# Further Results on thep,k−Analogue of Hypergeometric Functions Associated with Fractional Calculus Operators

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

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## Abstract

In a previous article, first and last researchers introduced an extension of the hypergeometric functions which is called “p,k-extended hypergeometric functions.” Motivated by this work, here, we derive several novel properties for these functions, including integral representations, derivative formula, k-Beta transform, Laplace and inverse Laplace transforms, and operators of fractional calculus. Relevant connections of some of the discussed results here with those presented in earlier references are outlined.


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## Body

## 1. Introduction

Recently, various extensions of the hypergeometric functions have been presented and investigated (see, for example, [1–7]).In particular, Diaz and Pariguan [8] introduced interesting generalizations of the gamma, beta, Pochhammer, and hypergeometric functions as follows.Definition 1.
Fork∈ℝ+, the k-gamma function Γkw is defined by(1)Γkw=∫0∞zw−1e−wk/kdz,w∈ℂ\kℤ−.
We note thatΓkw⟶Γw, for k⟶1, where Γw is the classical Euler’s gamma function and wm,k is the k-Pochhammer symbol given by(2)wm,k=Γkw+mkΓkw=ww+k…w+m−1k,m∈ℕ,w∈ℂ,1,m=0,k∈ℝ+,w∈ℂ\0.
The relation between theΓkw and the usual gamma function Γw (see, e.g., [9]) follows easily that(3)Γkw=kw/k−1Γwk,orΓτ=k1−τΓkkτ.Definition 2.
The k-beta functionBkz,w is defined by(4)Bkz,w=1k∫01uz/k−11−uw/k−1du,k∈ℝ+,minRez,Rew>0,ΓkzΓkwΓkz+w,k∈ℝ+,z,w∈ℂ\ℤ0−.
Clearly, the casek=1 in (4) reduces to the known beta function ℬz,w, and the relation between the k-beta function Bkz,w and the original beta function ℬz,w is (5)Bkz,w=1kℬzk,wk.Definition 3.
Letk∈ℝ+ and θ1,θ2,w∈ℂ and θ3∈ℂ\ℤ0−; then, k-hypergeometric function is defined in the form(6)F21kθ1,θ2θ3;w=∑s=0∞θ1s,kθ2s,kθ3s,k.wss!,w<1k,where θ1s,k is the k-Pochhammer symbol defined in (2).
Obviously, ifk=1, equation (6) is reduced to the hypergeometric function:(7)F21θ1,θ2,θ3;w=∑s=0∞θ1sθ2sθ3swss!,w<1,where(8)θ1s=Γθ1+sΓθ1=θ1θ1+1…θ1+n−1,s∈ℕ,θ1∈ℂ,1,s=0,θ1∈ℂ\0,is the usual Pochhammer symbol (or the shifted factorial) and Γ. is standard gamma function.
Since that period, many different outcomes concerning the k-analogue of hypergeometric function and related functions have been contributed by many researchers, for instance, Agarwal et al. [10], Mondal and Akel [11], Nisar et al. [12], Singh et al. [13], Li and Dong [14], Yilmaz et al. [15], Yilmazer and Ali [16], and Akdemir et al. [17].
Very recently, Abdalla and Hidan [18] introduced and investigated the following p,k-extend hypergeometric function:(9)ℌ21p,kθ1,θ2θ3;ξ=∑s=0∞θ1s,kθ2s,kθ3s,k.ξsps!,which is an entire function for p>1, where k∈ℝ+ and θ1,θ2,ξ∈ℂ and θ3∈ℂ\ℤ0−, and xs,k is the k-Pochhammer symbol defined in (2).
They studied several its properties such as the k-Euler and Laplace-type integral representations, differentiation type formulae, contiguous function relations, and differential equations (cf. [18]).Remark 1.
From important special cases ofℌ21p,k are equations (6) and (7). Furthermore, when k⟶1, we obtained the p-extend hypergeometric function in the following form (see Chapter 3 in [19]):(10)ℌ21p,1θ1,θ2θ3;ξ=∑s=0∞θ1sθ2sθ3s.ξsps!,ξ<∞,which is an entire function for p>1.
Motivated by some of these aforesaid studies of thep,k−extended hypergeometric function ℌ21p,k defined by (9), in this manuscript, we investigate certain integral representations, a derivative formula, k-Beta transform, Laplace and inverse Laplace transforms, and fractional calculus operators of the p,k-extended hypergeometric function which may be useful for carrying out further research studies to make more other developments and extensions of this field. In addition, some interesting special cases of our main results are also indicated.


