=2πln1+δ−1sδ−1χ−γ1−1Ψ221−χ−γ2,11−χ+ν+γ1,11−χ+ν−γ2,11−χ,1−ekln1+δ−1ξ/δ−1k−bke−kln1+δ−1ξ/δ−1k.Forβk =β in (69), the following generalized Erdélyi–Kober fractional derivative is found:
(72)Dβγ,νln1+δ−1sδ−1χ−1PδΓb,kz;s=2πln1+δ−1sδ−1χ−1Ψ11γ+ν+1+cβ,1βγ+1+cβ,1β−ekln1+δ−1ξ/δ−1k−bke−kln1+δ−1ξ/δ−1k.Similarly, one can obtain the corresponding left and right-handed E–K and R-L fractional derivatives [53–55] as follows:
(73)D0+γ,νln1+δ−1sδ−1χ−1PδΓb,kz;s=2πln1+δ−1sδ−1χ−1Ψ11γ+ν+χ,1γ+χ,1−ekln1+δ−1ξ/δ−1k−bke−kln1+δ−1ξ/δ−1k,(74)D−γ,νln1+δ−1sδ−1χ−1PδΓb,kz;s=2πln1+δ−1sδ−1χ−1Ψ111−χ+γ+ν,11−χ+γ,1−ekln1+δ−1ξ/δ−1k−bke−kln1+δ−1ξ/δ−1k(75)D0+νln1+δ−1sδ−1χ−1PδΓb,kz;s=2πln1+δ−1sδ−1χ−1−δΨ11χ,1χ−ν,1−ekln1+δ−1ξ/δ−1k−bke−kln1+δ−1ξ/δ−1k,(76)D−νln1+δ−1sδ−1χ−1PδΓb,kz;s=2πln1+δ−1sδ−1χ−1−δΨ11ν−χ+1,11−χ,1−ekln1+δ−1ξ/δ−1k−bke−kln1+δ−1ξ/δ−1k.Remark 30.
Special cases:(1)
By takingb=0 in the above results (69)-(76), the corresponding fractional derivatives of generalized gamma function can be attained(2)
By takingk=1 in the above results (69)-(76), the corresponding fractional derivatives of k-gamma function can be attained(3)
Similarly, by consideringb=0 and k=1; in the above results (69)-(76), the corresponding fractional derivatives of classical gamma function can be attainedRemark 31.
For the interest of large audience, the fractional integrals and derivatives involving the Laplace transform of the extendedk-gamma function are listed in the Appendices A and B.


### 3.3. Pathway Transforms and the Solution of Fractional Kinetic Equation Involving the Extendedk-Gamma Function

The use of non-integer operators has appeared recently in various technical [59–61] and science specialties [17–19]. For instance, the fractional kinetic equation is useful in studying gas theory, astrophysics, and aerodynamics [62–64]. A general non-integer order kinetic equation containing the extended k-gamma function was recently formulated and solved [34] using the Laplace transform. The main goal of this section is to formulate and solve this problem using the pathway transform and then validate the results using both methods.Sexana et al. [62] used subsequent kinetic equations to analyze the reaction and destruction to model changes in production rates:
(77)dΘdt=−dΘt+pΘt;Θtt∗=Θt−t∗,t∗>0Θ=Θt=changeinreactiond=dΘ=Changeindestructionp=pΘ=Changeinproduction.If we neglect the inhomogeneity and spatial fluctuation ofΘt with the concentration of species ,Θt=0=Θ0, then we can rewrite it as
(78)dΘjdt=−cjΘjt,and further to this, we ignore the subscript j and integrate (78) to get
(79)Θt−Θ0=−cI0+−1Θt,with c as a constant. Haubold and Mathai [59] developed the non-integer order kinetic equation:
(80)Θt−Θ0=−cδI0+δΘt,by using the Riemann–Liouville (R-L) fractional integral (I0+δ,δ>0). We now formulate and solve the fractional kinetic equation, as proposed by Haubold and Mathai [59] i-e for any integrable function ft:
(81)Θt−ftΘ0=−dδI0+νΘt.Now, we can formulate and solve the following fractional kinetic equation comprising of the extendedk-gamma function by using the above information.Theorem 32.
By using pathway transform, the solution of non-integer kinetic equation(82)Θt−Θ0Γb,kt=−dνI0+νΘt;b,k,d>0,ν>0
is computed as follows:(83)Θt=2πχ0t∑n,r,p=0∞−1/kn+rbkrkn−r/tpEν,−p−dνtνn!r!p!.Proof.
By applying the path way transformPδ on (82)
(84)PδΘt−Θ0PδΓb,kt=Pδ−dνI0+νΘt,
as well as by using (4) and (42) yield the following:
(85)Θs=2πχ0∑n,r,p=0∞−1/kn+rbkrkpn−rpn!r!p!δ−1ln1+δ−1sp−δ−1/ln1+δ−1sd−νΘs,Θs1+δ−1/ln1+δ−1sd−ν=2πΘ0∑n,r,p=0∞−1/kn+rbkrkpn−rpn!r!p!ln1+δ−1s/δ−1p.
After some simple calculation, one can obtain(86)Θs=2πΘ0∑n,r,p=0∞−1/kn+rbkrkpn−rpn!r!p!×ln1+δ−1s/δ−1p∑m=0∞−ln1+δ−1s/δ−1d−νmΓν.
The inverse path way transform of (86) is computed by using (3) and given as follows:
(87)Θt=2πΘ0∑n,r,p=0∞−1/kn+rbkrkpn−rpn!r!p!t−p−1×∑m=0∞−dνtνmΓνm−p;νm−p>0;ν>0.As a result, using (17) in (87) yields the desired result.Special cases:(1)
By takingb=0 in the above results (82)-(83), the corresponding non-integer kinetic equation containing Γkt (k-gamma function) as well as the solution can be obtained [34](2)
By takingk=1 in the above results (82)-(83), the corresponding non-integer kinetic equation containing Γbt (generalized gamma function) and its solution can be obtained [34](3)
Similarly, by consideringb=0 and k=1, in the above results (82)-(83), the corresponding non-integer kinetic equation containing Γt (gamma function) and its solution can be obtained [34]Remark 33.
Hence, we obtain the same solution [34] by using pathway transform. It validates that approach is consistent by using both transforms. Similarly, the results under various other related transforms like Sumudu transform [65], Natural transform [66], and Elzaki transform [67] can be obtained and validated.


