# Construction of Generalizedk-Bessel–Maitland Function with Its Certain Properties

Authors: Waseem Ahmad Khan; Hassen Aydi; Musharraf Ali; Mohd Ghayasuddin; Jihad Younis
Journal: Journal of Mathematics
 (2021)

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## Abstract

The main motive of this study is to present a new class of a generalizedk-Bessel–Maitland function by utilizing the k-gamma function and Pochhammer k-symbol. By this approach, we deduce a few analytical properties as usual differentiations and integral transforms (likewise, Laplace transform, Whittaker transform, beta transform, and so forth) for our presented k-Bessel–Maitland function. Also, the k-fractional integration and k-fractional differentiation of abovementioned k-Bessel–Maitland functions are also pointed out systematically.


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## Body

## 1. Introduction and Preliminaries

The computation of fragmentary integrals of special functions is significant from the mark of perspective on the value of these outcomes in the assessment of generalized integrals, and the solution of differential and integral equations. Fractional integral formulas involving the Bessel function have been created and assume a significant part in a few physical problems. The Bessel function is significant in examining the solutions of differential equations, and they are related to a wide scope of problems in numerous regions of mathematical physics, likewise radiophysics, fluid dynamics, and material sciences. These contemplations have driven different specialists in the field of special functions to investigating the possible expansions and also applications for the Bessel function. Valuable speculation of the Bessel function called thek-Bessel function has also been presented by Diaz et al. [1–3] and Suthar et al. [4]. They have presented k-beta, k-gamma, k-zeta functions, and Pochhammer k-symbol (rising factorial). Additionally, they demonstrated some of their properties and inequalities for the above-said functions. They have likewise considered k-hypergeometric functions based on k-rising factorial.Such functions play a discernible role in a variety of appropriate fields of science and engineering. During the past several years, several researchers have obtained variousk-type function (such as k-gamma, k-beta, and k-Pochhammer). This subject has received attention of various researchers and mathematicians during the last few decades. The k symbols are well known from many references related to finite difference calculus (see, [5–11], see additionally [12–16]). Recently, k-type functions and k-type operators have been considered in the literature by various authors. For this purpose, we start with the following properties in the literature.For our current assessment, we survey here the definition of some known functions and their generalizations. The integral representations ofk-gamma and k-beta functions are as follows (see [1–3]):(1)Γkx=kx/k−1Γxk=∫0∞tx−1e−tk/kdt,ℜx>0,k>0,(2)Bkx,y=1k∫01tx/k−11−ty/k−1dt,x>0,y>0,where(3)Bkx,y=1kBxk,yk,Bkx,y=ΓkxΓkyΓkx+y.The variety of the functions likewisek-Zeta function, k-Mittag–Leffler function for two and three parameters, k-Wright, and k-hypergeometric functions could be characterized by the following formulas (see also [4, 12, 13, 16–20]):(4)ξkz,p=∑n=0∞1Γz+nkp,k,z>0,p>1,Ek,α,βz=∑n=0∞znΓkαn+β,α,β>0,Ek,α,βγz=∑n=0∞γn,kΓkαn+βn!zn,k∈ℝ,α,β,γ∈ℂ,ℜα>0,ℜβ>0,Wk,α,βγz=∑n=0∞γn,kΓkαn+βn!2zn,k∈ℝ,α,β,γ∈ℂ;ℜα>0,ℜβ>0,ℜγ>0,Fkβ,k;γ,k;z=∑n=0∞βn,kγn,kn!zn,k∈ℝ,β,γ∈ℂ;ℜβ>0,ℜγ>0.Definition 1.
