Research article Special Issues

Certain k-fractional calculus operators and image formulas of k-Struve function

  • In this article, the Saigo's k-fractional order integral and derivative operators involving k-hypergeometric function in the kernel are applied to the k-Struve function; outcome are expressed in the term of k-Wright function, which are used to present image formulas of integral transforms including beta transform. Also special cases related to fractional calculus operators and Struve functions are considered.

    Citation: D. L. Suthar, D. Baleanu, S. D. Purohit, F. Uçar. Certain k-fractional calculus operators and image formulas of k-Struve function[J]. AIMS Mathematics, 2020, 5(3): 1706-1719. doi: 10.3934/math.2020115

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  • In this article, the Saigo's k-fractional order integral and derivative operators involving k-hypergeometric function in the kernel are applied to the k-Struve function; outcome are expressed in the term of k-Wright function, which are used to present image formulas of integral transforms including beta transform. Also special cases related to fractional calculus operators and Struve functions are considered.



    In recent years, the fractional calculus has become a significant instrument for the modeling analysis and assumed a important role in different fields, for example, material science, science, mechanics, power, science, economy and control theory. In addition, research on fractional differential equations (ordinary or partial) and other analogous topics is very active and extensive around the world. One may refer to the recent papers [1,2,3,4,5,6,7] on the subject. In continuous, a series of research publications in respect to the generalized classical fractional calculus operators, Mubeen and Habibullah [8] were bring-out k-fractional order integral of the Riemann-Liouville version and its application, Dorrego [9] was introduced an alternative definition for the k-Riemann-Liouville fractional derivative.

    In recent case, Gupta and Parihar [10] introduced the following Saigo k-fractional integral and derivative operators involving the k-hypergeometric function for xR+, ω,ξ,γC with (ω)>0,k>0, we have

    (Iω,ξ,γ0+,kf)(x)=xωξkkΓk(ω)x0(xt)ωk1
    ×2F1,k((ω+ξ,k),(γ,k);(ω,k);(1tx))f(t)dt, (1.1)
    (Iω,ξ,γ,kf)(x)=1kΓk(ω)x(tx)ωk1tωξk
    ×2F1,k((ω+ξ,k),(γ,k);(ω,k);(1xt))f(t)dt, (1.2)
    (Dω,ξ,γ0+,kf)(x)=(ddx)r(Iω+r,ξr,ω+γr0+,kf)(x),r=[(ω)+1], (1.3)
    =(ddx)rxω+ξkkΓk(ω+r)x0(xt)ωk+r1
    (×)2F1,k((ωξ,k),(γω+r,k);(ω+r,k);(1tx))f(t)dt,
    (Dω,ξ,γ,kf)(x)=(ddx)r(Iω+r,ξr,ω+γ,kf)(x),r=[(ω)+1], (1.4)
    =(ddx)r1kΓk(ω+r)x(tx)ω+rk1tω+ξk
    (×)2F1,k((ωξ,k),(γω,k);(ω+r,k);(1xt))f(t)dt,

    where [(ω)] is the integer part of (ω) and 2F1,k((ω,k),(ξ,k);(γ,k);x) defined by [11] for xC,|x|<1, (γ)>(ξ)>0 as:

    2F1,k((ω,k),(ξ,k);(γ,k);x)=r=0(ω)r,k(ξ)r,kxr(γ)r,kr!. (1.5)

    The k-hypergeometric function Fk is defined by Mubeen and Habibullah [11] in a power series form as:

    Fk((ξ,k);(γ,k);x)=r=0ξr,kxr(γ)r,kr!,kR+,ξ,γC,(ξ)>0,(γ)>0. (1.6)

    and its integral representation can be obtained as follows:

    1F1((ξ,k);(γ,k);x)=Γk(γ)kΓk(ξ)Γk(γξ)10tξk1(1t)γξk1extdt, (1.7)

    Also, if (γ)>(ξ)>0,k>0,m0,mR+ and |x|<1, then

    m+1Fm,k[(ω,k),(ξm,k),(ξ+km,k),,(γ+(m1)km,k);(γm,k),(γ+km,k),,(γ+(m1)km,k);x]
    =Γk(γ)kΓk(ξ)Γk(γξ)10tξk1(1t)γξk1(1kxt)ωkdt. (1.8)

    and if (γ)>(ξ)>0 and |x|<1, then

    2F1,k((ω,k),(ξ,k);(γ,k);x)=Γk(γ)kΓk(ξ)Γk(γξ)10tξk1(1t)γξk1(1kxt)ωkdt. (1.9)

    Remark 1.1. For k=1, Eqs. (1.1) to (1.4) reduces in to Saigo's fractional order integral and derivative operators stated in [12].