### 3.4. New Integrals of Products Involving Special Functions

It is worth noting that the subsequent results containing the products of a large class of special functions are evaluated by taking (42) and (45):
(88)∫01ln1+δ−1ξδ−1χ−1exp−ekln1+δ−1ξ/δ−1k−bke−kln1+δ−1ξ/δ−1kHm,mm,0ξγi+νi+1−1βi,1βi1mγi+1−1βi,1βi1mdξ=2πln1+δ−1sδ−1χ−1Ψmmγi+1+χ−1βi,1βi1mγi+νi+1+χ−1βi,1βi1m−ekln1+δ−1ξ/δ−1k−bke−kln1+δ−1ξ/δ−1k,and fork=1;b=0 special cases can be obtained:
(89)∫01ln1+δ−1ξδ−1χ−1exp−eln1+δ−1ξ/δ−1Hm,mm,0ξγi+νi+1−1βi,1βi1mγi+1−1βi,1βi1mdξ=2πln1+δ−1ξδ−1χ−1Ψmmγi+1+χ−1βi,1βi1mγi+νi+1+χ−1βi,1βi1m−eln1+δ−1ξ/δ−1.By making use of (13) and (23) alongwith the definition of Dirac delta function, subsequent new integrals of products of special functions can be computed:
(90)∫ξϵℂΓb,kzξHm,mm,0ξγi+νi+1−1βi,1βi1mγi+1−1βi,1βi1mdξ=2π∑n,r=0∞−1/kn+rbkrn!r!∫ξϵℂδzξ+nk−krHm,mm,0ξγi+νi+1−1βi,1βi1mγi+1−1βi,1βi1mdξ=2πz−1∑n,r=0∞−1/kn+rbkrn!r!Hm,mm,0kr−nzγi+νi+1−1βi,1βi1mγi+1−1βi,1βi1m=2πzHm,mm,0−ek/zk−bke−k/zkγi+νi+1−1βi,1βi1mγi+1−1βi,1βi1m.Hence, the following special cases involving the family of gamma function can be obtained:(91)∫ξϵℂΓbzξHm,mm,0ξγi+νi+1−1βi,1βi1mγi+1−1βi,1βi1mdξ=2πzHm,mm,0−e1/z−be−1/zγi+νi+1−1βi,1βi1mγi+1−1βi,1βi1m,∫ξϵℂΓkzξHm,mm,0ξγi+νi+1−1βi,1βi1mγi+1−1βi,1βi1mdξ=2πzHm,mm,0−ek/zkγi+νi+1−1βi,1βi1mγi+1−1βi,1βi1m,∫ξϵℂΓzξHm,mm,0ξγi+νi+1−1βi,1βi1mγi+1−1βi,1βi1mdξ=2πzHm,mm,0−e1/zγi+νi+1−1βi,1βi1mγi+1−1βi,1βi1m,and further new integrals of products of special functions are computable by using the relation of Fox-H functionHm,mm,0ξ/zγk+νk+1−1/βk,1/βk1mγk+1−1/βk,1/βk1m with other special functions as mentioned for example in Equations (19)-(20) for G-function, Fox-Wright function, and Mittag-Leffler function. For example,
(92)∫ξϵℂΓb,kzξGm,mm,0ξγi+νi1mγi1mdξ=2πzGm,mm,0−ek/zk−bke−k/zkγi+νi1mγi1m,∫ξϵℂΓbzξGm,mm,0ξγi+νi1mγi1mdξ=2πzGm,mm,0−e1/z−be−1/zγi+νi1mγi1m,∫ξϵℂΓkzξGm,mm,0ξγi+νi1mγi1mdξ=2πzGm,mm,0−ek/zkγi+νi1mγi1m,∫ξϵℂΓzξGm,mm,0ξγi+νi1mγi1mdξ=2πzGm,mm,0−e1/zγi+νi1mγi1m.Similarly, ([64], Equation (32)) can be rewritten as by using (23):
(93)∫01zωρ−1expezω−1Hm,mm,0ωγk+δk+1−1βk,1βk1mγk+1−1βk,1βk1mdω=ω−1∫0ωzρ−1expez−1Hm,mm,0zωγk+δk+1−1βk,1βk1mγk+1−1βk,1βk1mdz=ωρ−1∑n=0∞Ψmmγk+1−1βk,1βk1mγk+δk+1−1βk,1βk1m−n+1eω.