## 2. Integral Representations and Derivative Formula

Starting, we establish the following theorems in terms of the k-integral representations of thep,k-extended hypergeometric functions as follows.Theorem 1.
The following integral representation forℌ21p,k in (9) holds true:(11)Γkθ3+δΓkθ3Γkδ∫0ηuθ3/k−1η−uδ/k−1ℌ21p,kθ1,θ2θ3;ξudu=kηθ3+δ/k−1ℌ21p,kθ1,θ2θ3+δ;ξη,θ1,θ2,δ,ξ∈ℂ,θ3∈ℂ\ℤ0−,Reθ1>0,Reθ2>0,Reθ3>0,Reδ>0,ηξ<1,k∈ℝ+andp∈ℕ.Proof.
Inserting series (9) in the LHS of (11) and by using relation (2), we obtain(12)ℷ=Γkθ3+δΓkθ3Γkδ∫0ηuθ3/k−1η−uδ/k−1×∑m=0∞θ1m,kθ2m,kθ3m,k.uξmpm!du=∑m=0∞θ1m,kΓkθ2+mkΓkθ3+δΓkθ3+mkΓkθ2Γkδ.uξmpm!×∫0ηuθ3/k−1η−uδ/k−1du.
Lettingu=ηv and according to the k-beta function (4), we find that(13)ℷ=∑m=0∞θ1m,kΓkθ2+mkΓkθ3+δΓkθ3+mkΓkθ2Γkδ.ηξmpm!ηθ3+δ/k−1×∫01vθ3/k+m−11−vδ/k−1dv=∑m=0∞θ1m,kΓkθ2+mkΓkθ3+δΓkθ3+mkΓkθ2Γkδ.ηξmpm!ηθ3+δ/k−1kBkθ3+mk,δ.
The above equation gives the RHS of (11). We thus obtain result (11) in Theorem 1.Theorem 2.
The following integral representation forℌ21p,k in (9) holds true:(14)∫0∞exp−14kξ2ukξθ3−1ℌ21p,kθ1,k,θ2,kθ3,2k;ξ2kdξ=πkuθ3/2Γkθ3Γθ3+k/2ℌ21p,kθ1,θ2θ3+k/2;uk,θ1,θ2,ξ∈ℂ,θ3∈ℂ\ℤ0−,Reθ1>0,Reθ2>0,Reθ3>0,ξ<1,k∈ℝ+andp∈ℕ.Proof.
For convenience, let the left-hand side of (14) be denoted by T. Applying the series expression of (9) to T, we observe that(15)T=∫0∞exp−14kξ2uk∑m=0∞θ1m,kθ2m,kθ3m,2k.ξ2km+θ3−1pm!dξ=Γkθ3Γkθ2∑m=0∞θ1m,kΓkθ2+mkΓkθ3+2mk.ξ2km+θ3−1pm!×∫0∞exp−14kξ2ukdξ.
Settingξ2=41/kuv, we obtain(16)T=Γkθ3Γkθ2∑m=0∞θ1m,kΓkθ2+mkΓkθ3+2mk.1pm!×∫0∞exp−vkk21/kuv2mk+θ3−121/k−1u1/2v−1/2dv=Γkθ3Γkθ2∑m=0∞θ1m,kΓkθ2+mkΓkθ3+2mk.2θ3+2mk/k−1uθ3+2mk/2pm!×∫0∞exp−vkkvθ3+2mk/2−1dv=Γkθ3Γkθ2∑m=0∞θ1m,kΓkθ2+mkΓkθ3+2mk.2θ3+2mk/k−1uθ3+2mk/2pm!Γkθ3+2mk2.
Making an appeal to the following duplication k-gamma formula (cf. [8]),(17)Γk2w=kπ22w/k−1ΓkwΓkw+k2.
In the above series and after a simplification, we obtain(18)T=Γkθ3Γkθ2∑m=0∞θ1m,kΓkθ2+mkpm!.2θ3+2mk/kuθ3+2mk/2×πk1Γkθ3+2mk+k/22θ3+2mk/k−1=πkuθ3/2Γkθ3Γkθ2∑m=0∞θ1m,kΓkθ2+mkpm!umk1Γkθ3+k/2+mk=πkuθ3/2Γkθ3Γkθ3+k/2ℌ21p,kθ1,θ2θ3+k/2;uk.
We thus arrive at the desired result (14) asserted by Theorem 2.Theorem 3.
The following integral representation forℌ21p,k in (9) holds true:(19)Γkθ3kΓkθ2Γkθ3−θ2∫0∞uθ2/k−11+u−θ3/kℌ21p,kθ1,θ3θ3;ξuu+1du=ℌ21p,kθ1,θ2θ3;ξ,θ1,θ2,ξ∈ℂ,θ3∈ℂ\ℤ0−,Reθ1>0,Re2θ2>Reθ1>0,Reθ3>Reθ2>0,ξ<1,k∈ℝ+andp∈ℕ.Proof.