    Now, we recollect few notable formulas for the fractional integral and derivative operators (1.1), (1.2), (1.3) and (1.4) as in the leading Lemma (see [10]).

    Lemma 1.2. Let ω,ξ,γ,ϑC and (ω)>0,kR+(0,) such that (ϑ)>max [0,(ξγ)], then

    (Iω,ξ,γ0+,ktϑk1)(x)=r=0krΓk(ϑ)Γk(ϑξ+γ)Γk(ϑξ)Γk(ϑ+ω+γ)xϑξk1. (1.10)

    Lemma 1.3. Let ω,ξ,γ,ϑC and (ω)>0, kR+(0,) such that (ϑ)>max [(ξ),(γ)], then

    (Iω,ξ,γ,ktϑk)(x)=r=0krΓk(ϑ+ξ)Γk(ϑ+γ)Γk(ϑ)Γk(ϑ+ω+ξ+γ)xϑξk. (1.11)

    Lemma 1.4. Let ω,ξ,γ,ϑC, r=((ω))+1,kR+(0,) such that Re(ϑ)>max [0,(ωξγ)], then

    (Dω,ξ,γ0+,ktϑk1)(x)=r=0Γk(ϑ)Γk(ϑ+ξ+γ+ω)Γk(ϑ+γ)Γk(ϑ+ξ+rrk)xϑ+ξ+rkr1. (1.12)

    Lemma 1.5. Let ω,ξ,γ,ϑC, r=((ω))+1,kR+(0,) such that Re(ϑ)>max [(ωγ),(ξrk+r)], then

    (Dω,ξ,γ,ktϑk)(x)=r=0Γk(ϑξr+rk)Γk(ϑ+ω+γ)Γk(ϑ)Γk(ϑξ+γ)xϑ+ξ+rkr. (1.13)

    k-Struve function: The generalized k-Struve function defined by Nisar et al. [13] as:

    Skv,c(z)=r=0(c)rΓk(rk+v+3k2)Γ(r+32)(z2)2r+vk+1, (1.14)

    where kR+;v>1 and cR and Γk(z) is the k-gamma function defined in Dˊiaz and Pariguan [14] as:

    Γk(z)=0tz1etkkdt,zC. (1.15)

    By inspection the following relation holds:

    Γk(z+k)=zΓk(z), (1.16)

    and

    Γk(z)=k(z/k)1Γ(zk). (1.17)

    If k1 and c=1, reduces to yield the well-known Struve function of order v defined by Baricz [15] as

    Hv(z)=r=0(1)rΓ(r+v+32)Γ(r+32)(z2)2r+v+1. (1.18)

    For further detail about Struve function and its properties (see [16,17,18,19,20,21]). Also Dˊiaz et al. [22,23] introduced the k-gamma function, k-beta function and Pochhammer k-symbols, Mubeen and Rehman [24] have studied extension of k-gamma and Pochhammer k-symbol, Mubeen and Habibullah [11] introduced k-fractional integration with its application and an integral representation of k-hypergeometric functions m+1Fm,k within Pochhammer k-symbols, k-gamma and k-beta functions.

    k-Wright function: Gehlot and Prajapati [25] introduced the generalized k-Wright function pΨkq(z) defined for kR+;z,ai,bjC,Ai,BjR (Ai,Bj0) where i=1,2,,p; j=1,2,,q and (ai+Air),(bj+Bjr)CkZ

    pΨkq(z)=pΨkq[(ai,Ai)1,p(bj,Bj)1,q|z]=r=0pΠi=1Γk(ai+Air)qΠj=1Γk(bj+Bjr)zrr!, (1.19)

    satisfies the following condition

    qj=1Bjkpi=1Aik>1. (1.20)

    Here, we present formulas for the Saigo k-fractional integrals (1.1) and (1.2) associated with the generalized k-Struve function (1.14), which are verbalized in terms of the k-Wright function in (1.19).