## 3.1. New Fractional Image Formulae Containing the Extendedk-Gamma Function

Lemma 15.
Assumingb,k>0, prove that the subsequent relation for the Fox-Wright function holds true:
(38)∑n,r=0∞−1/kn+rbkrn!r!Ψ00−−kn−rξ=Ψ00−−−ekξk−bke−kξk.Proof.
Let us consider (14), then
(39)LΓb,kz;ξ=2π∑n,r=0∞−1/kn+rbkrn!r!Ψ00−−kn−rξ,(40)LΓb,kz;ξ=2πexp−ekξk−bke−kξk=Ψ00−−−ekξk−bke−kξk.
Hence, from both of the above Equations (39)–(40), the required result is proved.Remark 16.
It is to be remarked that a general result is obvious from (38) as follows:
(41)∑n,r=0∞−1/kn+rbkrn!r!Ψpqai,Aibj,Bjkn−rξ=Ψpqai,Aibj,Bj−ekξk−bke−kξk.
Similar results will also hold for Mittag-Leffler function and other related functions.Remark 17.
A similar result can be proved by using the path way transform. It can be deduced from (38) by using the following replacement on LHS:
(42)ξ⟶ln1+δ−1ξδ−1.
Henceforth,(43)PδΓb,kz;ξ=2π∑n,r,p=0∞−1/kn+rbkrkpn−rpn!r!p!ln1+δ−1ξδ−1p=2πexp−ekln1+δ−1ξ/δ−1k−bke−kln1+δ−1ξ/δ−1k=2πΨ00−−−ekln1+δ−1ξ/δ−1k−bke−kln1+δ−1ξ/δ−1k.
Or equivalently, in equation (43), 2\pi\psi should not be bold face(44)PδΓb,kz;ξ=2π∑n,r=0∞−1/kn+rbkrn!r!Ψ00−−kn−rln1+δ−1ξδ−1.Theorem 18.
The E-K fractional transform (of multiplicity m) involving the extendedk-gamma function is computed as
(45)Iβk,mγk,νkln1+δ−1sδ−1χ−1PδΓb,kz;s=2πln1+δ−1sδ−1χ−1Ψmmγi+1+χ−1βi,1βi1mγi+δi+1+χ−1βi,1βi1m−ekln1+δ−1ξ/δ−1k−bke−kln1+δ−1ξ/δ−1k−βk1+γk<p;δk≥0;k=1,⋯,m.Proof.
Let us first consider(46)Iβk,mγk,νkln1+δ−1sδ−1χ−1PδΓb,kz;sx=Iβk,mγk,δkln1+δ−1sδ−1χ−12π∑n,r,p=0∞−1/kn+rbkrkpn−rpn!r!p!ln1+δ−1sδ−1px,
then exchanging the summation and integration(47)Iβk,mγk,νkln1+δ−1sδ−1χ−1PδΓb,kz;sx=2π∑n,r,p=0∞−1/kn+rbkrkpn−rpn!r!p!Iβk,mγk,νkln1+δ−1sδ−1χ−1ln1+δ−1sδ−1px.
Next, to solve the RHS integralIβk,mγk,νkln1+δ−1s/δ−1χ−1ln1+δ−1s/δ−1p, we make substitution u=ln1+δ−1s/δ−1; then, using (27) with back substitution ln1+δ−1s/δ−1=u executes the following:
(48)Iβk,mγk,νkln1+δ−1sδ−1χ−1PδΓb,kz;sx=2π∑n,r,p=0∞−1/kn+rbkrkpn−rpn!r!p!∏i=1mΓγi+1+χ+p−1/βiΓγi+δi+1+χ+p−1/βiln1+δ−1sδ−1p+χ−1,which after modifications by using (18) leads to the following:
(49)Iβk,mγk,νkln1+δ−1sδ−1χ−1PδΓb,kz;s=2πln1+δ−1sδ−1χ−1∑n,r=0∞−1/kn+rbkrn!r!Ψmmγi+1+χ−1βi,1βi1mγi+δi+1+χ−1βi,1βi1mkn−rln1+δ−1sδ−1−βk1+γk<p;δk≥0;k=1,⋯,m.
Henceafter, by making use of Lemma15 leads to the required result.Corollary 19.
The E-K fractional transform (of multiplicity m) involving the extended gamma function is computed as(50)Iβk,mγk,νkln1+δ−1sδ−1χ−1PδΓbz;sx=2πln1+δ−1sδ−1χ−1Ψmmγi+1+χ−1βi,1βi1mγi+δi+1+χ−1βi,1βi1m−eln1+δ−1ξ/δ−1−be−ln1+δ−1ξ/δ−1.Proof.
It follows by consideringk=1 in (45).Corollary 20.
The E-K fractional transform (of multiplicity m) involving thek-gamma function is computed as
(51)Iβk,mγk,νkln1+δ−1sδ−1χ−1PδΓkz;sx=2πln1+δ−1sδ−1χ−1Ψmmγi+1+χ−1βi,1βi1mγi+δi+1+χ−1βi,1βi1m−ekln1+δ−1ξ/δ−1k.Proof.
It follows by consideringb=0 in (45).Corollary 21.
The E-K fractional transform (of multiplicity m) involving the classical gamma function is computed as(52)Iβk,mγk,νkln1+δ−1sδ−1χ−1PδΓz;sx=2πln1+δ−1sδ−1χ−1Ψmmγi+1+χ−1βi,1βi1mγi+δi+1+χ−1βi,1βi1m−eln1+δ−1ξ/δ−1.Proof.
It can be proved by consideringk=1;b=0 in (45).Corollary 22.
The E-K fractional transform of multiplicitym=3 or the Marichev–Saigo–Maeda fractional integral operator of the extended k-gamma function is given by