Letf be a sufficiently well-behaved function with support in ℝ+ and let α be a real number α>0. The k-Riemann–Liouville fractional integral of order α, I+αf is given by (see [21–23])(5)Ik,aαfz=1kΓkα∫azz−tα/k−1ftdt.
This definition unmistakably reduces the definition defined by Mubeen and Habibullah (see [14]):(6)Ikαfz=1kΓkα∫0zz−tα/k−1ftdt.
It is clear that the casek=1 of (6) yields the traditional Riemann–Liouville fractional integral:(7)Iαfz=1Γα∫0zz−tα−1ftdt.Definition 2.
Letβ be a real number. Then, k-Riemann–Liouville fractional derivative is defined by (see [21–23])(8)Dkβft=ddtIk1−βftdt,0<β≤1,where(9)Ik1−βfx=1kΓk1−β∫0zz−t1−β/k−1ftdt.Definition 3.
Foru∈φR, the fractional Fourier transform (FFT) of order α is defined as (see [21–23])(10)u^aw=Ϝauw=∫Reiw1/αtutdt,0<α≤1.
It is effectively observed that, forα=1, (10) reduces at the conventionally Fourier transform which is given by(11)Ϝφz=∫−∞+∞eiztφtdt.
Forw>0, (10) easily recovers the FFT presented by Luchko et al. [24].
In 2018, Ghayasuddin and Khan [25] presented generalized Bessel–Maitland functions by(12)Jν,γ,δμ,q,pz=∑n=0∞γqn−znΓμn+ν+1δpn,where μ,ν,γ,δ∈ℂ, ℜμ>0,ℜν>−1,ℜγ>0,ℜδ>0;p,q>0, and q<ℜμ+p.
Forbj,j=1,q¯ different from nonpositive integers, the series (see [26, 27])(13)Fqpa1,…,apb1,…,bq|z=∑n≥0a1na2n,…,apnb1nb2n,…,bqnznn!=∑n≥0∏j=1pajn∏j=1qbjnznn!is the generalized hypergeometric series, where the Pochhammer symbol(14)aμ≔Γa+μΓa=1,ifμ=0;a∈ℂ\0,aa+1,…,a+n−1,ifμ=n∈ℕ;a∈ℂ,and by convention a0=1. When p≤q, the generalized hypergeometric function converges for all complex values of z, that is, Fqpz is an entire function. When p>q+1, the series converges only for z=0, unless it terminates (as when one of the parameters aj,j=1,p¯ is a negative integer) in which case it is just a polynomial in z. When p=q+1, the series converges in the open unit disk z<1 and also for z=1 provided that(15)ℜ∑j=1qbj−∑j=1paj>0.
The summed upk-Wright function is addressed as follows (see details [7, 27]):(16)Ψqkpα1,A1,…,αp,Ap;β1,B1,…,βq,Bq;z=Ψqkpαj,Aj1,p;bj,βj1,q;z=∑n=0∞Γkα1+A1n,…,Γkαp+ApnΓkβ1+B1n,…,Γkβq+Bqnznn!,where k∈ℝ+; z∈ℂ and(17)ℜ∑j=1qBj−∑j=1pAj>0.
Motivated essentially by the demonstrated potential for applications of these extended generalizedk-Wright hypergeometric functions, we extend the generalized k-Bessel–Maitland function (18) by means of the generalized k-Pochhammer symbol (1) and investigate certain basic properties including differentiation formulas, integral representations, Euler-Beta, Laplace, Whittaker, and fractional Fourier transforms with their several special cases and relations with the k-Bessel–Maitland function. We also derive the k-fractional integration and differentiation of k-Bessel–Maitland function.


