# Composition Formulae for thek-Fractional Calculus Operators Associated with k-Wright Function

Authors: D. L. Suthar
Journal: Journal of Mathematics
 (2020)

---

## Abstract

In this article, thek-fractional-order integral and derivative operators including the k-hypergeometric function in the kernel are used for the k-Wright function; the results are presented for the k-Wright function. Also, some of special cases related to fractional calculus operators and k-Wright function are considered.


---

## Body

## 1. Introduction and Preliminaries

Fractional calculus was introduced in 1695, but in the last two decades researchers have been able to use it properly on the account of availability of computational resources. In many areas of application of fractional calculus, the researchers found significant applications in science and engineering. In the literature, many applications of fractional calculus are available in astrophysics, biosignal processing, fluid dynamics, nonlinear control theory, stochastic dynamical system, and so on. Also, a number of researchers [1–10] have studied in-depth level of properties, applications, and various directions of extensions of Gauss hypergeometric function of fractional integration.Recently, in a series of research publications on generalized classical fractional calculus operators, research by Mubeen and Habibullah [11] has been published on the integral part of the Riemann-Liouville version and its applications; an alternative definition for the k-Riemann–Liouville fractional derivative was introduced by Dorrego [12]. The left- and right-hand operators of Saigo k-fractional integration and differentiation associated with the k-Gauss hypergeometric function defined by Gupta and Parihar [13] (see also [14]) are as follows:(1)I0+,kτ,δ,γfx=x−τ−δ/kkΓkτ∫0xx−tτ/k−1F21,kτ+δ,k,−γ,k;τ,k;1−txftdt;ℜτ>0,k>0,(2)I−,kτ,δ,γfx=1kΓkτ∫x∞t−xτ/k−1t−τ−δ/kF21,kτ+δ,k,−γ,k;τ,k;1−xtftdt;ℜτ>0,k>0,where F21,kτ,k,δ,k;γ,k;x is the k-Gauss hypergeometric function defined by [11] for x∈ℂ,x<1,ℜγ>ℜδ>0:(3)F21,kτ,k,δ,k;γ,k;x=∑n=0∞τn,kδn,kxnγn,kn!.The corresponding fractional differential operators have their respective forms as(4)D0+,kτ,δ,γfx=ddxnI0+,k−τ+n,−δ−n,τ+γ−nfx;ℜτ>0,k>0;n=ℜτ+1D0+,kτ,δ,γfx=ddxnxτ+δ/kkΓk−τ+n∫0xx−t−τ/k+n−1×F21,k−τ−δ,k,−γ−τ+n,k;−τ+n,k;1−txftdt,(5)D−,kτ,δ,γfx=−ddxnI−,k−τ+n,−δ−n,τ+γfx;ℜτ>0,k>0;n=ℜτ+1D−,kτ,δ,γfx=−ddxn1kΓk−τ+n∫x∞t−x−τ+n/k−1tτ+δ/k×F21,k−τ−δ,k,−γ−τ,k;−τ+n,k;1−xtftdt,where x>0,τ∈ℂ,ℜτ>0,k>0, and ℜτ is the integer part of ℜτ.Remark 1.
If we setk=1 in equations (1), (2), (4), and (5), operators reduce to Saigo’s fractional integral and derivative operators stated in [5], respectively.
Now, we consider the following basic results for our study.Lemma 1 (see [13], pp. 497, Eq. 4.2).
Letτ,δ,γ,ζ∈ℂ,ℜζ>max0,ℜδ−γ; then,(6)I0+,kτ,δ,γtζ/k−1x=∑n=0∞knΓkζΓkζ−δ+γΓkζ−δΓkζ+τ+γxζ−δ/k−1.Lemma 2 (see [13], pp. 497, Eq. 4.3).
Letτ,δ,γ,ζ∈ℂ,ℜτ>0,k∈ℜ+0,∞, and ℜζ>maxℜ−δ,ℜ−γ; then,(7)I−,kτ,δ,γt−ζ/kx=∑n=0∞knΓkζ+δΓkζ+γΓkζΓkζ+τ+δ+γx−ζ−δ/k.Lemma 3 (see [13], pp. 500, Eq. 6.2).