(53)I0+γ1,γ1′,γ2,γ2′,νln1+δ−1sδ−1χ−1PδΓb,kz;s=2πln1+δ−1sδ−1δ+χ−γ1−γ1′−1Ψ33χ,1χ+ν−γ1−γ1′−γ2,1χ+γ2′−γ1′,1χ+γ2′,1χ+ν−γ1−γ1′,1χ+ν−γ1′−γ2−ekln1+δ−1ξ/δ−1k−bke−kln1+δ−1ξ/δ−1k.Proof.
It can be proved by using the casem=3 of Table 1 along with (29) in the main result (45).Corollary 23.
The E-K fractional transform of multiplicitym=3 or the Marichev–Saigo–Maeda fractional integral operator of the extended gamma function is given by
(54)I0+γ1,γ1′,γ2,γ2′,νln1+δ−1sδ−1χ−1PδΓbz;sx=2πln1+δ−1sδ−1δ+χ−γ1−γ1′−1Ψ33χ,1χ+ν−γ1−γ1′−γ2,1χ+γ2′−γ1′,1χ+γ2′,1χ+ν−γ1−γ1′,1χ+ν−γ1′−γ2−eln1+δ−1ξ/δ−1−be−ln1+δ−1ξ/δ−1.Proof.
It follows by consideringk=1 in (53).Corollary 24.
The E-K fractional transform of multiplicitym=3 or the Marichev–Saigo–Maeda fractional integral operator of the k-gamma function is given by
(55)I0+γ1,γ1′,γ2,γ2′,νln1+δ−1sδ−1χ−1PδΓkz;sx=2πln1+δ−1sδ−1δ+χ−γ1−γ1′−1Ψ33χ,1χ+ν−γ1−γ1′−γ2,1χ+γ2′−γ1′,1χ+γ2′,1χ+ν−γ1−γ1′,1χ+ν−γ1′−γ2−ekln1+δ−1ξ/δ−1k.Proof.
It follows by consideringb=0 in (53).Corollary 25.
The E-K fractional transform of multiplicitym=3 or the Marichev–Saigo–Maeda fractional integral operator of gammafunction is given by
(56)I0+γ1,γ1′,γ2,γ2′,νln1+δ−1sδ−1χ−1PδΓz;sx=2πln1+δ−1sδ−1δ+χ−γ1−γ1′−1Ψ33χ,1χ+ν−γ1−γ1′−γ2,1χ+γ2′−γ1′,1χ+γ2′,1χ+ν−γ1−γ1′,1χ+ν−γ1′−γ2−ekln1+δ−1ξ/δ−1k.Proof.
It follows by consideringk=b=0 in (53).Corollary 26.
The subsequent new result is valid containing the Laplace transform of extendedk-gamma function:
(57)I0−γ1,γ1′,γ2,γ2′,νln1+δ−1sδ−1χ−1PδΓb,kz;s=2πln1+δ−1sδ−1δ+χ−γ1−γ1′−1Ψ331−χ−ν+γ1+γ1′,11−χ+γ1+γ2′−ν,11−χ−γ11−χ,11−χ+γ1+γ1′+γ2+γ2′−ν,11−χ+γ1−γ2−ekln1+δ−1ξ/δ−1k−bke−kln1+δ−1ξ/δ−1k.Proof.
It can be proved by using casem=3 of Table 1 along with (31) in the main result (45).Corollary 27.
The subsequent new result is valid containing the Laplace transform of extended gamma function:(58)I0−γ1,γ1′,γ2,γ2′,νln1+δ−1sδ−1χ−1PδΓbz;sx=2πln1+δ−1sδ−1δ+χ−γ1−γ1′−1Ψ331−χ−ν+γ1+γ1′,11−χ+γ1+γ2′−ν,11−χ−γ11−χ,11−χ+γ1+γ1′+γ2+γ2′−ν,11−χ+γ1−γ2−eln1+δ−1ξ/δ−1−be−ln1+δ−1ξ/δ−1.Proof.
It follows by consideringk=1 in (57).Corollary 28.
The subsequent new result is valid containing the Laplace transform ofk-gamma function:
(59)I0−γ1,γ1′,γ2,γ2′,νln1+δ−1sδ−1χ−1PδΓkz;sx=2πln1+δ−1sδ−1δ+χ−γ1−γ1′−1,Ψ331−χ−ν+γ1+γ1′,11−χ+γ1+γ2′−ν,11−χ−γ11−χ,11−χ+γ1+γ1′+γ2+γ2′−ν,11−χ+γ1−γ2−ekln1+δ−1ξ/δ−1k.Proof.
It follows by consideringb=1 in (57).Corollary 29.
The subsequent new result is valid containing the Laplace transform of gamma function:(60)I0−γ1,γ1′,γ2,γ2′,νln1+δ−1sδ−1χ−1PδΓz;sx=2πln1+δ−1sδ−1δ+χ−γ1−γ1′−1Ψ331−χ−ν+γ1+γ1′,11−χ+γ1+γ2′−ν,11−χ−γ11−χ,11−χ+γ1+γ1′+γ2+γ2′−ν,11−χ+γ1−γ2−eln1+δ−1ξ/δ−1.Proof.
It follows by consideringk=1 in (57).For the benefit of the users, further deductions from the main result (45) and in the light of the discussion presented in Section 2.1 are given as follows:Ifγ1,γ2,ν∈ℂ∧Rν>0∧Rχ>max0,Rγ1−γ2, then the E-K fractional transform of multiplicity m=2 or the left handed Saigo integral operator containing the extended k-gamma function and its specific cases are given by