Taking left-hand side of equation (19) by S and v=u/u+1, we have(20)S=Γkθ3Γkθ2Γkθ3−θ2∫01v1−vθ2/k−11+v1−v−θ3/k×F21p,kθ1,θ3θ3;ξv11−v2dv=Γkθ3Γkθ3−θ2∫01vθ2/k−11−vθ3−θ2/k−1F21p,kθ1,θ3θ3;ξvdv=Γkθ3Γkθ2Γkθ3−θ2∫01vθ2/k−11−vθ3−θ2/k−1∑m=0∞θ1m,kθ3m,kθ3m,k.vξmpm!dv.
Changing the order of integration and summation in (20) and by using relation (4), we obtain(21)S=Γkθ3Γkθ2Γkθ3−θ2∑m=0∞θ1m,kξmpm!∫01vθ2/k+m−11−vθ3−θ2/k−1dv=Γkθ3kΓkθ2Γkθ3−θ2∑m=0∞θ1m,kξmpm!Γkθ2+mkΓkθ3−θ2Γkθ3+mk.
We thus get the required integral formula (19).
Similarly, we can easily obtain the following result without proof.Theorem 4.
The following integral representation forℌ21p,k in (9) holds true:(22)θ3∫0ξwθ3/k−11+2kξ−wℌ21p,kθ1,θ2θ3;wdw−∫0ξwθ3/kℌ21p,kθ1,θ2θ3+k;wdw,=kξθ3/kℌ21p,kθ1,θ2θ3+k;ξ,θ1,θ2,w,ξ∈ℂ,θ3∈ℂ\ℤ0−,Reθ1>0,Reθ2>0,Reθ3>0,w<1,k∈ℝ+andp∈ℕ.Remark 2.
Atp=1 in Theorems 1–4, we obtain integral formulae of the k-analogue of hypergeometric functions defined in (6).Remark 3.
The substitutionk=1 in Theorems 1–4 leads to the integral representations of the p−extended hypergeometric functions defined in (10).Remark 4.
If we takep=1 and k=1 in the abovementioned theorems, we obtain the corresponding results for the classical hypergeometric functions (see, e.g., [20]).Theorem 5.
The following derivative formula holds true:(23)dmdξmξθ3/kℌ21p,kθ1,θ2θ3+k;ξ=ξθ3/k−mΓkθ3km−1Γkθ3−m−1kℌ21p,kθ1,θ2θ3−m−1k;ξ,θ1,θ2,ξ∈ℂ,θ3∈ℂ\ℤ0−,Reθ1>0,Reθ2>0,Reθ3>0,ξ<1,k∈ℝ+andp∈ℕ.Proof.
Result (23) is obviously valid in the trivial case when m=0. For m=1, by the power series representation (9) of ℌ21p,k, we see from (23) that(24)ddξξθ3/k∑m=0∞θ1m,kθ2m,kθ3+km,k.ξmpm!=∑m=0∞θ1m,kθ2m,kθ3+km,k.θ3+mkkξθ3/k+m−1pm!.
Replacing the k-pochhammer symbolθ3+kn,k in relation (2) by its k-gamma function and using the k-gamma function property given in [8] by Γkθ3+k=θ3Γkθ3, we arrive at(25)ddξξθ3/kℌ21p,kθ1,θ2θ3+k;ξ=Γkθ3kξθ3/k−1ℌ21p,kθ1,θ2θ3;ξ.
Therefore, the general result (23) can now be easily derived by using the principle of mathematical induction on m∈ℕ0.Remark 5.
p=1 in (23) leads naturally to the differentiation formula for F21k in (6).Remark 6.
k=1 in (23) leads naturally to the differentiation formula for ℌ21p,1 in (10).Remark 7.
Forp=1 and k=1 in (23), we obtain the corresponding result for the classical hypergeometric functions (see, e.g., [20]).