    Theorem 2.1. Let ω,ξ,γ,ϑ,ςC,(v)>1, kR+ such that (ω)>0, (ϑ)>max[0,(ξγ)]. If condition (1.20) satisfied and Iω,ξ,γ0+,k be the left sided operator of the generalized k-fractional integration involving k-hypergeometric function, thereupon the subsequent result true:

    (Iω,ξ,γ0+,k(tϑk1Skv,c[tςk]))(x)=kxϑ+ςξk+vςk21(1/2)vk+1
    ×3Ψk4[(ϑ+ς+vςk,2ς),(ϑ+ς+vςkξ+γ,2ς),(k,k)(v+3k2,k),(3k2,k),(ϑ+ς+vςkξ,2ς),(ϑ+ς+vςk+ω+γ,2ς)|ckx2ςk4]. (2.1)

    Proof. By applying (1.14) on the left side of (2.1), we have

    =Iω,ξ,γ0+,k(tϑk1r=0(c)rΓk(rk+v+3k2)Γ(r+32)(tςk2)2r+vk+1)(x),
    =r=0(c)rΓk(rk+v+3k2)Γ(r+32)(12)2r+vk+1Iω,ξ,γ0+,k(tϑ+(2r+1)ς+vςkk1), (2.2)

    which upon Lemma (1.2), yields

    =xϑ+ςξ+vςkk1r=01Γk(v+3k2+rk)Γ(r+32)(12)vk+1
    ×Γk(ϑ+ς+vςk+2rς)Γk(ϑ+ς+vςkξ+γ+2rς)Γk(ϑ+ς+vςkξ+2rς)Γk(ϑ+ς+vςk+ω+γ+2rς)(ckx2ςk4)r, (2.3)

    On using Γ(r+1)=krΓk(rk+k), we get

    =xϑ+ςξ+vςkk1r=0k12Γk(rk+k)Γk(v+3k2+rk)Γk(3k2+rk)r!(12)vk+1
    ×Γk(ϑ+ς+vςk+2rς)Γk(ϑ+ς+vςkξ+γ+2rς)Γk(ϑ+ς+vςkξ+2rς)Γk(ϑ+ς+vςk+ω+γ+2rς)(ckx2ςk4)r, (2.4)

    Using the definition of (1.19) in the right-hand side of (2.4), we arrive at the result (2.1).

    Theorem 2.2. Let ω,ξ,γ,ϑ,ςC,(v)>1, kR+ such that (ω)>0, (ω+ϑ)> max[(ξ),(γ)]. If condition (1.20) satisfied and Iω,ξ,γ0+,k be the right sided operator of the generalized k-fractional integration involving k-hypergeometric function, hence the leading result true:

    (Iω,ξ,γ,k(tωϑkSkv,c[tςk]))(x)=xωϑςξkvςk21(1/2)vk+1k
    ×3Ψk4[(ω+ϑ+ς+ξ+vςk,2ς),(ω+ϑ+ς+vςk+γ,2ς),(k,k)(v+3k2,k),(3k2,k),(ω+ϑ+ς+vςk,2ς),(ϑ+ς+2ω+ξ+γ+vςk,2ς)|ckx2ςk4]. (2.5)

    Proof. The proof is parallel to that of Theorem 2.1. Therefore, we omit the details.

    The results given in (2.1) and (2.5), being very general, can yield a huge number of specific cases by allotting some suited values to the involved parameters. Now, we demonstrate some Corollaries as below. If we take k=1 and c=1 in (2.1) and (2.5), we obtain the following two formulas in Corollaries 2.3 and 2.4.