(61)I0+γ1,γ2,νln1+δ−1sδ−1χ−1PδΓb,kz;sx=2πln1+δ−1sδ−1χ−γ1−1Ψ22χ,1χ+γ2−γ1,1χ−γ2,1χ+ν+γ2−ekln1+δ−1ξ/δ−1k−bke−kln1+δ−1ξ/δ−1k,I0+γ1,γ2,νln1+δ−1sδ−1χ−1PδΓbz;sx=2πln1+δ−1sδ−1χ−γ1−1Ψ22χ,1χ+γ2−γ1,1χ−γ2,1χ+ν+γ2−eln1+δ−1ξ/δ−1−be−ln1+δ−1ξ/δ−1,I0+γ1,γ2,νln1+δ−1sδ−1χ−1PδΓkz;sx=2πln1+δ−1sδ−1χ−γ1−1Ψ22χ,1χ+γ2−γ1,1χ−γ2,1χ+ν+γ2−ekln1+δ−1ξ/δ−1k,I0+γ1,γ2,νln1+δ−1sδ−1χ−1PδΓz;sx=2πln1+δ−1sδ−1χ−γ1−1Ψ22χ,1χ+γ2−γ1,1χ−γ2,1χ+ν+γ2−eln1+δ−1ξ/δ−1.Ifγ1,γ2,ν∈ℂ∧Rν>0∧Rχ<1+minRγ1,Rγ2, then the E-K fractional transform of multiplicity m=2 or the right handed Saigo integral operator containing the extended k-gamma function, and its special cases are given by
(62)I−γ1,γ2,νln1+δ−1sδ−1χ−1PδΓb,kz;sx=2πln1+δ−1sδ−1χ−γ1−1Ψ22γ1−χ+1,1γ2−χ+1,11−χ,1γ1+γ2+ν−χ+1,1−ekln1+δ−1ξ/δ−1k−bke−kln1+δ−1ξ/δ−1k,I−γ1,γ2,νln1+δ−1sδ−1χ−1PδΓbz;sx=2πln1+δ−1sδ−1χ−γ1−1Ψ22γ1−χ+1,1γ2−χ+1,11−χ,1γ1+γ2+ν−χ+1,1−eln1+δ−1ξ/δ−1−be−ln1+δ−1ξ/δ−1,I−γ1,γ2,νln1+δ−1sδ−1χ−1PδΓkz;sx=2πln1+δ−1sδ−1χ−γ1−1Ψ22γ1−χ+1,1γ2−χ+1,11−χ,1γ1+γ2+ν−χ+1,1−ekln1+δ−1ξ/δ−1k,I−γ1,γ2,νln1+δ−1sδ−1χ−1PδΓz;sx=2πln1+δ−1sδ−1χ−γ1−1Ψ22γ1−χ+1,1γ2−χ+1,11−χ,1γ1+γ2+ν−χ+1,1−eln1+δ−1ξ/δ−1.Ifγ,ν∈ℂ,Rν>0,Rν+p>−Rγ, then the E-K fractional transform of multiplicity m=1 or the left handed Erdélyi–Kober integral operator containing the extended k-gamma function, and its special cases are given by
(63)I0+γ,νln1+δ−1sδ−1χ−1PδΓb,kz;sx=2πln1+δ−1sδ−1χ−1Ψ11χ+γ,1χ+γ+ν,1−ekln1+δ−1ξ/δ−1k−bke−kln1+δ−1ξ/δ−1k,I0+γ,νln1+δ−1sδ−1χ−1PδΓbz;sx=2πln1+δ−1sδ−1χ−1Ψ22χ+γ,1χ+γ+ν,1−eln1+δ−1ξ/δ−1−be−ln1+δ−1ξ/δ−1,I0+γ,νln1+δ−1sδ−1χ−1PδΓkz;sx=2πln1+δ−1sδ−1χ−1Ψ11χ+γ,1χ+γ+ν,1−ekln1+δ−1ξ/δ−1k,I0+γ,νln1+δ−1sδ−1χ−1PδΓz;sx=2πln1+δ−1sδ−1χ−1Ψ11χ+γ,1χ+γ+ν,1−eln1+δ−1ξ/δ−1.Ifγ,ν∈ℂ;Rχ+p<1+Rγ, then the E-K fractional transform of multiplicity m=1 or the right handed Erdélyi–Kober integral operator containing the extended k-gamma function, and its special cases are given by
(64)I0−γ,νln1+δ−1sδ−1χ−1PδΓb,kz;sx=2πln1+δ−1sδ−1χ−1Ψ11γ−χ+1,−1γ+ν−χ+1,−1−ekln1+δ−1ξ/δ−1k−bke−kln1+δ−1ξ/δ−1k,I0−γ,νln1+δ−1sδ−1χ−1PδΓbz;sx=2πln1+δ−1sδ−1χ−1Ψ11γ−χ+1,−1γ+ν−χ+1,−1−eln1+δ−1ξ/δ−1−be−ln1+δ−1ξ/δ−1,I0−γ,νln1+δ−1sδ−1χ−1PδΓkz;sx=2πln1+δ−1sδ−1χ−1γ−χ+1,−1γ+ν−χ+1,−1−ekln1+δ−1ξ/δ−1k,I0−γ,νln1+δ−1sδ−1χ−1PδΓz;sx=2πln1+δ−1sδ−1χ−1Ψ11γ−χ+1,−1γ+ν−χ+1,−1−eln1+δ−1ξ/δ−1.Ifν,χ∈ℂ,Rν>0,Rχ>0, then the E-K fractional transform of multiplicity m=1 or the left handed Riemann-Liouville (R-L) integral operator containing the extended k-gamma function, and its special cases are given by
(65)I0+νln1+δ−1sδ−1χ−1PδΓb,kz;sx=2πln1+δ−1sδ−1χ+δ−1Ψ11χ,1ν+χ,1−ekln1+δ−1ξ/δ−1k−bke−kln1+δ−1ξ/δ−1k,I0+νln1+δ−1sδ−1χ−1PδΓbz;sx=2πln1+δ−1sδ−1χ+δ−1Ψ11χ,1ν+χ,1−eln1+δ−1ξ/δ−1−be−ln1+δ−1ξ/δ−1,I0+νln1+δ−1sδ−1χ−1PδΓkz;sx=2πln1+δ−1sδ−1χ+δ−1Ψ11χ,1ν+χ,1−ekln1+δ−1ξ/δ−1k,I0+νln1+δ−1sδ−1χ−1PδΓz;sx=2πln1+δ−1sδ−1χ+δ−1Ψ11χ,1ν+χ,1−eln1+δ−1ξ/δ−1.Ifδ,χ∈ℂ,0<Rδ<1−Rχ+p, then the E-K fractional transform of multiplicity m=1 or the right handed Riemann-Liouville (R-L) integral operator containing the extended k-gamma function, and its special cases are given by
(66)I−νln1+δ−1sδ−1χ−1PδΓb,kz;sx=2πln1+δ−1sδ−1χ+δ−1Ψ111−ν−χ,−11−χ,−1−ekln1+δ−1ξ/δ−1k−bke−kln1+δ−1ξ/δ−1k,I−νln1+δ−1sδ−1χ−1PδΓbz;sx=2πln1+δ−1sδ−1χ+δ−1Ψ111−ν−χ,−11−χ,−1−eln1+δ−1ξ/δ−1−be−ln1+δ−1ξ/δ−1,I−νln1+δ−1sδ−1χ−1PδΓkz;sx=2πln1+δ−1sδ−1χ+δ−1Ψ111−ν−χ,−11−χ,−1−ekln1+δ−1ξ/δ−1k,I−νln1+δ−1sδ−1χ−1PδΓz;sx=2πln1+δ−1sδ−1χ+δ−1Ψ111−ν−χ,−11−χ,−1−eln1+δ−1ξ/δ−1.It is to be remarked that the results obtained for this case i-em=1 can be expressed in terms of Mittag-Leffler function by using the relation
(67)Ψ11χ,1ν+χ,1kn−rln1+δ−1sδ−1=ΓχE1,ν+χχkn−rln1+δ−1sδ−1.


