# Investigation of Extendedk-Hypergeometric Functions and Associated Fractional Integrals

Authors: Mohamed Abdalla; Muajebah Hidan; Salah Mahmoud Boulaaras; Bahri-Belkacem Cherif
Journal: Mathematical Problems in Engineering
 (2021)

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## Abstract

Hypergeometric functions have many applications in various areas of mathematical analysis, probability theory, physics, and engineering. Very recently, Hidan et al. (Math. Probl. Eng., ID 5535962, 2021) introduced the (p, k)-extended hypergeometric functions and their various applications. In this line of research, we present an expansion of the k-Gauss hypergeometric functions and investigate its several properties, including, its convergence properties, derivative formulas, integral representations, contiguous function relations, differential equations, and fractional integral operators. Furthermore, the current results contain several of the familiar special functions as particular cases, and this extension may enrich the theory of special functions.


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## Body

## 1. Introduction

Special functions are important tools in solving certain problems arising from many different research areas in mathematical physics, astronomy, chemistry, applied statistics, and engineering (see, e.g., [1–3]). Hypergeometric functions are among most important special functions mainly because they have a lot of applications in a variety of research branches such as (for example) quantum mechanics, electromagnetic field theory, probability theory, analytic number theory, and data analysis (see, e.g., [1, 2, 4–6]). Also, a number of elementary functions and special polynomials are expressed in terms of hypergeometric functions. Accordingly, a number of various extensions of hypergeometric functions have been introduced and investigated.Throughout this research,ℕ:=1,2,3,… denotes the set of natural numbers, ℕ0=ℕ∪0, ℤ−:=−1,−2,−3,… denotes the set of negative integers, ℤ0−=ℤ−∪0, ℝ+ denotes the set of positive real numbers, and ℂ denotes the set of complex numbers.Traditionally, the hypergeometric function known as Gauss function is defined by(1)Wv=Ϝδ1,δ2,δ3;v=∑n=0∞δ1nδ2nδ3nvnn!,v∈ℂ,which is absolutely and uniformly convergent if v<1, divergent when v>1, and absolutely convergent when v=1, if Reδ3−δ1−δ2>0, where δ1,δ2,andδ3 are complex parameters with δ3∈ℂ\ℤ0−, and(2)δ1n=Γδ1+nΓδ1=δ1δ1+1,…,δ1+n−1,n∈ℕ,δ1∈ℂ,1,n=0;δ1∈ℂ\0,is the Pochhammer symbol (or the shifted factorial) and Γ. is gamma function. The function in (1) satisfies the following differential equation:(3)v1−vW″+δ3−δ1+δ2+1vW′−δ1δ2W=0.Nowadays, numerous investigations, for example, in recent works of Srivastava et al. [7, 8], Jana et al. [9, 10], Goswami et al. [11, 12], Fuli et al. [13], and Abdalla and Bakhet [14, 15] to introduce extensions and generalizations of the hypergeometric functions, defined by Euler-type integrals, are associated with properties and applications.In particular, Diaz and Pariguan [16] introduced the k-analogue of gamma, beta, and hypergeometric functions and proved a number of their properties. Since that period, many different results concerning the k-hypergeometric function and related functions have been considered by many researchers, for instance, Agarwal et al. [17], Mubeen et al. [18–20], Rahman et al. [21], Chinra et al. [22], Korkmaz-Duzgun and Erkus-Duman [23], Nisar et al. [24], Li and Dong [25], Yilmaz et al. [26], Hidan et al. [27], and Yilmazer and Ali [28].Motivated by some of these aforesaid studies of the k-hypergeometric functions and related functions, we introduce thep,k-extended Gauss and Kummer hypergeometric functions and their properties. Relevant connections of some of the discussed results here with those presented in earlier references are outlined.The manuscript is organized as follows. In Section2, we list some basic definitions and terminologies that are needed in the paper. In Section 3, we introduce the p,k-extended Gauss and Kummer (or confluent) hypergeometric functions and discuss their regions of convergence. In Section 4, we obtain integral and differentiation formulas of the (p, k)-extended Gauss and Kummer hypergeometric functions. In addition, contiguous function relations and differential equations connecting these functions are established in Section 5. Compositions of the k-Riemann–Liouville fractional integral operators of these functions are presented in Section 6. Finally, we point out outlook and observations in Section 7.