Letτ,δ,γ,ζ∈ℂ,n=ℜτ+1,k∈ℜ+0,∞ such that ℜζ>max0,ℜ−τ−δ−γ; then,(8)D0+,kτ,δ,γtζ/k−1x=∑n=0∞knΓkζΓkζ+δ+γ+τΓkζ+γΓkζ+δ+n−nkxζ+δ+n/k−n−1.Lemma 4 (see [13], pp. 500, Eq. 6.3).
Letτ,δ,γ,ζ∈ℂ, and n=ℜτ+1,k∈ℜ+, and ℜζ>maxℜ−τ−γ,ℜδ−nk+n; then,(9)D−,kτ,δ,γt−ζ/kx=∑n=0∞knΓkζ−δ−n+nkΓkζ+τ+γΓkζΓkζ−δ+γx−ζ+δ+n/k−n.Recently, Gehlot and Prajapati [15] studied the concept of the generalized k-Wright function, which is presented in the following definition, and its connection with other special functions. It is the generalization of Mittag-Leffler function and many other special functions (see also, [16–21]). These special functions have found many important applications in solving problems of physics, biology, engineering, and applied sciences.Thek-Wright function is defined for k∈ℝ+;x,ai,bj∈ℂ, εi,ςj∈ℝεi,ςj≠0;i=1,2,…,p;j=1,2,…,q, and ai+εin,bj+ςjn∈ℂ\kz− as(10)Ψpqkx=Ψpqka1,ε1,…,ap,εp;b1,ς1,…,bq,ςq;x=Ψpqkai,εi1,p;bj,ςj1,q;x=∑n=0∞∏i=1pΓkai+εin∏j=1qΓkbj+ςjnxn/n!,with the convergence conditions described as(11)∑j=1qςjk−∑i=1pεik>−1.Remark 2.
When we putk=1 in (10), the k-Wright function reduces to Wright function which is stated in [22].
The following relation of thek-Wright function in terms of the generalized k-Mittag-Leffler function, k-Bessel function, k-hypergeometric function, and Mittag-Leffler family function is defined as follows by giving the appropriate values of the parameters:(1)
Forp=1,q=2, the generalized k-Mittag-Leffler function from Gehlot [17] is(12)Ψ12kx=Ψ12kϑ,ωk;ς,ε,ϑ,0;x=∑n=0∞ϑnω,kΓkεn+ςxnn!=Ek,ε,ςϑ,ωx,Here, Díaz and Pariguan [23] introduced the k-Pochhammer symbol and k-gamma function as follows:(13)ℑn,k=Γkℑ+nkΓkℑ,k∈ℜ,ℑ∈ℂ/0;ℑℑ+k…ℑ+n−1k,n∈ℕ,ℑ∈ℂ,and the relation with classical Euler’s gamma function is as follows:(14)Γkℑ=kℑ/k−1Γℑk,whereℑ∈ℂ,k∈ℜ, and n∈ℕ. For more information on the k-Pochhammer symbol, k-special functions, and fractional Fourier transforms, refer to Romero and Cerutti’s [24] articles.(2)
Forp=1,q=2, the generalized k-Mittag-Leffler function from Dorrego and Cerutti [16] is(15)Ψ12kx=Ψ12kϑ,k;ς,ε,ϑ,0;x=∑n=0∞ϑn,kΓkεn+ςxnn!=Ek,ε,ςϑx.(3)
Forp=1,q=3, the k-Bessel function of the first kind from Cerutti [25] is(16)Ψ13kx=Ψ13kϑ,k;ς+1,ε,ϑ,0k,k;−xk2=∑n=0∞−1nϑn,kΓkεn+ς+1x/2nn!2=Jk,ςϑ,εx.(4)
Forp=3,q=3, the k-hypergeometric function with three parameters from Mubeen et al. [26] is(17)Ψ33kx=Ψ33kϑ,k,φ,k,ς,0;ς,k,ϑ,0φ,0;x=∑n=0∞ϑn,kφn,kxnςn,kn!=F21,kϑ,k,φ,k;ς,k;x.(5)
Fork=1,p=1,q=2, the generalized Mittag-Leffler function from Shukla and Prajapati [20] is(18)Ψ121x=Ψ121ϑ,ω;ς,ε,ϑ,0;x=∑n=0∞ϑnωΓεn+ςxnn!=Eε,ςϑ,ωx.(6)
Fork=1,p=1,q=2, the generalized Mittag-Leffler function from Prabhakar [19] is(19)Ψ121x=Ψ121ϑ,1;ς,ε,ϑ,0;x=∑n=0∞ϑnΓεn+ςxnn!=Eε,ςϑx.(7)
Fork=1,p=1,andq=2, the Mittag-Leffler function from Wiman [21] is(20)Ψ111x=Ψ1111,1;ς,ε;x=∑n=0∞1Γεn+ςxnn!=Eε,ςx.(8)
Fork=1,p=1,andq=2, the Mittag-Leffler function from Mittag-Leffler [18] is(21)Ψ111x=Ψ1111,1;1,ε;x=∑n=0∞1Γεn+1xnn!=Eεx.