## 3.2. Generalized Fractional Derivatives Containing the Extendedk-Gamma Function

By using the methodology of Theorem18 and new representation of the extended k-gamma function, we can find the multiple fractional derivatives concerning the extended k-gamma function. Here, we obtain them directly by using the general result [46, Theorem 4] also given as
(68)Dβk,mγk1m,νkzcΨpqai,αi1pbj,βj1q;λzμ=zcΨp+mq+mai,αi1p,γk+νk+1+cβk,1βk1mbj,βj1q,γk+1+cβk,1βk1m;λzμ,Generalized fractional derivatives containing the extendedk-gamma function is obtained by applying (68) on (42):
(69)Dβk,mγk1m,νkln1+δ−1sδ−1χ−1PδΓb,kz;s=2πln1+δ−1sδ−1χ−1Ψmmγk+νk+1+cβk,1βk1mγk+1+cβk,1βk1m−ekln1+δ−1ξ/δ−1k−bke−kln1+δ−1ξ/δ−1k.Similarly, one can obtain the corresponding Marichev–Saigo–Maeda fractional derivatives [53–55] as follows:
(70)D0+γ1,γ1′,γ2,γ2′,νln1+δ−1sδ−1χ−1PδΓb,kz;s=2πln1+δ−1sδ−1χ−1Ψ33χ,1χ−γ2+γ1,1χ+γ1+γ1′+γ2′−ν,1χ−γ2,1χ−ν+γ1++γ2′,1χ−ν+γ1′+γ1,1−ekln1+δ−1ξ/δ−1k−bke−kln1+δ−1ξ/δ−1k,D−γ1,γ1′,γ2,γ2′,νln1+δ−1sδ−1χ−1PδΓb,kz;s=2πωχ−1Ψ331−χ+γ2′,11+γ2′−χ−γ2+γ1,11−χ−γ1−γ1′+ν,11−χ,11−χ−γ1′+γ2′,11−χ+ν−γ1′−γ1−γ2,1−ekln1+δ−1ξ/δ−1k−bke−kln1+δ−1ξ/δ−1k.The corresponding Saigo fractional derivatives [53–55] are given as follows:
(71)D0+γ1,γ2,νln1+δ−1sδ−1χ−1PδΓb,kz;s=2πln1+δ−1sδ−1χ−γ1−1Ψ22χ,1χ+ν+γ2+γ1,1χ+γ2,1χ+ν,1−ekln1+δ−1ξ/δ−1k−bke−kln1+δ−1ξ/δ−1k,D−γ1,γ2,νln1+δ−1sδ−1χ−1PδΓb,kz;s=2πln1+δ−1sδ−1χ−γ1−1Ψ221−χ−γ2,11−χ+ν+γ1,11−χ+ν−γ2,11−χ,1−ekln1+δ−1ξ/δ−1k−bke−kln1+δ−1ξ/δ−1k.Forβk =β in (69), the following generalized Erdélyi–Kober fractional derivative is found:
(72)Dβγ,νln1+δ−1sδ−1χ−1PδΓb,kz;s=2πln1+δ−1sδ−1χ−1Ψ11γ+ν+1+cβ,1βγ+1+cβ,1β−ekln1+δ−1ξ/δ−1k−bke−kln1+δ−1ξ/δ−1k.Similarly, one can obtain the corresponding left and right-handed E–K and R-L fractional derivatives [53–55] as follows:
(73)D0+γ,νln1+δ−1sδ−1χ−1PδΓb,kz;s=2πln1+δ−1sδ−1χ−1Ψ11γ+ν+χ,1γ+χ,1−ekln1+δ−1ξ/δ−1k−bke−kln1+δ−1ξ/δ−1k,(74)D−γ,νln1+δ−1sδ−1χ−1PδΓb,kz;s=2πln1+δ−1sδ−1χ−1Ψ111−χ+γ+ν,11−χ+γ,1−ekln1+δ−1ξ/δ−1k−bke−kln1+δ−1ξ/δ−1k(75)D0+νln1+δ−1sδ−1χ−1PδΓb,kz;s=2πln1+δ−1sδ−1χ−1−δΨ11χ,1χ−ν,1−ekln1+δ−1ξ/δ−1k−bke−kln1+δ−1ξ/δ−1k,(76)D−νln1+δ−1sδ−1χ−1PδΓb,kz;s=2πln1+δ−1sδ−1χ−1−δΨ11ν−χ+1,11−χ,1−ekln1+δ−1ξ/δ−1k−bke−kln1+δ−1ξ/δ−1k.Remark 30.
Special cases:(1)
By takingb=0 in the above results (69)-(76), the corresponding fractional derivatives of generalized gamma function can be attained(2)
By takingk=1 in the above results (69)-(76), the corresponding fractional derivatives of k-gamma function can be attained(3)
Similarly, by consideringb=0 and k=1; in the above results (69)-(76), the corresponding fractional derivatives of classical gamma function can be attainedRemark 31.
For the interest of large audience, the fractional integrals and derivatives involving the Laplace transform of the extendedk-gamma function are listed in the Appendices A and B.