## 2. Generalizedk-Bessel–Maitland Function

This section deals with the new development ofk-Bessel–Maitland function Jk,ν,q,pμ,γ,δz and its associated properties.Definition 4.
Letk∈ℝ, μ,ν,γ,δ∈ℂ, ℜμ>0,ℜν≥−1,ℜγ>0,ℜδ>0;p,q>0, and q<ℜμ+p. The generalized k-Bessel–Maitland function is defined as(18)Jk,ν,q,pμ,γ,δz=∑n=0∞γqn,k−znΓkμn+ν+1δpn,k.Remark 1.
We note that the casek=1 in (18) leads to the generalized Bessel–Maitland function defined by Ghayasuddin and Khan [25], which further for δ=p=1 gives the Bessel–Maitland function given by Singh et al. [20].Theorem 1.
Ifk∈ℝ, μ,ν,γ,δ∈ℂ,ℜμ>0,ℜν≥−1,ℜγ>0,ℜδ>0; p,q>0 and q<ℜμ+p, then we have(19)ν+1Jk,ν+k,q,pμ,γ,δz+μzddzJk,ν+k,q,pμ,γ,δz=Jk,ν,q,pμ,γ,δzand(20)Jk,ν,q,pμ,γ+k,δ+kz−Jk,ν,q,pμ,γ,δz=zδqγq,kγpδp,kJk,μ+ν,q,pμ,γ+qk,δ+pkz.Proof.
With the help of (18) on the L.H.S of (19), we get(21)ν+1Jk,ν+k,q,pμ,γ,δz+μzddzJk,ν+k,q,pμ,γ,δz,=ν+1∑n=0∞γqn,kΓkμn+ν+k+1δpn,k−zn+μzddz∑n=0∞γqn,kΓkμn+ν+k+1δpn,k−zn=ν+1∑n=0∞γqn,kΓkμn+ν+k+1δpn,k−zn+μz∑n=0∞γqn,kΓkμn+ν+k+1δpn,kn−zn−1ν+1Jk,ν+k,q,pμ,γ,δz+μzddzJk,ν+k,q,pμ,γ,δz=∑n=0∞γqn,kΓkμn+ν+k+1δpn,k−znμn+ν+1.
In view ofΓkz+k=zΓkz, we acquire at our stated result (19).
Using Definition3 on the L.H.S of (20), we get (22)Jk,ν,q,pμ,γ+k,δ+kz−Jk,ν,q,pμ,γ,δz=∑n=0∞γ+kqn,kΓkμn+ν+1δ+kpn,k−zn−∑n=0∞γqn,kΓkμn+ν+1δpn,k−zn=∑n=0∞−znΓkμn+ν+1γ+kqn,k−γqn,kδ+kpn,k−δpn,k.