## 2. Preliminaries

In this section, we give some basic definitions and terminologies which are used further in this manuscript.Definition 1.
(see [16, 26]). For k∈ℝ+, the k-gamma function Γku is defined by(4)Γku=∫0∞yu−1e−yk/kdy,where u∈ℂ\kℤ−. We note that Γku⟶Γu, for k⟶1, where Γu is the classical Euler’s gamma function and um,k is the k-Pochhammer symbol given in the form(5)um,k=Γku+mkΓku=uu+k,…,u+m−1k,m∈ℕ,u∈ℂ1,m=0,k∈ℝ+,u∈ℂ0.
The relation betweenΓku and gamma function Γu follows easily that(6)Γku=ku/k−1Γuk.Definition 2.
(see [16, 26]). For u,v∈ℂ and k∈ℝ+, the k-beta function Bku,v is defined by(7)Bku,v=1k∫01yu/k−11−yv/k−1dy=ΓkuΓkvΓku+v,where Reu>0 and Rev>0.
Clearly, the casek=1 in (7) reduces to the known beta function Bu,v, and the relation between the k-beta function Bku,v and the original beta function Bu,v is(8)Bku,v=1kBuk,vk.Definition 3.
(see [16, 26, 28]) Let k∈ℝ+ and s1,s2,η∈ℂ and s3∈ℂ\ℤ0−; then, k-Gauss hypergeometric function is defined in(9)Uη=2Ϝ1ks1,s2s3;η=∑m=0∞s1m,ks2m,ks3m,k⋅ηmm!,η<1k,where s1m,k is the k-Pochhammer symbol defined in (5). Obviously, if k=1, equation (9) is reduced to (1).Proposition 1.
(see [16, 26]) For any δ∈ℂ and k∈ℝ+, the following identity holds(10)Ϝ10kδ−;η=∑n=0∞δn,kηnn!=1−kη−δ/k,η<1k.
The k-hypergeometric differential equation of second order is defined in [18, 25, 26, 28] by(11)kη1−kηU″+s3−s1+s2+kkηU′−s1s2U=0.
Particular choices of the parameterss1,s2,s3, and k in the linearly independent solutions of the differential equation (11) yield more than 24 special cases. Also, the k-hypergeometric function can be given an integral representation in the following result [20, 26]:Theorem 1.
Assume thatη,s1,s2,s3∈ℂ such that Res3>Res2>0 and k∈ℝ+; then, the integral formula of the k-hypergeometric function is given by(12)2Ϝ1ks1,s2s3;η=Γks3kΓks2Γks3−s2∫01ys2/k−11−ys3−s2/k−11−kyη−s1/kdy.
Furthermore, thek-Kummer (confluent) hypergeometric function 1Ϝ1k is defined in [24] in the form(13)1Ϝ1ks1s2;η=∑m=0∞s1m,ks2m,k.ηmm!.