## 2. Saigok-Fractional Integration in terms of k-Wright Function

In this section, we present the composition formulas ofk-fractional integrals (1) and (2), involving the k-Wright function.Theorem 1.
Letτ,δ,γ,∈ℂ;k∈ℝ+, c,εi,ςj∈ℝεi,ςj≠0;i=1,2,…,p;j=1,2,…,q, and v>0 such that ℜτ>0, ℜζ>max0,ℜδ−γ,andℜζ+γ−δ>0. If condition (11) is satisfied and I0+,kτ,δ,γ be the left-sided integral operator of the generalized k-fractional integration associated with k-Wright function, then the following equation holds true:(22)I0+,kτ,δ,γtζ/k−1Ψpqkai,εi1,p;bj,ςj1,q;ctv/kx=xζ−δ/k−1Ψp+2q+2ka1,ε1,…,ap,εp,ζ,v,ζ+γ−δ,v;b1,ς1,…,bq,ςq,ζ−δ,v,ζ+τ+γ,v;kcxv/k.Proof.
We indicate the R.H.S. of equation (22) by I1, and invoking equation (10), we obtain(23)I1=I0+,kτ,δ,γtζ/k−1∑n=0∞∏i=1pΓkai+εin∏j=1qΓkbj+ςjnctv/knn!x=∑n=0∞∏i=1pΓkai+εin∏j=1qΓkbj+ςjncnn!I0+,kτ,δ,γtζ+vn/kx.
Now applying equation (6), we get(24)I1=xζ−δ/k−1∑n=0∞∏i=1pΓkai+εinΓkζ+vnΓkζ+γ−δ+vn∏j=1qΓkbj+ςjnΓkζ−δ+vnΓkζ+τ+γ+vnkcxv/knn!.
Now, interpreting definition (10) on the aforementioned equation, we arrive at the desired result (22).Theorem 2.
Letτ,δ,γ,∈ℂ;k∈ℝ+, c,εi,ςj∈ℝεi,ςj≠0;i=1,2,…,p;j=1,2,…,q, and v>0 such that ℜτ>0 and ℜζ+τ>max−ℜδ,−ℜγ, with ℜδ≠ℜγ. If condition (11) is satisfied and I−,kτ,δ,γ be the right-sided integral operator of the generalized k-fractional integration associated with k-Wright function, then the following equation holds true:(25)I−,kτ,δ,γt−τ−ζ/kΨpqkai,εi1,p;bj,ςj1,q;ct−v/kx=x−τ−ζ−δ/kΨp+2q+2ka1,ε1,…,ap,εpτ+ζ+δ,v,τ+ζ+γ,v;b1,ς1,…,bq,ςq,τ+ζ,v,2τ+ζ+δ+γ,v;kcx−v/k.Proof.
The finding is similar to that of Theorem1. So, we omit the details.Corollary 1.
By assumingk=1 in (22) and (25), the result becomes(26)I0+τ,δ,γtζ−1Ψpqai,εi1,p;bj,ςj1,q;ctvx=xζ−δ−1Ψp+2q+2a1,ε1,…,ap,εp,ζ,v,ζ+γ−δ,v;b1,ς1,…,bq,ςq,ζ−δ,v,ζ+τ+γ,v;cxv,and(27)I−τ,δ,γt−τ−ζΨpqai,εi1,p;bj,ςj1,q;ct−vx=x−τ−ζ−δΨp+2q+2a1,ε1,…,ap,εpτ+ζ+δ,v,τ+ζ+γ,v;b1,ς1,…,bq,ςq,τ+ζ,v,2τ+ζ+δ+γ,v;cx−v.