Now, by using the result given in [6], we get(23)Jk,ν,q,pμ,γ+k,δ+kz−Jk,ν,q,pμ,γ,δz=∑n=1∞−znΓkμn+ν+1δqnkγqn,kγpnkδpn,k=∑n=0∞−zn+1Γkμn+1+ν+1δqγqn+1,kγpδpn+1,k.
Using the result (see [6]), we get(24)Jk,ν,q,pμ,γ+k,δ+kz−Jk,ν,q,pμ,γ,δz=zδqγp∑n=0∞−znΓkμn+1+ν+1γq,kγ+qkqn,kδp,kδ+pkpn,k=zδqγq,kγpδp,k∑n=0∞−znΓkμn+1+ν+1γ+qkqn,kδ+pkpn,kJk,ν,q,pμ,γ+k,δ+kz−Jk,ν,q,pμ,γ,δz=zδqγq,kγpδp,kJk,μ+ν,q,pμ,γ+qk,δ+pkz,which is our stated result (20).Theorem 2.
Letk∈ℝ, μ,ν,γ,δ∈ℂ,ℜμ>0,ℜν≥−1,ℜγ>0,ℜδ>0; p,q>0 and q<ℜμ+p, then for m∈ℕ, we have(25)ddzJk,ν,q,pμ,γ,δz=γq,kδp,kn+1Jk,μ+ν,q,pμ,γ+qk,δ+pkzand(26)ddzmJk,ν,q,pμ,γ,δz=γqm,kδpm,kn+1mJk,μm+ν,q,pμ,γ+qm,δ+pmz.Proof.
With the help of (18) on the L.H.S of (25), we get(27)ddzJk,ν,q,pμ,γ,δz=ddz∑n=0∞γqn,k−znΓkμn+ν+1δpn,k=∑n=1∞γqn,k−n−zn−1Γkμn+ν+1δpn,k=∑n=0∞γqn+1,k−1n+1n+1znΓkμn+μ+ν+1δpn+1,kddzJk,ν,q,pμ,γ,δz=γq,kδp,k∑n=0∞−1n+1γ+qkqn,kn+1znΓkμn+μ+ν+1δ+pkpn,k,=γq,kδp,kn+1Jk,μ+ν,q,pμ,γ+qk,δ+pkz,which is our stated result (25).
Now, by using Definition3 on the L.H.S of (26), we get(28)ddzmJk,ν,q,pμ,γ,δz=ddzm∑n=0∞γqn,kΓkμn+ν+1δpn,k−zn=∑n=m∞nn−1,…,n−m+1γqn,k−1nzn−mΓkμn+ν+1δpn,k=∑n=0∞n+mn+m−1,…,n+1γqn+m,k−1n+mznΓkμn+m+ν+1δpn+m,kddzmJk,ν,q,pμ,γ,δz=γqm,kδpm,k∑n=0∞n+1mγ+qmqn,k−znΓkμn+m+ν+1δ+pmpn,k=γqm,kδpm,kn+1mJk,μm+ν,q,pμ,γ+qm,δ+pmz,which is our stated result (26).