## 3. Thep,k-Extended Hypergeometric Functions

In this section, we introduce and discuss thep,k-extended Gauss hypergeometric function Wp,k;ξ and p,k-extended Kummer (or confluent) hypergeometric function Yp,k;ξ as follows:(14)Wp,k;ξ=2F1p,kζ1,ζ2ζ3;ξ=∑m=0∞ζ1m,kζ2m,kζ3m,k⋅ξmpm!,(15)Yp,k;ξ=1F1p,kζ1ζ3;ξ=∑m=0∞ζ1m,kζ3m,k⋅ξmpm!,where k∈ℝ+ and ζ1,ζ2,ξ∈ℂ and ζ3∈ℂ\ℤ0−, p is a positive integer, and ζm,k is the k-Pochhammer symbol defined in (5).Remark 1.
Some important special cases ofWp,k;ξ and Yp,k;ξ for some particular choice of the parameters p and k are enumerated below:(1)
Puttingp=1, we produce the k-analogue of Gauss and Kummer hypergeometric functions which are given in (9) and (13), respectively.(2)
Settingk=1, we obtain a p-extension of the Gauss and Kummer hypergeometric functions in the following forms, respectively (see Chapter 3 in [29]):(16)Wp;ξ=2F1p,1ζ1,ζ2ζ3;ξ=∑m=0∞ζ1mζ2mζ3m⋅ξmpm!,(17)Yp,ξ=1F1p,1ζ1ζ3;ξ=∑m=0∞ζ1mζ3m⋅ξmpm!.(3)
Takingk=1 and p=1 in (11), we produce the standard Gauss hypergeometric function in (1).(4)
Whenk=1 and p=1, (12) yields the following special case (see, e.g., [1, 2]).
The following theorem shows the convergence property of series (14).Theorem 2.
For allk∈ℝ+ and p>1, the p,k-extended Gauss hypergeometric function Wp,k;ξ given by (14) is an entire function.Proof.
For this prove, we relabel and write (14) as(18)2F1p,kζ1,ζ2ζ3;ξ=∑m=0∞Umξ,where(19)Umξ=ζ1m,kζ2m,kζ3m,k⋅ξmpm!.
By using the ratio test and according to the identityζm+1,k=ζ+mkζm,k, we see that(20)limm⟶∞Um+1ξUmξ=limm⟶∞ζ1m+1,kζ2m+1,kξm+1pm+1!ζ3m+1,k⋅pm!ζ3m,kζ1m,kζ2m,kξm=limm⟶∞ζ1+mkζ2+mkpm!ξζ3+mkpm+ppm+p−1…pm+1pm!=limm⟶∞ζ1+mkζ2+mkζ3+mkpm+ppm+p−1…pm+1ξ=limm⟶∞m2ζ1/m+kζ2/m+kmp+1ζ3/m+kp+p/mp+p−1/m…p+1/mξ=0.
Thus, the power series (14) is convergent for all ξ<∞, under the hypothesis p>1,k∈ℝ+, and ζ3∈ℂ\ℤ0−. Thus, it yields our desired result.
The following result can be verified in a similar way.Theorem 3.
For allk∈ℝ+ and p>1, the p,k-extended Kummer hypergeometric function Yp,k;ξ given by (15) is an entire function.Corollary 1.
For allp>1, the power series (16) and (17) are an entire function.Remark 2.