    Corollary 2.3. Let ω,ξ,γ,ϑ,ςC,(v)>1, such that (ω)>0 and (ϑ)>max[0,(ξγ)], then the subsequent result true:

    (Iω,ξ,γ0+(tϑ1Hv[tς]))(x)=xϑ+(v+1)ςξ1(1/2)v+1
    ×3Ψ4[(ϑ+(v+1)ς,2ς),(ϑ+(v+1)ςξ+γ,2ς),(1,1)(v+32,1),(32,1),(ϑ+(v+1)ςξ,2ς),(ϑ+(v+1)ς+ω+γ,2ς)|x2ς4]. (2.6)

    Corollary 2.4. Let ω,ξ,γ,ϑ,ςC,(v)>1, such that (ω)>0 and (ω+ϑ)> max[(ξ),(γ)], then the following result true:

    (Iω,ξ,γ(tωϑHv[tς]))(x)=xωϑ(v+1)ςξ1(1/2)v+1
    ×3Ψ4[(ω+ϑ+(v+1)ς+ξ,2ς),(ω+ϑ+(v+1)ς+γ,2ς),(1,1)(v+32,1),(32,1),(ω+(v+1)ς+ϑ,2ς),(ϑ+(v+1)ς+2ω+ξ+γ,2ς)|x2ς4]. (2.7)

    If we substitute ξ=ω in Eqs. (2.1) and (2.5), Saigo k-fractional integral operators reduce to k-Riemann-Liouville integral operators as follows:

    Corollary 2.5. Let ω,γ,ϑ,ςC,(v)>1, kR+ such that (ω)>0, then the pursuing result true:

    (Iω,0+,k(tϑk1Skv,c[tςk]))(x)=kxϑ+ς+ωk+vςk21(1/2)vk+1
    ×2Ψk3[(ϑ+ς+vςk,2ς),(k,k)(v+3k2,k),(3k2,k),(ϑ+ς+vςk+ω,2ς)|ckx2ςk4]. (2.8)

    Corollary 2.6. Let ω,γ,ϑ,ςC,(v)>1, kR+ such that (ω)>0, then the following result true:

    (Iω,k(tωϑkSkv,c[tςk]))(x)=xϑςkvςk21(1/2)vk+1k
    ×2Ψk3[(ϑ+ς+vςk,2ς),(k,k)(v+3k2,k),(3k2,k),(ϑ+ς+ω+γ+vςk,2ς)|ckx2ςk4]. (2.9)

    At this point, we present formulas for the Saigo k-fractional derivative (1.1) and (1.2) associated with the generalized k -Struve function (1.14), which are suggested in terms of the k-Wright function in (1.19).

    Theorem 3.1. Let ω,ξ,γ,ϑ,ςC,(v)>1, r=((ω))+1, kR+ be such that (ω)>0, (ϑ)>max[0,(ωξγ)], If condition (1.20) is satisfied and Dω,ξ,γ0+,k be the left sided operator of the generalized k-fractional differentiation involving k-Gauss hypergeometric function, and so succeeding result true:

    (Dω,ξ,γ0+,k(tϑk1Skv,c[tςk]))(x)=xϑ+ξ+ςk+vςk21(1/2)vk+1k
    ×3Ψk4[(ϑ+ς+vςk,2ς),(ϑ+ς+vςk+ξ+γ+ω,2ς),(k,k)(v+3k2,k),(3k2,k),(ϑ+ς+vςk+γ,2ς),(ϑ+ς+vςk+ξ,2ςk+1)|cx2ς+1k14]. (3.1)

    Proof. By applying Eq. (1.14) in the left-side of (3.1), we get

    =Dω,ξ,γ0+,k(tϑk1r=0(c)rΓk(rk+v+3k2)Γ(r+32)(tςk2)2r+vk+1)(x),
    =r=0(c)rΓk(rk+v+3k2)Γ(r+32)(12)2r+vk+1Dω,ξ,γ0+,k(tϑ+(2r+1)ς+vςkk1), (3.2)

    Using Lemma (2.5), in the above equation can be written as

    =xϑ+ξ+ςk+vςk21r=0(c)rΓk(rk+v+3k2)Γ(r+32)(12)2r+vk+1
    ×r=0Γk(ϑ+ς+(vς/k)+2rς)Γk(ϑ+ς+(vς/k)+ξ+γ+ω+2rς)Γk(ϑ+ς+(vς/k)+γ+2rς)Γk(ϑ+ς+(vς/k)+ξ+2rς+rrk)x2rς+rkr, (3.3)

    On using Γ(r+1)=krΓk(rk+k), we get

    =xϑ+ξ+ςk+vςk21r=0(c)rΓk(rk+v+3k2)Γk(rk+k)Γk(rk+3k2)(12)2r+vk+1k
    ×r=0Γk(ϑ+ς+(vς/k)+2rς)Γk(ϑ+ς+(vς/k)+ξ+γ+ω+2rς)Γk(ϑ+ς+(vς/k)+γ+2rς)Γk(ϑ+ς+(vς/k)+ξ+2rς+rrk)x2rς+rkr, (3.4)

    Using the definition of (1.19) in the right-hand side of (3.4), we arrive at the result (3.1).