## 3. Integral Transform of a Generalizedk-Bessel–Maitland Function

This section manages with some integral transforms likewise Laplace transform, Whittaker transform, beta transform, Hankel transform,K-transform, and fractional Fourier transform as follows.Theorem 3 (k-beta transform).
Letk∈ℝ, μ,ν,β,γ,δ∈ℂ,ℜμ>0,ℜν≥−1,ℜγ>0,ℜδ>0,ℜβ>0, p,q>0, and q<ℜμ+p, then we have(29)1Γkβ∫01tv/k1−tβ/k−1Jk,ν,q,pμ,γ,δztμ/kdt=Jk,ν+β,q,pμ,γ,δzand(30)1Γkβ∫txx−sβ/k−1s−tv/kJk,ν,q,pμ,γ,δzs−tμ/kds=x−tβ+ν/kJk,ν+β,q,pμ,γ,δzx−tμ/k.Proof.
By using (18) on the L.H.S of (29) and rearranging in reference to integration and summation (which is ensured under the condition), we acquire(31)1Γkβ∫01tv/k1−tβ/k−1Jk,ν,q,pμ,γ,δztμ/kdt=1Γkβ∑n=0∞γqn,kΓkμn+ν+1δpn,k−zn∫01tν+μn/k1−tβ/k−1dt=1Γkβ∑n=0∞γqn,kΓkμn+ν+1δpn,k−znBkμn+ν+1,β=Jk,ν+β,q,pμ,γ,δz,which is our stated result (29).
In the event that we set the transformationw=s−t/x−t on the L.H.S of equation (30) and using Definition 3, we acquire(32)x−tβ+ν/k∫01wv/k1−wβ/k−1Jk,ν,q,pμ,γ,δzwx−tμ/kdw=x−tβ+ν/k∑n=0∞γqn,k−zx−tμ/knΓkμn+ν+1δpn,k∫01wμn+ν/k1−wβ/k−1dw=x−tβ+ν/k∑n=0∞γqn,k−zx−tμ/knΓkμn+ν+1δpn,kBkμn+ν+1,β=x−tβ+ν/k∑n=0∞γqn,k−zx−tμ/knΓkμn+ν+1δpn,kΓkμn+ν+1ΓkβΓkμn+β+ν+1,which is our stated result (30).Theorem 4 (Laplace Transform).
Letk∈ℝ, μ,ν,α,β,γ,δ∈ℂ,ℜμ>0,ℜν≥−1,ℜγ>0,ℜδ>0,ℜα>0,ℜβ>0, p,q>0, and q<ℜμ+p, then we have(33)∫0∞zα−1e−szJk,ν,q,pμ,γ,δxzβdz=Γδ/ksαΓγ/k3Ψ2γ/k,q,α,β,1,1−xkq−p−μ/ksβν+1k,μk,δk,p.Proof.
By using (18) and the definition of Laplace transform(34)Lft=∫0∞e−stftdt,we get(35)∫0∞zα−1e−szJk,ν,q,pμ,γ,δxzβdz=∑n=0∞γqn,k−xnΓkμn+ν+1δpn,k∫0∞e−szzα+βn−1dz=∑n=0∞γqn,k−xnΓkμn+ν+1δpn,kΓα+βnsα+βn∫0∞zα−1e−szJk,ν,q,pμ,γ,δxzβdz=Γδ/ksαΓγ/k∑n=0∞kqnΓγ/k+qn−xnkμn/kΓμn+ν+1/kkpnΓδ/k+pnΓα+βnsβn.
Summing up the above the series with the help of (1), we easily arrive at our stated result (33).Theorem 5 (Hankel transform).