Forp=1 in Theorems 2 and 3, we get the convergence property of the k-Gauss hypergeometric function W1,k;ξ and the k-Kummer hypergeometric function Y1,k;ξ, provided that k∈ℝ+ and ζ3∈ℂ\ℤ0− (see [16]).Remark 3.
Forp=1 in Corollary 1, we obtain the convergence property of the usual Gauss and Kummer hypergeometric series (see [1, 2]).


## 4. Integral Representations and Derivative Formulae

### 4.1. Integral Representations

In the following, we establish the following theorems in terms of thek-integral representations of the p,k-extended Gauss and Kummer hypergeometric functions.Theorem 4.
The following integral representation for2F1p,k in (14) holds true:(21)2F1p,kζ1,ζ2ζ3;ξ=Γkζ3kΓkζ2Γkζ3−ζ2∫01tζ2/k−11−tζ3−ζ2/k−11F0p,kζ1,−−;tξdt,where ζ1,ζ2∈ℂ,ζ3∈ℂ\ℤ0−,Reζ3>Reζ2>0,Reζ1>0,k∈ℝ+,andp>1.Proof.
Considering the following elementary identity involving thek-Beta function Bku,v,(22)ζ2m,kζ3m,k=Γkζ2+mkΓkζ2Γkζ3Γkζ3+mk=Γkζ3Γkζ2Γkζ3−ζ2Bkζ2+mk,ζ3−ζ2=Γkζ3Γkζ2Γkζ3−ζ21k∫01tζ2/k+m−11−tζ3−ζ2/k−1dt,Reζ3>Reζ2,in (14), and using relation (17), we get the required integral formula (21).Theorem 5.
The following integral representation for2F1p,k in (14) holds true:(23)2F1p,kζ1,ζ2ζ3;ξ=1Γkζ1∫0∞tζ1−1e−tk/k1F1p,kζ2ζ3;tξdt,where ζ1,ζ2∈ℂ,ζ3∈ℂ\ℤ0−,Reζ1>0,p>1,andk∈ℝ+whenp=1.Proof.
Inserting thek-Pochhammer symbol α1n,k from (5) in definition (14) by its integral form given by (4) and from relation (15), we thus obtain the desired result (23).
By virtue of the same theorems, we give the following theorem:Theorem 6.
The following integral representations for1F1p,k in (15) hold true:(24)1F1p,kζ2ζ3;ξ=Γkζ3kΓkζ3−ζ2∫01tζ2/k−11−tζ3−ζ2/k−10F0p,k−−;tξdt,where ζ2∈ℂ,ζ3∈ℂ\ℤ0−,Reζ3>Reζ2>0,p>1,andk∈ℝ+whenp=1,(25)1F1p,kζ2ζ3;ξ=1Γkζ2∫0∞tζ2−1e−tk/k0F1p,k−ζ3;tξdt,where ζ2∈ℂ,ζ3∈ℂ\ℤ0−,Reζ1>0,p>1,andk∈ℝ+whenp=1.Remark 4.
The substitutionp=1 in (21)–(25) leads to the integral representations of the k-analogue of Gauss and Kummer hypergeometric functions (see [20, 24, 26]).Remark 5.
The substitutionp=1 in (21)–(25) leads to the integral formulas of the k-analogue of Gauss and Kummer hypergeometric functions (see [20, 24, 26]).
Also, the special cases of (3.6)–(3.9) whenk=1 and p=1 are seen to yield the classical integral representations of the Gauss and Kummer hypergeometric functions (see, e.g., [1, 2, 6]).