## 3. Saigok-Fractional Differentiation in terms of k-Wright Function

In this section, we present the composition formulas ofk-fractional derivatives (4) and (5), involving the k-Wright function.Theorem 3.
Letτ,δ,γ∈ℂ;k∈ℝ+, c,εi,ςj∈ℝεi,ςj≠0;i=1,2,…,p;j=1,2,…,q, and v>0 such that ℜτ>0, ℜζ>max0,ℜ−τ−δ−γ,ℜζ+γ+δ>0. If condition (11) is satisfied and D0+,kτ,δ,γ be the left-sided differential operator of the generalized k-fractional differentiation associated with k-Wright function, then the following equation holds true:(28)D0+,kτ,δ,γtζ/k−1Ψpqkai,εi1,p;bj,ςj1,q;ctv/kx=xζ+δ/k−1Ψp+2q+2ka1,ε1,…,ap,εp,ζ,v,ζ+δ+γ+τ,v;b1,ς1,…,bq,ςq,ζ+γ,v,ζ+δ,1−k+v;cxv+1/k−1.Proof.
For simplicity, letI2 denote the left side of (28). Using definition (10), we obtain(29)I2=∑n=0∞∏i=1pΓkai+εin∏j=1qΓkbj+ςjncnn!D0+,kτ,δ,γtζ+vn/k−1x.
Now, applying equation (8), we obtain(30)I2=xζ+δ/k−1∑n=0∞∏i=1pΓkai+εinΓkζ+vnΓkζ+δ+γ+τ+vn∏j=1qΓkbj+ςjnΓkζ+γ+vnΓkζ+δ+n−nk+vnn!cxv+1/k−1n.
In accordance with (10), the required result is (28). This completes the proof of Theorem 3.Theorem 4.
Letτ,δ,γ,∈ℂ;k∈ℝ+, c,εi,ςj∈ℝεi,ςj≠0;i=1,2,…,p;j=1,2,…,q, and v>0 such that ℜτ>0, ℜζ>maxℜτ+δ+n−ℜγ,ℜτ+δ−γ+n≠0, where n=ℜτ+1. If condition (11) is satisfied and D−,kτ,δ,γ be the right-sided differential operator of the generalized k-fractional differentiation associated with k-Wright function, then thefollowing equation holds true:(31)D−,kτ,δ,γtτ−ζ/kΨpqkai,εi1,p;bj,ςj1,q;ct−v/kx=xτ−ζ+δ/kΨp+2q+2ka1,ε1,…,ap,εp,ζ−τ−δ,v+k−1,ζ+γ,v;b1,ς1,…,bq,ςq,ζ−τ,v,ζ−τ−δ+γ,v;cx−v+1/k−1.Proof.
The proof is parallel to that of Theorem3. Therefore, we omit the details.Corollary 2.
By lettingk=1 in (28) and (31), the equation becomes(32)D0+τ,δ,γtζ−1Ψpqkai,εi1,p;bj,ςj1,q;ctvx=xζ+δ−1Ψp+2q+2ka1,ε1,…,ap,εp,ζ,v,ζ+δ+γ+τ,v;b1,ς1,…,bq,ςq,ζ+γ,v,ζ+δ,v;cxv,(33)D−τ,δ,γtτ−ζΨpqkai,εi1,p;bj,ςj1,q;ct−vx=xτ−ζ+δΨp+2q+2ka1,ε1,…,ap,εp,ζ−τ−δ,v,ζ+γ,v;b1,ς1,…,bq,ςq,ζ−τ,v,ζ−τ−δ+γ,v;cx−v.