Ifk∈ℝ, μ,ν,β,γ,δ,η,λ∈ℂ,ℜμ>0,ℜν≥−1,ℜγ>0,ℜδ>0,ℜβ>0,ℜη>0,ℜλ>0; a,b>0, p,q>0, and q<ℜμ+p, then we have(36)∫0∞zη−1JλazJk,ν,q,pμ,γ,δbzβdz=2η−1Γδ/kaηΓγ/kkν+1/k−1×3Ψ3γk,q,λ+η2,β2,1,1kq−p−μ/k−b2aβν+1k,μk,δk,p,1+λ−η2,−β2.Proof.
Applying Definition3, we have(37)∫0∞zη−1JλazJk,ν,q,pμ,γ,δbzβdz=∑n=0∞γqn,k−bnΓkμn+ν+1δpn,k∫0∞zη+βn−1Jλazdz.
By following the given formula [13],(38)∫0∞ts−1Jναtdt=2s−1α−sΓν+s/2Γ1+ν−s/2,ℜν<ℜs<32,α>0,we get(39)∫0∞zη−1JλazJk,ν,q,pμ,γ,δbzβdz=2η−1Γδ/kaηΓγ/kkν+1/k−1∑n=0∞kqnΓγ/k+qn−bnkμn/kΓμn+ν+1/kkpnΓδ/k+pn×2βnΓλ+η+βn/2aβnΓ1+λ−η−βn/2.
In view of (16), we get our stated result (36).Theorem 6 (K-transform).
Letk∈ℝ, μ,ν,γ,δ,η,λ∈ℂ,ℜμ>0,ℜν≥−1,ℜγ>0,ℜδ>0,ℜη±λ>0; a>0,b>0,w>0, p,q>0, and q<ℜμ+p, then we have(40)∫0∞zη−1KλazJk,ν,q,pμ,γ,δbzwdz=2η−2Γδ/kaηΓγ/kkν+1/k−1×3Ψ2γk,q,η±λ2,w2,1,1−bkq−p−μ/k2awν+1k,μk,δk,p.Proof.
Applying Definition3, we have(41)∫0∞zη−1KλazJk,ν,q,pμ,γ,δbzwdz=∑n=0∞γqn,k−bnΓkμn+ν+1δpn,k∫0∞zη+wn−1Kλazdz.
By using the following integral (given in [13])(42)∫0∞xρ−1Kνxdx=2ρ−2Γρ±ν2,in the above equation, we arrive at(43)∫0∞zη−1KλazJk,ν,q,pμ,γ,δbzwdz=2η−2Γδ/kaηΓγ/kkν+1/k−1×∑n=0∞kqnΓγ/k+qn−bnkμn/kΓμn+ν+1/kkpnΓδ/k+pn2wnawnΓη+wn±λ2.
In view of (16), we get our stated result (40).Theorem 7 (Whittaker transform).
Letk∈ℝ, μ,ν,α,γ,δ,λ,ρ∈ℂ,ℜμ>0,ℜν≥−1,ℜγ>0,ℜδ>0,ℜα±m>−1/2; p,q>0; and q<ℜα+p, then we have(44)∫0∞tα−1e−st/2Wλ,mstJk,ν,q,pμ,γ,δwtρdt=Γδ/ksαΓγ/kkν+1/k−1×4Ψ3γk,q,12±m+α,ρ,1,1kq−p−μ/k−wsρν+1k,μk,δk,p,1−λ+α,ρ.Proof.
Applying Definition3 on the L.H.S of (44) and by setting st=z, we get(45)∫0∞e−z/2zsα−1Wλ,mzJk,ν,q,pμ,γ,δwzsρdzs=s−α∑n=0∞γqn,k−w/sρnΓkμn+ν+1δpn,k×∫0∞e−z/2zα+ρn−1Wλ,mzdz.
By using the following formula (given in [11])(46)∫0∞e−x/2xν−1Wλ,mxdx=Γ1/2±m+νΓ1−λ+ν,we get(47)∫0∞e−z/2zsα−1Wλ,mzJν,q,pμ,γ,δwzsρdzs=Γδ/ksαΓγ/kkν+1/k−1×∑n=0∞kqnΓγ/k+qn−w/sρnkμn/kΓμn+ν+1/kkpnΓδ/k+pnΓ1/2±m+α+ρnΓ1−λ+α+ρn.
In view of (16), we get our stated result.Theorem 8.