## 3. Integral Transforms

In this section, we prove three theorems, which exhibit the connection between the integral transforms such as k-Beta transform and Laplace and inverse Laplace transforms forℌ21p,k given in (8).Theorem 6.
The k-Beta transform forℌ21p,k in (8) is given in the following form:(26)Bkℌ21p,kθ1+ϑ,θ2θ3;ξη:θ1,ϑ=Bkθ1,ϑℌ21p,kθ1,θ2θ3;η,θ1,θ2,ϑ,η,ξ∈ℂ,θ3∈ℂ\ℤ0−,Reθ1>0,Reθ2>0,Reθ3>0,η<1,k∈ℝ+andp∈ℕ,where the k-Beta transform of Φξ is defined as(27)BkΦξ:δ1,δ2=1k∫01ξδ1/k−11−ξδ2/k−1Φξdξ,Reδ1>0,Reδ2>0,k∈ℝ+.Proof.
By invoking definition (27) and applying (9) to the k-Beta transform of (26), we obtain(28)1k∫01ξθ1/k−11−ξϑ/k−1ℌ21p,kθ1+ϑ,θ2θ3;ξηdξ=1k∫01ξθ1/k−11−ξϑ/k−1∑m=0∞θ1+ϑm,kθ2m,kθ3m,k.ξηmpm!.
Changing the order of integration and summation, we have(29)1k∫01wθ1/k−11−wϑ/k−1ℌ21p,kθ1+ϑ,θ2θ3;ξηdξ=∑m=0∞θ1+ϑm,kθ2m,kθ3m,k.Γkθ1+kmΓkϑΓkθ1+ϑ+kmηmpm!,=Γkθ1ΓkϑΓkθ1+ϑ∑m=0∞θ1m,kθ2m,kθ3m,k.ηmpm!,which, upon using (4) and (9), yields our desired result (26) in Theorem 6.Theorem 7.
The following Laplace transform and inverse Laplace transform formulae hold, respectively:(30)Lℌ21p,kθ1,θ2θ3;ξz:w=1wℌ31p,kθ1,θ2,kθ3;ξkw,θ1,θ2,w,ξ∈ℂ,θ3∈ℂ\ℤ0−,Reθ1>0,Reθ2>0,Rew>0,ξkw<1,k∈ℝ+andp∈ℕ,(31)L−1w−θ1ℌ21p,kθ1,θ2θ3;ξwk:w=uθ1−1Γkθ1ℌ11p,kθ2θ3;ξuk,θ1,θ2,ξ,w∈ℂ,θ3∈ℂ\ℤ0−,Reθ1>0,Reθ2>0,Reθ3>0,ξwk<1,k∈ℝ+andp∈ℕ,where the Laplace transform and is defined as (see Chapter 3 in [21]):(32)LΦu:w=Φw=∫0∞e−wuΦudu,Rew>0.
For simplicity, we can write the inverse Laplace transform as(33)Φu=L−1Φw,provided that both sides above results exist.Proof.
To prove (30), we use the power series expansion (9), and applying definition (32), we observe that(34)Lℌ21p,kθ1,θ2θ3;ξz:w=∫0∞e−wz∑m=0∞θ1m,kθ2m,kθ3m,k.ξzmpm!dz.
By change the order of integration and summation with settingwz=τk/k and then applying the k-gamma function (1) to the last integral, we obtain(35)Lℌ21p,kθ1,θ2θ3;ξz:w=∑m=0∞θ1m,kθ2m,kθ3m,k.Γkmk+kξmkmwm+1pm!,which leads to the desired result (30).
Now, in order to demonstrate (31), making use of (8) in the left-hand side of (31), we find that(36)L−1w−θ1ℌ21p,kθ1,θ2θ3;ξwk:w=∑m=0∞θ1m,kθ2m,kθ3m,k.ξmpm!L−11wkm+θ1=∑m=0∞θ1m,kθ2m,kθ3m,k.ξmpm!.uθ1+mk−1Γkθ1+km.
After a simplification, we get the required result in (31).Remark 8.
The special cases of (30) and (31) when p=1 are easily seen to reduce to the known integral transforms of the k-hypergeometric functions in (6). Also, k=1, in (30) and (31), leads to the integral transforms for ℌ21p,1 in (10). Furthermore, for p=1 and k=1, in (30) and (31), we get the corresponding results for the usual hypergeometric function in (7) (see, e.g., [20]).