## 4. Special Cases and Concluding Remarks

Being very general, the results given in (22), (25), (28), and (31) can yield a wide number of special cases by assigning some appropriate values to the parameters involved. Now, as shown in the following, we are explaining a few corollaries.Corollary 3.
If we putp=1 and q=2 in Theorems 1 and 2, then we get the following interesting results on the right known as k-Mittag-Leffler function:(34)I0+,kτ,δ,γtζ/k−1Ek,ε,ςϑ,ωctv/kx=xζ−δ/k−1Ψ34kϑ,ωk,ζ,v,ζ+γ−δ,v;ς,ε,ϑ,0,ζ−δ,v,ζ+τ+γ,v;kcxv/k,and(35)I−,kτ,δ,γt−τ−ζ/kEk,ε,ςϑ,ωct−v/kx=x−τ−ζ−δ/kΨ34kϑ,ωk,τ+ζ+δ,v,τ+ζ+γ,v;ς,ε,ϑ,0,τ+ζ,v,2τ+ζ+δ+γ,v;kcx−v/k.Corollary 4.
If we putp=1 and q=3 in Theorems 1 and 2, then we get the following interesting results on the right known as k-Bessel function of the first kind:(36)I0+,kτ,δ,γtζ/k−1Jk,ςϑ,εctv/kx=xζ−δ/k−1Ψ35kϑ,k,ζ,v,ζ+γ−δ,v;ς+1,ε,ϑ,0,ζ−δ,v,ζ+τ+γ,v,k,k;−ck2xv/k2,and(37)I−,kτ,δ,γt−τ−ζ/kJk,ςϑ,εct−v/kx=x−τ−ζ−δ/kΨ35kϑ,k,τ+ζ+δ,v,τ+ζ+γ,v;ς+1,ε,ϑ,0,τ+ζ,v,2τ+ζ+δ+γ,v,k,k;−ck2x−v/k2.Corollary 5.
If we putp=3 and q=3 in Theorems 1 and 2, then we get the following results on the right known as k-hypergeomrtric function:(38)I0+,kτ,δ,γtζ/k−1F21,kϑ,k,φ,k;ς,k;ctv/kx=xζ−δ/k−1Ψ55kϑ,k,φ,k,ς,0,ζ,v,ζ+γ−δ,v;ς,k,φ,0,ϑ,0,ζ−δ,v,ζ+τ+γ,v;kcxv/k,(39)I−,kτ,δ,γt−τ−ζ/kF21,kϑ,k,φ,k;ς,k;ct−v/kx=x−τ−ζ−δ/kΨ55kϑ,k,φ,k,ς,0,τ+ζ+δ,v,τ+ζ+γ,v;ς,k,φ,0,ϑ,0,τ+ζ,v,2τ+ζ+δ+γ,v;kcx−v/k.Corollary 6.
If we putp=1 and q=2 in Theorems 3 and 4, then we get the following results on the right known as k-Mittag-Leffler function:(40)D0+,kτ,δ,γtζ/k−1Ek,ε,ςϑ,ωctv/kx=xζ+δ/k−1Ψ34kϑ,ωk,ζ,v,ζ+δ+γ+τ,v;ς,ε,ϑ,0,ζ+γ,v,ζ+δ,1−k+v;cxv+1/k−1,(41)D−,kτ,δ,γtτ−ζ/kEk,ε,ςϑ,ωct−v/kx=xτ−ζ+δ/kΨ34kϑ,ωk,ζ−τ−δ,v+k−1,ζ+γ,v;ς,ε,ϑ,0,ζ−τ,v,ζ−τ−δ+γ,v;cx−v+1/k−1.Corollary 7.
If we putp=1 and q=3 in Theorems 3 and 4, then we get the following results on the right known as k-Bessel function of the first kind:(42)D0+,kτ,δ,γtζ/k−1Jk,ςϑ,εctv/kx=xζ+δ/k−1Ψ35kϑ,k,ζ,v,ζ+δ+γ+τ,v;ς+1,ε,ϑ,0,ζ+γ,v,ζ+δ,1−k+v,k,k;−ckxv+1/k−12,(43)D−,kτ,δ,γtτ−ζ/kJk,ςϑ,εct−v/kx=xτ−ζ+δ/kΨ35kϑ,k,ζ−τ−δ,v+k−1,ζ+γ,v;ς+1,ε,ϑ,0,ζ−τ,v,ζ−τ−δ+γ,v,k,k;−ckx−v+1/k−12.Corollary 8.
If we putp=3 and q=3 in Theorems 3 and 4, then we get the following results on the right known as k-hypergeomrtric function:(44)D0+,kτ,δ,γtζ/k−1F21,kϑ,k,φ,k;ς,k;ctv/kx=xζ+δ/k−1Ψ55kϑ,k,φ,k,ς,0,ζ,v,ζ+δ+γ+τ,v;ς,k,ϑ,0,φ,0,ζ+γ,v,ζ+δ,1−k+v;cxv+1/k−1.(45)D−,kτ,δ,γtτ−ζ/kF21,kϑ,k,φ,k;ς,k;ct−v/kx=xτ−ζ+δ/kΨ55kϑ,k,φ,k,ς,0ζ−τ−δ,v+k−1,ζ+γ,v;ς,k,ϑ,0,φ,0,ζ−τ,v,ζ−τ−δ+γ,v;cx−v+1/k−1.The advantage of the generalizedk-fractional calculus operators, which are also called by many authors as the general operator, is that they generalize Saigo’s fractional calculus operators and classical Riemann–Liouville (R-L) operators. For k⟶1, operators (1), (2), (4), and (5) reduce to Saigo’s [5] fractional integral and differentiation operators. If we take δ=−τ, (1), (2), (4), and (5) reduce the operators to k-Riemann–Liouville as follows:(46)I0+,kτ,−τ,γfx=I0+,kτfx,I−,kτ,−τ,γfx=I−,kτfx,D0+,kτ,−τ,γfx=D0+,kτfx,D−,kτ,−τ,γfx=D−,kτfx.Due to the most general character of thek-Wright function, numerous other interesting special cases from (22), (25), (28), and (31) can be given in the form of k-Struve function, k-Wright-type function, and many more, but due to lack of space, they are not represented here.



---

*Source: 5471715-2020-07-21.xml*