    Theorem 3.2. Let ω,ξ,γ,ϑ,ςC,(v)>1, and kR+ be such that (ω)>0, (ϑ)>max[(ωγ),(ξrk+r)], where (r=[(ω+1)]) and Dω,ξ,γ,k be the left sided operator of the generalized k-fractional differentiation then the succeeding formula preserves true:

    (Dω,ξ,γ,k(tωϑkwkv,c[atςk]))(x)=xωϑς+ξvςkk1(1/2)vk+1k
    ×3Ψk4[(ϑ+ςωξ+vςk,2ς+k1),(ϑ+ς+vςk+γ,2ς),(k,k)(v+3k2,k),(3k2,k),(ϑω+ς+vςk,2ς),(ϑ+ςωξ+γ+vςk,2ς)|cx2ς+1k14]. (3.5)

    Proof. The proof is similar of Theorem 3.1. Therefore, we omit the details.

    If we take k=1, c=1 in (3.1) and (3.5), we obtain the following two formulas as:

    Corollary 3.3. Let ω,ξ,γ,ϑ,ςC,(v)>1, such that (ω)>0, and (ϑ)>max[0,(ωξγ)], then the following result true:

    (Dω,ξ,γ0+(tϑ1Hv[tς]))(x)=xϑ+ξ+(v+1)ς1(1/2)v+1
    ×3Ψ4[(ϑ+(v+1)ς,2ς),(ϑ+(v+1)ς+ξ+γ+ω,2ς),(1,1)(v+32,1),(32,1),(ϑ+(v+1)ς+γ,2ς),(ϑ+(v+1)ς+ξ,2ς),(ϑ+(v+1)ς+ξ,2ς)|x2ς4]. (3.6)

    Corollary 3.4. Let ω,ξ,γ,ϑ,ςC,(v)>1 such that (ω)>0, (ϑ)>max [(ωγ),(ξrk+r)], then the following result true:

    (Dω,ξ,γ(tωϑHv[atς]))(x)=xωϑ(v+1)ς+ξ1(1/2)v+1
    ×3Ψ4[(ϑ+(v+1)ςωξ,2ς),(ϑ+(v+1)ς+γ,2ς),(1,1)(v+32,1),(32,1),(ϑ+(v+1)ςω,2ς),(ϑ+(v+1)ςωξ+γ,2ς)|x2ς4]. (3.7)

    If we substitute ξ=ω in Eqs. (3.5) and (3.8), Saigo k-fractional derivative operators reduce to k-Riemann-Liouville derivative operators as follows:

    Corollary 3.5. Let ω,γ,ϑ,ςC,(v)>1, kR+ be such that (ω)>0, then leading result true:

    (Dω0+,k(tϑk1Skv,c[tςk]))(x)=xϑω+ςk+vςk21(1/2)vk+1k
    ×2Ψk3[(ϑ+ς+vςk,2ς),(k,k)(v+3k2,k),(3k2,k),(ϑ+ς+vςkω,2ςk+1)|cx2ς+1k14]. (3.8)

    Corollary 3.6. Let ω,γ,ϑ,ςC,(v)>1, and kR+be such that (ω)>0, then the succeeding formula holds true:

    (Dω,k(tωϑkwkv,c[atςk]))(x)=xϑςvςkk1(1/2)vk+1k
    ×2Ψk3[(ϑ+ς+vςk,2ς+k1),(k,k)(v+3k2,k),(3k2,k),(ϑω+ς+vςk,2ς)|cx2ς+1k14]. (3.9)

    In this segment, we establish some theorems associated with the results obtained in previous sections pertaining to the integral transform.