## 4. Fractional Calculus Approach

The k-Riemann–Liouville fractional integral is using k-gamma function defined in [22](37)IkυΦτξ=1kΓkυ∫0ξΦτξ−τυ/k−1dτ,υ,k∈ℝ+.Also, the k-Riemann–Liouville fractional derivative of orderυ introduced in [23, 24] by(38)DkυΦξ=DIk1−υΦξ;D=ddξ.Nowadays, various studies and extensions of k-fractional calculus operators were presented by several researchers (see, for example, [22–28]).Here, we consider the k-fractional differentiation of thep,k-extend hypergeometric functions using define the extended Riemann–Liouville k-fractional derivative with the parameters δ,μ,ν∈ℂ.Definition 4 (see [27]).
Forn−1<Reμ,Reν≤n,Reδ≤1,k∈ℝ+,n∈ℕ, we have(39)Dkδ,μ,νfξ=Ikδn−νdndξnIk1−δn−μfξ.Remark 9.
Some important special cases of this definition for some particular choice of the parametersδ,μ,ν, and k are enumerated below:(1)
Puttingk=1, we produce the generalized fractional derivative is defined in [26] in the form(40)Dδ,μ,νfξ=Ikδn−νdndξnIk1−δn−μfξ.(2)
Ifν=μ and n=1, then we obtain the k-fractional Hilfer derivative is defined [25] as(41)Dkδ,μfξ=Ikδn−νddξIk1−δn−μfξ.(3)
Takingδ=0,μ=ν, and n=1 in (39), we reduce relation (38).
We need the following lemma.Lemma 1 (see [27]).
The following formula holds true:(42)Dkδ,μ,νξλ/k−1=ΓkλknΓkλ−nk+1−δn−μ+δn−νξ1−δn−μ+δn−ν+λ/k−n+1,n−1<Reμ,Reν≤n,Reδ≤1,Reξ>0,k∈ℝ+,n∈ℕ.Proof.
From (39), we have(43)Dkδ,μ,νξλ/k−1=Ikδn−νdndξnIk1−δn−μξλ/k−1.
Consider the following k-fractional integral operator:(44)Ik1−δn−μξλ/k−1=1kΓk1−δn−μ∫0ξξ−τ1−δn−μ/k−1τλ/k−1dτ,=1kΓk1−δn−μ∫0ξξ1−δn−μ/k−11−τξ1−δn−μ/k−1τλ/k−1dτ.
Substitutingτ=wξ in (44), we obtain(45)Ik1−δn−μξλ/k−1=1kΓk1−δn−μ∫01ξ1−δn−μ/k−11−w1−δn−μ/k−1wλ/k−1dw.
According to (4) into the above equation, we obtain(46)Ik1−δn−μξλ/k−1=ΓkλΓkλ+1−δn−μξ1−δn−μ/k−1.
Direct computation using differentiating equation (46) with respect to ξ gives(47)ddξIk1−δn−μξλ/k−1=ΓkλkΓkλ−k+1−δn−μξ1−δn−μ+λ/k−2.
Recursive application of this procedure finally gives(48)dndξnIk1−δn−μξλ/k−1=ΓkλknΓkλ−nk+1−δn−μξ1−δn−μ+λ/k−n+1.