### 4.2. Derivative Formulae

Theorem 7.
The following derivative formulas hold true:(26)dndξn2F1p,kζ1,ζ2ζ3;ξ=ζ1n,kζ2n,kppζ3n,k2F1p,kζ1+nk,ζ2+nkζ3+nk;ξand(27)dndξn1F1p,kζ1ζ3;ξ=ζ1n,kppζ3n,k1F1p,kζ1+nkζ3+nk;ξ,where k∈ℝ+,p>1,andn∈ℕ0.Proof.
Result (26) is obviously valid in the trivial case when n=0. For n=1, by the power series representation (14) of 2F1p,k, we see from (26) that(28)ddξ2F1p,kζ1,ζ2ζ3;ξ=∑m=1∞ζ1m,kζ2m,kζ3m,kppm−1!ξm−1.
Replacing thek-Pochhammer symbols ζ1+km,k by relation (5), we arrive at(29)ddξ2F1p,kζ1,ζ2ζ3;ξ=1pζ1ζ2ζ32F1p,kζ1+k,ζ2+kζ3+k;ξ.
Therefore, the general result (26) can now be easily derived by using the principle of mathematical induction on n∈ℕ0.
A similar procedure yields the desired representation (27).Theorem 8.
The following derivative formulas hold true:(30)dndξnξζ3−12F1p,kζ1,ζ2ζ3;ξk=ξζ3−n−1Γkζ3Γkζ3−n2F1p,kζ1,ζ2ζ3−n;ξkand(31)dndξnξζ3−11F1p,kζ1ζ3;ξ=ξζ3−n−1Γkζ3Γkζ3−n1F1p,kζ1ζ3−n;ξk,where k∈ℝ+,p>1,andn∈ℕ0.Proof.
By using series (14) in (30) and differentiating term by term under the sign of summation, we observe that(32)dndξnξζ3−12F1p,kζ1,ζ2ζ3;ξk=∑m=0∞ζ1m,kζ2m,kζ3m,kdndξnξkn+ζ3−1pm!=Γkζ3∑m=0∞ζ1m,kζ2m,kΓkζ3+mk−npm!ξkm+ζ3−n−1=ξζ3−n−1ζ3Γkζ3−nΓkζ3−n∑m=0∞ζ1m,kζ2m,kΓkζ3+mk−npm!ξkm,which, in view of series (14), yields the coveted formula (30).
Similarly, we can derive the derivative formula (31).Remark 6.
The special cases of (30) and (31) when p=1 are easily seen to reduce to the known derivative formulas of the k-Gauss and Kummer hypergeometric functions (see [18]).Remark 7.
If we takep=1 and k=1 in the abovementioned theorems, we obtain the corresponding results for the classical hypergeometric functions 2F11,1 and 1F11,1 (cf. [6]).