    k-Beta function:The k-beta function [22] is defined as

    Bk(g,h)=1k10tgk1(1t)hk1dt,g>0,h>0. (4.1)

    and they have the following important identities

    Bk(g,h)=1kB(gk,hk)=Γk(g)Γk(h)Γk(g+h). (4.2)

    Now, we are defined k-beta function defined in the form:

    Bk(f(t);g,h)=1k10tgk1(1t)hk1f(t)dt,g>0,h>0. (4.3)

    Theorem 4.1. Let ω,ξ,γ,ϑ,ςC,(v)>1, kR+ such that (ω)>0, (ϑ)>max[0,(ξγ)], then the leading fractional order integral holds true:

    Bk((Iω,ξ,γ0+,k(tϑk1Skv,c[(zt)ςk]))(x);g,h)=xϑ+ςξk+vςk21(1/2)vk+1Γk(h)k
    ×4Ψk5[(ϑ+ς+vςk,2ς),(ϑ+ς+vςkξ+γ,2ς),(v+3k2,k),(3k2,k),(ϑ+ς+vςkξ,2ς),
    (g+ς+vςk,2ς),(k,k)(ϑ+ς+vςk+ω+γ,2ς),(g+h+ς+vςk,2ς)|ckx2ςk4] (4.4)

    Proof. Let be the left-hand side of (4.4) and using (4.3), we have

    =1k10zgk1(1z)hk1(Iω,ξ,γ0+,k(tϑk1Skv,c[(zt)ςk]))(x)dz, (4.5)

    which, using (1.14) and changing the order of integration and summation, which is valid under the conditions of Theorem 2.1, yields

    =r=0(c)rΓk(rk+v+3k2)Γ(r+32)(12)2r+vk+1(Iω,ξ,γ0+,k(tϑ+ς+2rς+vς/kk1))(x)
    ×1k10zg+ς+2nς+vς/kk1(1z)hk1dz, (4.6)

    which upon Lemma (1.2) and Eq. (4.2) in (4.6), we get

    =xϑ+ςξk+vςk21r=0(ck)rΓk(rk+v+3k2)Γ(r+32)(12)2r+vk+1
    ×Γk(ϑ+ς+(vς/k)+2rς)Γk(ϑ+ς+(vς/k)ξ+γ+2rς)Γk(ϑ+ς+(vς/k)ξ+2rς)Γk(ϑ+ς+(vς/k)+ω+γ+2rς)
    ×Γk(g+ς+2rς+vς/k)Γk(h)Γk(g+h+ς+2rς+vς/k)(x)2rςk, (4.7)

    Using the definition of (1.19) in the right-hand side of (4.7), we arrive at the result (4.4).

    Theorem 4.2. Let ω,ξ,γ,ϑ,ςC,(v)>1, kR+ such that (ω)>0, (ω+ϑ)> max[(ξ),(γ)], then the following fractional integral holds true:

    Bk((Iω,ξ,γ,k(tωϑkSkv,c[(z/t)ςk]))(x);g,h)=xωϑςξkvςk2(1/2)vk+1kΓk(h)
    ×4Ψk5[(ω+ϑ+ς+γ+vςk,2ς),(ω+ϑ+ς+vςk+ξ,2ς),(v+3k2,k),(3k2,k),(ω+ϑ+ς+vςk,2ς),
    (g+ς+vςk,2ς),(k,k)(ϑ+ς+2ω+γ+ξ+vςk,2ς),(g+h+ς+vςk,2ς)|ckx2ςk4]. (4.8)

    Proof. The proof is similar of Theorem 4.1. Therefore we omit the details.