Letk∈ℝ, μ,ν,α,γ,δ,λ,ρ∈ℂ,ℜμ>0,ℜν≥−1,ℜγ>0,ℜδ>0,ℜα+m>−1/2, p,q>0, and q<ℜμ+p, then we have(48)∫0∞tα−1e−st/2Mλ,mstJk,ν,q,pμ,γ,δwtρdt=Γ2m+1Γδ/ksαΓγ/kΓm+λ+1/2kν+1/k−1×4Ψ3γk,q,m+α+12,ρ,λ−α,−ρ,1,1kq−p−μ/k−wsρν+1k,μk,δk,p,m−α+12,−ρ.Proof.
Applying Definition3 on the L.H.S of (48) and by setting st=z, we get(49)∫0∞e−z/2zsα−1Mλ,mzJk,ν,q,pμ,γ,δwzsρdzs=s−α∑n=0∞γqn,k−w/sρnΓkμn+ν+1δpn,k∫0∞e−z/2zα+ρn−1Mλ,mzdz.
By using the following integral (given in [13])(50)∫0∞e−x/2xν−1Mλ,mxdx=Γ2m+1Γm+ν+1/2Γλ−νΓm−ν+1/2Γm+λ+1/2,we get(51)∫0∞e−z/2zsα−1Mλ,mzJk,ν,q,pμ,γ,δwzsρdzs=s−α∑n=0∞γqn,k−w/sρnΓkμn+ν+1δpn,kΓ2m+1Γm+α+ρn+1/2Γλ−α−ρnΓm−α−ρn+1/2Γm+λ+1/2∫0∞e−z/2zsα−1Mλ,mzJk,ν,q,pμ,γ,δwzsρdzs=Γ2m+1Γδ/ksαΓγ/kΓm+λ+1/2kν+1/k−1∑n=0∞kqnΓγ/k+qn−w/sρnkμn/kΓμn+ν+1/kkpnΓδ/k+pn×Γm+α+ρn+1/2Γλ−α−ρnΓm−α−ρn+1/2.
Finally, by applying Definition 1.17, we get our stated result.Theorem 9.
Letk∈ℝ, μ,ν,α,γ,δ,ρ∈ℂ,ℜμ>0,ℜν≥−1,ℜγ>0,ℜδ>0,ℜα>−1,ℜα/2±m>−1,ℜα/2±λ>−1; p,q>0; and q<ℜμ+p, then we have(52)∫0∞tα−1Wλ,mstW−λ,mstJk,ν,q,pμ,γ,δwtρdt=Γδ/ksαΓγ/kkν+1/k−1Ψ45γk,q,α2±m+1,ρ2,α+1,ρ,1,1kq−p−μ/k−wsρν+1k,μk,δk,p,1+α2±λ,ρ2.Proof.
Applying Definition3 on the L.H.S of (44) and by setting st=z, we get(53)∫0∞zsα−1Wλ,mzW−λ,mzJk,ν,q,pμ,γ,δwzsρdzs=s−α∑n=0∞γqn,k−w/sρnΓkμn+ν+1δpn,k∫0∞zα+ρn−1Wλ,mzW−λ,mzdz.
By using the integral given in [11](54)∫0∞xν−1Wλ,mxW−λ,mxdx=Γν+1/2±mΓν+1Γ1+v/2±λ,we get(55)=Γδ/ksαΓγ/kkν+1/k−1∑n=0∞kqnΓγ/k+qn−w/sρnkμn/kΓμn+ν+1/kkpnΓδ/k+pnΓα+ρn+1/2±mΓα+ρn+1Γ1+α+ρn/2±λ.
Now, by summing up the above series with the help of (16), we get our stated result.Theorem 10 (fractional Fourier transform).
The FFT of the generalizedk-Bessel–Maitland function for t<0 is given by(56)ϜαJk,ν,q,pμ,γ,δz=∑n=0∞n!γqn,ki−n−1w−n+1/αΓkμn+ν+1δpn,k.Proof.
From (11) and (18), we have(57)ϜαJk,ν,q,pμ,γ,δz=∫Reiw1/αz∑n=0∞γqn,kΓkμn+ν+1δpn,k−zndz=∑n=0∞γqn,k−1nΓkμn+ν+1δpn,k∫−∞0eiw1/αzzndz.
On changing variablesiw1/αz=−t and iw1/αdz=−dt, we arrive at(58)ϜαJk,ν,q,pμ,γ,δz=∑n=0∞γqn,ki−n−1w−n+1/αΓkμn+ν+1δpn,k∫0∞e−ttndt=∑n=0∞n!γqn,ki−n−1w−n+1/αΓkμn+ν+1δpn,k,which is our stated result (56).