## 4.1. Integral Representations

In the following, we establish the following theorems in terms of thek-integral representations of the p,k-extended Gauss and Kummer hypergeometric functions.Theorem 4.
The following integral representation for2F1p,k in (14) holds true:(21)2F1p,kζ1,ζ2ζ3;ξ=Γkζ3kΓkζ2Γkζ3−ζ2∫01tζ2/k−11−tζ3−ζ2/k−11F0p,kζ1,−−;tξdt,where ζ1,ζ2∈ℂ,ζ3∈ℂ\ℤ0−,Reζ3>Reζ2>0,Reζ1>0,k∈ℝ+,andp>1.Proof.
Considering the following elementary identity involving thek-Beta function Bku,v,(22)ζ2m,kζ3m,k=Γkζ2+mkΓkζ2Γkζ3Γkζ3+mk=Γkζ3Γkζ2Γkζ3−ζ2Bkζ2+mk,ζ3−ζ2=Γkζ3Γkζ2Γkζ3−ζ21k∫01tζ2/k+m−11−tζ3−ζ2/k−1dt,Reζ3>Reζ2,in (14), and using relation (17), we get the required integral formula (21).Theorem 5.
The following integral representation for2F1p,k in (14) holds true:(23)2F1p,kζ1,ζ2ζ3;ξ=1Γkζ1∫0∞tζ1−1e−tk/k1F1p,kζ2ζ3;tξdt,where ζ1,ζ2∈ℂ,ζ3∈ℂ\ℤ0−,Reζ1>0,p>1,andk∈ℝ+whenp=1.Proof.
Inserting thek-Pochhammer symbol α1n,k from (5) in definition (14) by its integral form given by (4) and from relation (15), we thus obtain the desired result (23).
By virtue of the same theorems, we give the following theorem:Theorem 6.
The following integral representations for1F1p,k in (15) hold true:(24)1F1p,kζ2ζ3;ξ=Γkζ3kΓkζ3−ζ2∫01tζ2/k−11−tζ3−ζ2/k−10F0p,k−−;tξdt,where ζ2∈ℂ,ζ3∈ℂ\ℤ0−,Reζ3>Reζ2>0,p>1,andk∈ℝ+whenp=1,(25)1F1p,kζ2ζ3;ξ=1Γkζ2∫0∞tζ2−1e−tk/k0F1p,k−ζ3;tξdt,where ζ2∈ℂ,ζ3∈ℂ\ℤ0−,Reζ1>0,p>1,andk∈ℝ+whenp=1.Remark 4.
The substitutionp=1 in (21)–(25) leads to the integral representations of the k-analogue of Gauss and Kummer hypergeometric functions (see [20, 24, 26]).Remark 5.
The substitutionp=1 in (21)–(25) leads to the integral formulas of the k-analogue of Gauss and Kummer hypergeometric functions (see [20, 24, 26]).
Also, the special cases of (3.6)–(3.9) whenk=1 and p=1 are seen to yield the classical integral representations of the Gauss and Kummer hypergeometric functions (see, e.g., [1, 2, 6]).


## 4.2. Derivative Formulae

Theorem 7.
The following derivative formulas hold true:(26)dndξn2F1p,kζ1,ζ2ζ3;ξ=ζ1n,kζ2n,kppζ3n,k2F1p,kζ1+nk,ζ2+nkζ3+nk;ξand(27)dndξn1F1p,kζ1ζ3;ξ=ζ1n,kppζ3n,k1F1p,kζ1+nkζ3+nk;ξ,where k∈ℝ+,p>1,andn∈ℕ0.Proof.
Result (26) is obviously valid in the trivial case when n=0. For n=1, by the power series representation (14) of 2F1p,k, we see from (26) that(28)ddξ2F1p,kζ1,ζ2ζ3;ξ=∑m=1∞ζ1m,kζ2m,kζ3m,kppm−1!ξm−1.
Replacing thek-Pochhammer symbols ζ1+km,k by relation (5), we arrive at(29)ddξ2F1p,kζ1,ζ2ζ3;ξ=1pζ1ζ2ζ32F1p,kζ1+k,ζ2+kζ3+k;ξ.
Therefore, the general result (26) can now be easily derived by using the principle of mathematical induction on n∈ℕ0.
A similar procedure yields the desired representation (27).Theorem 8.
The following derivative formulas hold true:(30)dndξnξζ3−12F1p,kζ1,ζ2ζ3;ξk=ξζ3−n−1Γkζ3Γkζ3−n2F1p,kζ1,ζ2ζ3−n;ξkand(31)dndξnξζ3−11F1p,kζ1ζ3;ξ=ξζ3−n−1Γkζ3Γkζ3−n1F1p,kζ1ζ3−n;ξk,where k∈ℝ+,p>1,andn∈ℕ0.Proof.
By using series (14) in (30) and differentiating term by term under the sign of summation, we observe that(32)dndξnξζ3−12F1p,kζ1,ζ2ζ3;ξk=∑m=0∞ζ1m,kζ2m,kζ3m,kdndξnξkn+ζ3−1pm!=Γkζ3∑m=0∞ζ1m,kζ2m,kΓkζ3+mk−npm!ξkm+ζ3−n−1=ξζ3−n−1ζ3Γkζ3−nΓkζ3−n∑m=0∞ζ1m,kζ2m,kΓkζ3+mk−npm!ξkm,which, in view of series (14), yields the coveted formula (30).
Similarly, we can derive the derivative formula (31).Remark 6.
The special cases of (30) and (31) when p=1 are easily seen to reduce to the known derivative formulas of the k-Gauss and Kummer hypergeometric functions (see [18]).Remark 7.
If we takep=1 and k=1 in the abovementioned theorems, we obtain the corresponding results for the classical hypergeometric functions 2F11,1 and 1F11,1 (cf. [6]).