    Theorem 4.3. Let ω,ξ,γ,ϑ,ςC,(v)>1, kR+ be such that (ω)>0, (ϑ)>max[0,(ωξγ)], then the following fractional derivative holds true:

    Bk((Dω,ξ,γ0+,k(tϑk1Skv,c[(zt)ςk]))(x);g,h)=xϑ+ς+ξk+vςk21(1/2)vk+1kΓk(h)
    ×4Ψk5[(ϑ+ς+vςk,2ς),(ϑ+ς+vςk+ξ+γ+ω,2ς),(v+3k2,k),(3k2,k),(ϑ+ς+vςk+γ,2ς),
    (g+ς+vςk,2ς),(k,k)(ϑ+ς+vςk+ξ,2ςk+1),(g+h+ς+vςk,2ς)|cx2ς+1k14]. (4.9)

    Proof. Let be the left-hand side of (4.9) and using the definition of Beta transform, we have

    =1k10zgk1(1z)hk1(Dω,ξ,γ0+,k(tϑk1Skv,c[(zt)ςk]))(x)dz, (4.10)

    which, using (1.14) and changing the order of integration and summation, which is reasonable under the conditions of Theorem 3, yields

    =r=0(c)rΓk(rk+v+3k2)Γ(r+32)(12)2r+vk+1(Dω,ξ,γ0+,k(tϑ+ς+2rς+vς/kk1))(x)
    ×1k10zg+ς+2nς+vς/kk1(1z)hk1dz, (4.11)

    which upon Lemma (1.4) and Eq. (4.2) in (4.11), we get

    =xϑ+ς+ξk+vςk21r=0(ck)rΓk(rk+v+3k2)Γ(r+32)(12)2r+vk+1
    ×Γk(ϑ+ς+(vς/k)+2rς)Γk(ϑ+ς+(vς/k)+ξ+γ+ω+2rς)Γk(ϑ+ς+(vς/k)+γ+2rς)Γk(ϑ+ς+(vς/k)+ξ+2rς+rrk)
    ×Γk(g+ς+2rς+vς/k)Γk(h)Γk(g+h+ς+2rς+vς/k)(x)2nς+rkr, (4.12)

    Using the definition of (1.14) in the right-hand side of (4.12), we arrive at the result (4.9).

    Theorem 4.4. Let ω,ξ,γ,ϑ,ςC,(v)>1, and kR+ be such that (ω)>0, (ϑ)>max[(ωγ),(ξrk+r)], then the following formula holds true:

    Bk((Dω,ξ,γ,k(tωϑkSkv,c[(z/t)ςk]))(x);g,h)=xωϑςξkvςk2k(1/2)vk+1Γk(h)
    ×4Ψk5[(ϑ+ςω+vςkξ,2ς+k1),(ϑ+ς+vςk+γ,2ς),(v+3k2,k),(3k2,k),(ϑ+ςω+vςk,2ς),
    (g+ς+vςk,2ς),(k,k)(ϑ+ςω+vςkξ+γ,2ς),(g+h+ς+vςk,2ς)|cx2ς+1k14]. (4.13)

    Proof. The proof is parallel to that of Theorem 4.3. Therefore, we omit the details.

    Setting k=1, c=1 in (4.4), (4.7), (4.9) and (4.13), we obtain the following new formulas as:

    Corollary 4.5. Let ω,ξ,γ,ϑ,ςC,(v)>1, such that (ϑ)>max [0,(ξγ)], (ω)>0; then

    B((Iω,ξ,γ0+(tϑ1Hv[(zt)ς]))(x);g,h)=xϑξ+(v+1)ς1(1/2)v+1Γ(h)
    ×4Ψ5[(ϑ+(v+1)ς,2ς),(ϑ+(v+1)ςξ+γ,2ς),(v+32,1),(32,1),(ϑ+(v+1)ςξ,2ς),
    (g+(v+1)ς,2ς),(1,1)(ϑ+(v+1)ς+ω+γ,2ς),(g+(v+1)ς+h,2ς)|x2ς4]. (4.14)

    Corollary 4.6. Let ω,ξ,γ,ϑ,ςC,(v)>1, such that (ω+ϑ)> max[(ξ),(γ)], (ω)>0; then

    B((Iω,ξ,γ,(tωϑHv[(z/t)ς]))(x);g,h)=xωϑξ(v+1)ς(1/2)v+1Γ(h)
    ×4Ψ5[(ω+ϑ+(v+1)ς+γ,2ς),(ω+ϑ+(v+1)ς+ξ,2ς),(v+32,1),(32,1),(ω+(v+1)ς+ϑ,2ς),
    (g+(v+1)ς,2ς),(1,1)(ϑ+(v+1)ς+2ω+γ+ξ,2ς),(g++(v+1)ς+h,2ς)|x2ς4]. (4.15)