## 3.3. Pathway Transforms and the Solution of Fractional Kinetic Equation Involving the Extendedk-Gamma Function

The use of non-integer operators has appeared recently in various technical [59–61] and science specialties [17–19]. For instance, the fractional kinetic equation is useful in studying gas theory, astrophysics, and aerodynamics [62–64]. A general non-integer order kinetic equation containing the extended k-gamma function was recently formulated and solved [34] using the Laplace transform. The main goal of this section is to formulate and solve this problem using the pathway transform and then validate the results using both methods.Sexana et al. [62] used subsequent kinetic equations to analyze the reaction and destruction to model changes in production rates:
(77)dΘdt=−dΘt+pΘt;Θtt∗=Θt−t∗,t∗>0Θ=Θt=changeinreactiond=dΘ=Changeindestructionp=pΘ=Changeinproduction.If we neglect the inhomogeneity and spatial fluctuation ofΘt with the concentration of species ,Θt=0=Θ0, then we can rewrite it as
(78)dΘjdt=−cjΘjt,and further to this, we ignore the subscript j and integrate (78) to get
(79)Θt−Θ0=−cI0+−1Θt,with c as a constant. Haubold and Mathai [59] developed the non-integer order kinetic equation:
(80)Θt−Θ0=−cδI0+δΘt,by using the Riemann–Liouville (R-L) fractional integral (I0+δ,δ>0). We now formulate and solve the fractional kinetic equation, as proposed by Haubold and Mathai [59] i-e for any integrable function ft:
(81)Θt−ftΘ0=−dδI0+νΘt.Now, we can formulate and solve the following fractional kinetic equation comprising of the extendedk-gamma function by using the above information.Theorem 32.
By using pathway transform, the solution of non-integer kinetic equation(82)Θt−Θ0Γb,kt=−dνI0+νΘt;b,k,d>0,ν>0
is computed as follows:(83)Θt=2πχ0t∑n,r,p=0∞−1/kn+rbkrkn−r/tpEν,−p−dνtνn!r!p!.Proof.
By applying the path way transformPδ on (82)
(84)PδΘt−Θ0PδΓb,kt=Pδ−dνI0+νΘt,
as well as by using (4) and (42) yield the following:
(85)Θs=2πχ0∑n,r,p=0∞−1/kn+rbkrkpn−rpn!r!p!δ−1ln1+δ−1sp−δ−1/ln1+δ−1sd−νΘs,Θs1+δ−1/ln1+δ−1sd−ν=2πΘ0∑n,r,p=0∞−1/kn+rbkrkpn−rpn!r!p!ln1+δ−1s/δ−1p.
After some simple calculation, one can obtain(86)Θs=2πΘ0∑n,r,p=0∞−1/kn+rbkrkpn−rpn!r!p!×ln1+δ−1s/δ−1p∑m=0∞−ln1+δ−1s/δ−1d−νmΓν.
The inverse path way transform of (86) is computed by using (3) and given as follows:
(87)Θt=2πΘ0∑n,r,p=0∞−1/kn+rbkrkpn−rpn!r!p!t−p−1×∑m=0∞−dνtνmΓνm−p;νm−p>0;ν>0.As a result, using (17) in (87) yields the desired result.Special cases:(1)
By takingb=0 in the above results (82)-(83), the corresponding non-integer kinetic equation containing Γkt (k-gamma function) as well as the solution can be obtained [34](2)
By takingk=1 in the above results (82)-(83), the corresponding non-integer kinetic equation containing Γbt (generalized gamma function) and its solution can be obtained [34](3)
Similarly, by consideringb=0 and k=1, in the above results (82)-(83), the corresponding non-integer kinetic equation containing Γt (gamma function) and its solution can be obtained [34]Remark 33.
Hence, we obtain the same solution [34] by using pathway transform. It validates that approach is consistent by using both transforms. Similarly, the results under various other related transforms like Sumudu transform [65], Natural transform [66], and Elzaki transform [67] can be obtained and validated.