## 5. Contiguous Function Relations and Differential Equations

The k-analogue of theta operatorkΘ, as given in [18, 19, 25], takes the form kΘ=kξd/dξ. This operator has the particularly pleasant property that kΘξm=kmξm, which makes it handy to be used on power series. In this section, relying on Definition 1, we present some results concerning contiguous function relations and differential equations for the p,k-extended Gauss hypergeometric function 2F1p,k and p,k-extended Kummer hypergeometric function 1F1p,k.To realize that, we increase or decrease one or more of the parameters of thep,k-extended Gauss hypergeometric function:(33)W=Wp,k;ξ=2F1p,kζ1,ζ2ζ3;ξ,k∈ℝ+,p>1.By±k, then the resultant function is said to be contiguous to Wp,k;ξ. For simplicity, we use the following notations:(34)Wp,k;ζ1±=2F1p,kζ1±k,ζ2ζ3;ξ.Now, we consider(35)Wp,k;ζ1+=∑m=0∞ζ1+km,kζ2m,kζ3m,k⋅ξmpm!=∑m=0∞ζ1+mkζ1Um,kξ,k∈ℝ+,p>1,where α1α1+kn,k=α1+nkα1n,k and Um,kξ is defined in (19).Similarly, we can writeWp,k;α1− as(36)Wp,k;ζ1−=∑n=0∞ζ1−kζ1m−1kUm,k,where ζ1m−1kζ1−km,k=ζ1−kζ1m,k. Similarly, for Wp,k;ζ2±, and Wp,k;ζ3±.By the help of differential operatorkΘ=kξd/dξ, we get the following relations:(37)kΘ+ζ1Wp,k;ξ=ζ1Wp,k;ζ1+,kΘ+ζ2Wp,k;ξ=ζ2Wp,k;ζ2+,kΘ+ζ3−kWp,k;ξ=ζ1−kWp,k;ζ3−.From the above relations, we can easily obtain the following results:(38)ζ1−ζ2Wp,k;ξ=ζ1Wp,k;ζ1+−ζ2Wp,k;ζ2+,(39)ζ1−ζ2Wp,k;ξ=ζ1Wp,k;ζ1+,ζ2+−ζ2Wp,k;ζ2+,(40)ζ1−ζ3+kWp,k;ξ=ζ1Wp,k;ζ1+−ζ3−kWp,k;ζ1+2k,(41)ζ2−ζ3+kWp,k;ξ=ζ2Wp,k;ζ2+−ζ3−kWp,k;ζ2+2k.Remark 8.
Other contiguous function relations for thek-Gauss hypergeometric function may be derived from the relations in (38) to (41) and the same manner, and other results can also be obtained.Remark 9.
We can easily obtain many known results in [19, 22] by setting the parameters in our main findings. Therefore, the obtained results here extend to those results.Remark 10.
It is easy to see that, in (38) to (41), if we take k=1, we get hypergeometric contiguous function relations (see [6]).
Furthermore, the operatorkΘ=kξd/dξ, which is used in the derivation of the contiguous function relations, is also used in deriving the differential equations satisfied by Wp,k;ξ and Yp,k;ξ as follows:(42)ΘΘ−1pΘ−2p…Θ−p−1pWp,k;ξ=∑m=1∞mm−1/pm−2/p…m−p−1/pζ1m,kζ2m,kζ3m,kpm!ξm=ξpp∑m=0∞ζ1m+1,kζ2m+1,kpm!ζ3m+1,kξm.
Using the following identityan+1,k=an,ka+nk, we find that(43)ΘΘ−1pΘ−2p…Θ−p−1pWp,k;ξ=ξpp∑m=0∞ζ1+mkζ2+mkζ3+mkUm,kξ=ξpp∑m=0∞mk+ζ1+ζ2−ζ3+ζ3−ζ1ζ3−ζ2ζ3+mkUm,kξ=ξppkΘWp,k;ξ+ζ1+ζ2−ζ3Wp,k;ξ+ζ3−ζ1ζ3−ζ2ζ3Wp,k;ζ3+.
Thus, we get the following differential equation:(44)ΘΘ−1pΘ−2p…Θ−p−1pWp,k;ξ−ξppkΘWp,k;ξ+ζ1+ζ2−ζ3Wp,k;ξ+ζ3−ζ1ζ3−ζ2ζ3Wp,k;ζ3+=0.
Similarly, we can derive the following result of thep,k-extended Kummer hypergeometric function Yp,k;ξ:(45)ΘΘ−1pΘ−2p…Θ−p−1pYp,k;ξ−ξζ1−ζ3ppYp,k;ζ3+=0.
In addition, we consider(46)ΘΘ−1pΘ−2p…Θ−p−1pkΘ+ζ3−kWp,k;ξ=1pp∑m=1∞km−ζ3−kζ1m,kζ2m,kpm−p!ζ3m,kξm.
Replacingm by m+1 and according to the identity ζ1m+1,k=ζ1+mkζ1m,k, we have(47)ΘΘ−1pΘ−2p…Θ−p−1pkΘ+ζ3−kWp,k;ξ=1pp∑m=0∞km−ζ3ζ1m+1,kζ2m+1,kmp!ζ3m+1,kξm+1=ξpp∑m=0∞ζ1m,kζ1+mkζ2m,kζ2+mkmp!ζ3m,kξm=ξppζ1+mkζ2+mkWp,k;ξ.
We thus get the following differential equation:(48)ΘΘ−1pΘ−2p…Θ−p−1pkΘ+ζ3−k−ξppζ1+mkζ2+mkWp,k;ξ=0.Remark 11.
Forp=1 in (48), it obviously reduces to the usual differential equation of the k-Gauss hypergeometric function in (11) (see, [18]).
A similar procedure yields differential equation of thep,k-extended Gauss hypergeometric function Wp,k,ξ, by using the operator kΘ in (15). We thus obtain differential equation of the p,k-extended Kummer hypergeometric function Yp,k,ξ in the form(49)ΘΘ−1pΘ−2p…Θ−p−1pkΘ+ζ3−k−ξppζ1+mkYp,k;ξ=0.Remark 12.
The special cases of (48) and (49) when k=1 and p=1 are seen to yield the classical differential equations of Gauss and Kummer hypergeometric functions (see, for details, [6]).