Now, applying the k-Riemann fractional integral in (37) with the order δn−ν to the above equation, we arrive the desired result in (42).Theorem 8.
The following relation holds true:(49)Dkδ,μ,ντλ/k−1ℌ21p,kθ1,θ2θ3;στ=ΓkλknΓkλ−μ+n1−k+δμ−νξλ−μ+n1−k+δμ−ν/k−1×ℌ31p,kθ1,θ2,λθ3,λ−μ+n1−k+δμ−ν;σξ,θ1,θ2,τ,σ∈ℂ,θ3∈ℂ\ℤ0−,Reθ1>0,Reθ2>0,Reτ>0,στ<1,n−1<Reμ,Reν≤n,Reδ≤1,k∈ℝ+andn,p∈ℕ.Proof.
According to (39) and by invoking (9) into the left-hand side of equation (49), we find that(50)Υ=Dkδ,μ,ντλ/k−1∑m=0∞θ1m,kθ2m,kθ3m,k.στmpm!=∑m=0∞θ1m,kθ2m,kσmθ3m,kpm!Dkδ,μ,ντλ/k+m−1.
Applying Lemma1 and after simple computations lead to(51)Υ=1kn∑m=0∞θ1m,kθ2m,kσmθ3m,kpm!×Γkλ+mkΓkλ+mk−nk+1−δn−μ+δn−νξ1−δn−μ+mk+λ+δn−ν/k−n+1=Γkλkn∑m=0∞θ1m,kθ2m,kσmθ3m,kpm!×Γkλ+mkΓkλΓkλ+mk−nk+1−δn−μ+δn−νξ1−δn−μ+mk+λ+δn−ν/k−n+1=Γkλkn1Γkλ−nk+1−δn−μ+δn−νξ1−δn−μ+λ+δn−ν/k−n+1×∑m=0∞θ1m,kθ2m,kλ−nk+1−δn−μ+δn−νm,kθ3m,kσξmpm!.
We thus get the required result in (49).


## 5. Concluding Remarks

As a matter of fact, this manuscript is a continuation of the recent author’s paper [18], where we have introduced the p,k-analogues of hypergeometric functions. The researchers did not state the properties of integral transforms as well as the fractional calculus operators in the mentioned work, which are the main objective of the current work.Also, we have detected that, by settingp⟶1, the various outcomes considered in this manuscript reduce to the corresponding outcomes in [22–25, 27]. Also, for k⟶1, we get many interesting new outcomes for the p−extended hypergeometric functions. Furthermore, if we take both k⟶1 and p⟶1, then the obtained results reduce to the results analogous to the classical hypergeometric functions.More importantly, all the results reported here are expected to find some applications in control theory and to the solutions of fractional-order systems in many works (see, e. g., [29–34]).



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