## 3.4. New Integrals of Products Involving Special Functions

It is worth noting that the subsequent results containing the products of a large class of special functions are evaluated by taking (42) and (45):
(88)∫01ln1+δ−1ξδ−1χ−1exp−ekln1+δ−1ξ/δ−1k−bke−kln1+δ−1ξ/δ−1kHm,mm,0ξγi+νi+1−1βi,1βi1mγi+1−1βi,1βi1mdξ=2πln1+δ−1sδ−1χ−1Ψmmγi+1+χ−1βi,1βi1mγi+νi+1+χ−1βi,1βi1m−ekln1+δ−1ξ/δ−1k−bke−kln1+δ−1ξ/δ−1k,and fork=1;b=0 special cases can be obtained:
(89)∫01ln1+δ−1ξδ−1χ−1exp−eln1+δ−1ξ/δ−1Hm,mm,0ξγi+νi+1−1βi,1βi1mγi+1−1βi,1βi1mdξ=2πln1+δ−1ξδ−1χ−1Ψmmγi+1+χ−1βi,1βi1mγi+νi+1+χ−1βi,1βi1m−eln1+δ−1ξ/δ−1.By making use of (13) and (23) alongwith the definition of Dirac delta function, subsequent new integrals of products of special functions can be computed:
(90)∫ξϵℂΓb,kzξHm,mm,0ξγi+νi+1−1βi,1βi1mγi+1−1βi,1βi1mdξ=2π∑n,r=0∞−1/kn+rbkrn!r!∫ξϵℂδzξ+nk−krHm,mm,0ξγi+νi+1−1βi,1βi1mγi+1−1βi,1βi1mdξ=2πz−1∑n,r=0∞−1/kn+rbkrn!r!Hm,mm,0kr−nzγi+νi+1−1βi,1βi1mγi+1−1βi,1βi1m=2πzHm,mm,0−ek/zk−bke−k/zkγi+νi+1−1βi,1βi1mγi+1−1βi,1βi1m.Hence, the following special cases involving the family of gamma function can be obtained:(91)∫ξϵℂΓbzξHm,mm,0ξγi+νi+1−1βi,1βi1mγi+1−1βi,1βi1mdξ=2πzHm,mm,0−e1/z−be−1/zγi+νi+1−1βi,1βi1mγi+1−1βi,1βi1m,∫ξϵℂΓkzξHm,mm,0ξγi+νi+1−1βi,1βi1mγi+1−1βi,1βi1mdξ=2πzHm,mm,0−ek/zkγi+νi+1−1βi,1βi1mγi+1−1βi,1βi1m,∫ξϵℂΓzξHm,mm,0ξγi+νi+1−1βi,1βi1mγi+1−1βi,1βi1mdξ=2πzHm,mm,0−e1/zγi+νi+1−1βi,1βi1mγi+1−1βi,1βi1m,and further new integrals of products of special functions are computable by using the relation of Fox-H functionHm,mm,0ξ/zγk+νk+1−1/βk,1/βk1mγk+1−1/βk,1/βk1m with other special functions as mentioned for example in Equations (19)-(20) for G-function, Fox-Wright function, and Mittag-Leffler function. For example,
(92)∫ξϵℂΓb,kzξGm,mm,0ξγi+νi1mγi1mdξ=2πzGm,mm,0−ek/zk−bke−k/zkγi+νi1mγi1m,∫ξϵℂΓbzξGm,mm,0ξγi+νi1mγi1mdξ=2πzGm,mm,0−e1/z−be−1/zγi+νi1mγi1m,∫ξϵℂΓkzξGm,mm,0ξγi+νi1mγi1mdξ=2πzGm,mm,0−ek/zkγi+νi1mγi1m,∫ξϵℂΓzξGm,mm,0ξγi+νi1mγi1mdξ=2πzGm,mm,0−e1/zγi+νi1mγi1m.Similarly, ([64], Equation (32)) can be rewritten as by using (23):
(93)∫01zωρ−1expezω−1Hm,mm,0ωγk+δk+1−1βk,1βk1mγk+1−1βk,1βk1mdω=ω−1∫0ωzρ−1expez−1Hm,mm,0zωγk+δk+1−1βk,1βk1mγk+1−1βk,1βk1mdz=ωρ−1∑n=0∞Ψmmγk+1−1βk,1βk1mγk+δk+1−1βk,1βk1m−n+1eω.


## 4. Conclusion

The novel fractional transitions of the extendedk-gamma function are procured using the generalized fractional calculus’ multiple E-K operators. As a result, specific new images are acquired as special cases for the numerous different other popular fractional transforms. It is only possible because recent research [34] investigates the Laplace transform of various extensions of the gamma function. Furthermore, the pathway transform is used to solve the fractional kinetic equation encompassing these extensions. As corollaries, specific cases involving the gamma function family are discussed. The findings are supported by the previously obtained solution of the fractional kinetic equation involving such functions, which was solved using the Laplace transform. A newly discovered representation of the generalizations of the gamma function and their Laplace transform played an important role. Novel identities containing the Fox-Wright function were found to be extremely useful in simplifying the results. Consequently, numerous results, including [46, Lemmas 7–15, Theorems 2-4], are applicable to the Laplace transform of the gamma function, and the key result (45) along with important cases are completely provable using the known theory and techniques. Given the known representations of the gamma function (as well as its extensions), it is clear that this theory and techniques cannot be applied. As a result, it is concluded that the results of this article are only possible because of a new representation [34] of as a series of complex delta functions, and this research is significant in expanding the applicability of the gamma function (as well as its extensions) beyond its original domain. By means of this novel definition of the k-gamma functions, we can discover more integrals in a simple way. For example, considering the domain 0<Rz<1, and for k=2, we have corrected an integral of Gaussian function [35, Section 4]:
(94)Γ2z,ζz=2π∑n=0∞−1/2nn!δz+2n,ζz=2π∑n=0∞−1/2nn!ζ−2n=−π.It can be concluded that this research is substantial to enhance the application of extendedk-gamma beyond its original domain.



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*Source: 9698299-2022-07-22.xml*