## 6. Thek-Fractional Integral Operators

Nowadays, computations of images of thek-analogues of special functions under operators of k-fractional calculus have found significant importance and applications by many references (for instance, see, [20, 21, 30–34]).Thek-Riemann–Liouville fractional integral using k-gamma function is defined in [20, 21](50)Ikυfτx=1kΓkυ∫ϱxfτx−τυ/k−1dτ,υ,ϱ>0,k∈ℝ+.In current section, we consider compositions of thek-Riemann–Liouville fractional integral operators of the p,k-extended Gauss and Kummer hypergeometric functions.Theorem 9.
Assume thatϱ,k∈ℝ+,p∈ℕ, ζ1,ζ2∈ℂ,ζ3∈ℂ\ℤ0−,Reζ1>0,and Reζ2>0. Then, for n>0,x>ϱ, the following formula holds true:(51)1ΓknΓkζ3∫ϱxx−ρn/k−1ρ−ϱζ3/k−1F21p,kζ1,ζ2ζ3;ϱ−ρdρ=kx−ϱn+ζ3/k−1Γkn+ζ3F21p,kζ1,ζ2n+ζ3;ϱ−x.Proof.
By virtue of relations (14) and (50) and putting w=ρ−ϱ/x−ϱ, we arrive at(52)1ΓknΓkζ3∫ϱxx−ρn/k−1ρ−ϱζ3/k−12F1p,kζ1,ζ2ζ3;ϱ−ρdρ=1ΓknΓkζ3∫ϱxx−ϱ−ρ−ϱn/k−1ρ−ϱζ3/k−12F1p,kζ1,ζ2ζ3;−ρ−ϱdρ=x−ρn+ζ3/k−1ΓknΓkζ3∫011−wn/k−1wζ3/k−12F1p,kζ1,ζ2ζ3;wϱ−xdw=x−ρn+ζ3/k−1ΓknΓkζ3∫011−wn/k−1wζ3/k−1∑m=0∞ζ1m,kζ2m,kζ3m,k⋅wϱ−xmpm!dw.
Changing order of summation and integration and applying relation (7), we obtain(53)x−ρn+ζ3/k−1ΓknΓkζ3∫011−wn/k−1wζ3/k−1∑m=0∞ζ1m,kζ2m,kζ3m,k⋅wϱ−xmpm!dw=x−ρn+ζ3/k−1ΓknΓkζ3∑m=0∞ζ1m,kζ2m,kζ3m,k⋅ϱ−xmpm!∫011−wn/k−1wζ3/k+m−1dw=kx−ρn+ζ3/k−1ΓknΓkζ3∑m=0∞ζ1m,kζ2m,kζ3m,k⋅ϱ−xmpm!ΓknΓkζ3+mkΓkn+ζ3+mk.
We thus get the required formula (51).Theorem 10.
Assume thatϱ,k∈ℝ+,p∈ℕ, ζ1∈ℂ,ζ3∈ℂ\ℤ0−,and Reζ1>0. Then, for n>0,x>ϱ, the following formula holds true:(54)1ΓknΓkζ3∫ϱxx−ρn/k−1ρ−ϱζ3/k−11F1p,kζ1ζ3;ϱ−ρdρ=kx−ϱn+ζ3/k−1Γkn+ζ32F1p,kζ1n+ζ3;ϱ−x.Proof.
The proof here would run in parallel with that of Theorem9. The details are omitted.Theorem 11.
Letk∈ℝ+,p∈ℕ, ζ1,ζ2∈ℂ,ζ3∈ℂℤ0−,Reζ1>0,Reζ2>0. Then, for n>0,x>0,ϱ<∞, the following formula holds true:(55)∫0xx−ϱζ3/k−1Γkζ32F1p,kζ1,ζ2ζ3;ϱ−xΦϱdϱ=Ψx,where Φx and Ψx have the relation(56)Ik−ζ3Ikζ2Ψx=∑m=0∞k−1mζ1m,kζ2m,kpm!Ik−ζ2+mkΦx.Proof.
Assume that(57)Ykζ1,ζ2;ζ3Φx=∫0xx−ϱζ3/k−1Γkζ32F1p,kζ1,ζ2ζ3;ϱ−xΦϱdϱ=Ψx.
Then, the proof would flow along the lines of that of Theorem 2.4 in [31]. The details are omitted.Remark 13.
The present study in the above theorems is assumed to be extensions of the results in [31].


## 7. Concluding Remarks

Recently, many studies and extensions of the well-known special functions have been considered by various researchers.In this paper, we obtained a new extension of the Gauss and Kummer hypergeometric functions, so-calledp,k-extended Gauss and Kummer hypergeometric functions. Also, we gave some of their main properties, namely, the convergence properties, integral representations, differential formulas, contiguous function relations, differential equations, and fractional integral operators.We have spotted that, by settingp=1, the various outcomes presented in this article will reduce to the corresponding outcomes derived earlier in [16, 18, 19, 24, 25, 31]. Furthermore, if we let k=1, then we obtain several interesting new outcomes for the p-extended Gauss and Kummer hypergeometric functions. Finally, we have spotted that if p=1 and k=1, then we obtain some known results for the usual Gauss and Kummer hypergeometric functions defined and established in [1, 2, 6]. Additional research and application on this topic is now under preparation and will be presented in forthcoming articles.



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