## 4.K-Fractional Integration and K-Fractional Differentiation

Recently,k-fractional calculus gained more attention due to its wide variety of applications in various fields [14, 17]. The k-fractional calculus of various types of special functions is used in many research papers [4, 28]. For more details about the recent works in the field of dynamic system theory, stochastic systems, non-equilibrium statistical mechanics, and quantum mechanics, we refer the interesting readers to [9, 17, 24]. In this section, we deduce the outcomes for k-fractional integration and k-fractional differentiation of the above-said function in an orderly way.Theorem 11 (K-fractional integration).
Ifk,η∈ℝ; γ,δ,μ,ν,∈ℂ, ℜγ>0,ℜδ>0,ℜμ>0,ℜν≥−1, p,q>0, and q<ℜμ+p, then(59)Ikηzv/k−1Jk,ν−1,q,pμ,γ,δzμ/k=zν+η/k−1Jk,ν+η−1,q,pμ,γ,δzμ/k.Proof.
From (9) and (18), we have(60)Ikηzv/k−1Jk,ν−1,q,pμ,γ,δzμ/k=1kΓkη∫0zz−tη/k−1∑n=0∞γqn,kΓkμn+νδpn,ktμn+ν/k−1dt=1kΓkη∑n=0∞γqn,kΓkμn+νδpn,k∫0zz−tη/k−1tμn+ν/k−1dt.
Puttingt=zx and dt=zdx in the above equation, we get(61)=1kΓkη∑n=0∞−1nγqn,kΓkμn+νδpn,kzμn+ν+η/k−1∫011−xη/k−1xμn+ν/k−1dx=1kΓkη∑n=0∞−1nγqn,kΓkμn+νδpn,kzμn+ν+η/k−1Bηk,μn+νk=1kΓkη∑n=0∞−1nγqn,kΓkμn+νδpn,kzμn+ν+η/k−1Γη/kΓμn+ν/kΓμn+ν+η/k.
By using Definition2, we have(62)Ikηzv/k−1Jk,ν−1,q,pμ,γ,δzμ/k=1kΓkη∑n=0∞−1nγqn,kΓkμn+νδpn,kzμn+ν+η/k−1kμn+ν+η/k−1ΓkηΓkμn+νkμn+ν+η/k−2Γkμn+ν+ηIkηzv/k−1Jk,ν−1,q,pμ,γ,δzμ/k=zν+η/k−1Jk,ν+η−1,q,pμ,γ,δzμ/k,which is our stated result.Theorem 12 (K-fractional differentiation).
Ifk,η∈ℝ; γ,δ,μ,ν∈ℂ, ℜγ>0,ℜδ>0,ℜμ>0,ℜν≥−1, p,q>0, and q<ℜμ+p, then(63)Dkηzv/k−1Jk,ν−1,q,pμ,γ,δzμ/k=zν−η+1/k−2kJk,ν−η−k−1,q,pμ,γ,δzμ/k.Proof.
From (8), (9), and (18), we have(64)Dkηzν/k−1Jk,ν−1,q,pμ,γ,δzμ/k=1kΓk1−ηddz∫0zz−t1−η/k−1∑n=0∞−1nγqn,kΓkμn+νδpn,ktμn+ν/k−1dt=1kΓk1−ηddz∑n=0∞−1nγqn,kΓkμn+νδpn,k∫0zz−t1−η/k−1tμn+ν/k−1dt.
Puttingt=zx and dt=zdx in the above equation, we get(65)=1kΓk1−η∑n=0∞−1nγqn,kΓkμn+νδpn,kddzzμn+ν−η+1/k−1∫011−x1−η/k−1xμn+ν/k−1dx=1kΓk1−η∑n=0∞−1nγqn,kΓkμn+νδpn,kddzzμn+ν−η+1/k−1B1−ηk,μn+νk=1kΓk1−η∑n=0∞−1nγqn,kΓkμn+νδpn,kμn+ν−η+1k−1zμn+ν−η+1/k−2Γ1−η/kΓμn+ν/kΓμn+ν−η+1/k−1+1.
Using Definition2 and the result Γn+1=nΓn in the above expression, we get(66)=1Γk1−η∑n=0∞−1nγqn,kΓkμn+νδpn,kμn+ν−η+1/k−1μn+ν−η+1/k−1zμn+ν−η+1/k−2kμn+ν−η+1−k/k−1Γk1−ηΓkμn+νkμn+ν−η+1−k/k−1Γkμn+ν−η+1−k=zν−η+1/k−2k∑n=0∞−1nγqn,kΓkμn+ν−η−k+1δpn,kzμn/kDkηzν/k−1Jk,ν−1,q,pμ,γ,δzμ/k=zν−η+1/k−2kJk,ν−η−k−1,q,pμ,γ,δzμ/k.
This completes the proof.


## 5. Concluding Remarks

In the present article, we have established generalizedk-Bessel–Maitland function Jk,ν,δ,pμ,γ,qz and its intriguing properties. Also, we have pointed out several integral transform such as beta transform, Laplace transform, Whittaker transform, K-transform, and fractional Fourier transform. In the last section, we deduced the outcomes for k-fractional integration and k-fractional differentiation of k-Bessel–Maitland function. Various special cases of the papers related results may be analyzed by taking appropriate values of the relevant parameters. For example, as given in Remarks 1.5, 1.6, and 1.7, we obtain the undeniable result due to Nisar et al. [15]. For several other special cases, we refer to [4, 12, 23, 24, 26, 28, 29] and leave the findings to interested readers.



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*Source: 5386644-2021-11-20.xml*