    Corollary 4.7. Let ω,ξ,γ,ϑ,ςC,(v)>1, be such that (ϑ)>max[0,(ωξγ)], (ω)>0; then

    B((Dω,ξ,γ0+(tϑ1Hv[(zt)ς]))(x);g,h)=xϑ+ξ+(v+1)ς1(1/2)v+1Γ(h)
    ×4Ψ5[(ϑ+(v+1)ς,2ς),(ϑ+(v+1)ς+ξ+γ+ω,2ς),(v+32,1),(32,1),(ϑ+(v+1)ς+γ,2ς),
    (g+(v+1)ς,2ς),(1,1)(ϑ+(v+1)ς+ξ,2ς),(g+(v+1)ς+h,2ς)|x2ς4]. (4.16)

    Corollary 4.8. Let ω,ξ,γ,ϑ,ςC,(v)>1, be such that (ω)>0, (ϑ)>max[(ωγ),(ξ)], then the following formula holds true:

    B((Dω,ξ,γ(tωϑHv[(z/t)ς]))(x);g,h)=xωϑςξvς(1/2)v+1Γ(h)
    ×4Ψ5[(ϑ+(v+1)ςωξ,2ς),(ϑ+(v+1)ς+γ,2ς),(v+32,1),(32,1),(ϑ+(v+1)ςω,2ς),
    (g+(v+1)ς,2ς),(1,1)(ϑ+(v+1)ςωξ+γ,2ς),(g+(v+1)ς+h,2ς)|x2ς4]. (4.17)

    Similarly, If we put ξ=ω in Eqs. (4.4), (4.7), (4.9) and (4.13), Saigo k-fractional calculus operators reduce to k-Riemann-Liouville calculus operators as follows:

    Corollary 4.9. Let ω,γ,ϑ,ςC,(v)>1, kR+ such that (ω)>0, then

    Bk((Iω0+,k(tϑk1Skv,c[(zt)ςk]))(x);g,h)=xϑ+ς+ωk+vςk21(1/2)vk+1Γk(h)k
    ×3Ψk4[(ϑ+ς+vςk,2ς),(g+ς+vςk,2ς),(k,k)(v+3k2,k),(3k2,k),(ϑ+ς+vςk+ω,2ς),(g+h+ς+vςk,2ς)|ckx2ςk4]. (4.18)

    Corollary 4.10. Let ω,γ,ϑ,ςC,(v)>1, kR+ such that (ω)>0, then

    Bk((Iω,k(tωϑkSkv,c[(z/t)ςk]))(x);g,h)=xϑςkvςk2(1/2)vk+1kΓk(h)
    ×3Ψk4[(ϑ+ς+vςk,2ς),(g+ς+vςk,2ς),(k,k)(v+3k2,k),(3k2,k),(ϑ+ς+ω+γ+vςk,2ς),(g+h+ς+vςk,2ς)|ckx2ςk4]. (4.19)

    Corollary 4.11. Let ω,γ,ϑ,ςC,(v)>1, kR+ be such that (ω)>0, then

    Bk((Dω0+,k(tϑk1Skv,c[(zt)ςk]))(x);g,h)=xϑ+ςωk+vςk21(1/2)vk+1kΓk(h)
    ×3Ψk4[(ϑ+ς+vςk,2ς),(g+ς+vςk,2ς),(k,k)(v+3k2,k),(3k2,k),(ϑ+ς+vςk+ξ,2ςk+1),(g+h+ς+vςk,2ς)|cx2ς+1k14]. (4.20)

    Corollary 4.12. Let ω,γ,ϑ,ςC,(v)>1, and kR+be such that (ω)>0, then

    Bk((Dω,k(tωϑkSkv,c[(z/t)ςk]))(x);g,h)=x2ωϑςkvςk2k(1/2)vk+1Γk(h)
    ×3Ψk4[(ϑ+ς+vςk,2ς+k1),(g+ς+vςk,2ς),(k,k)(v+3k2,k),(3k2,k),(ϑ+ςω+vςk,2ς),(g+h+ς+vςk,2ς)|cx2ς+1k14]. (4.21)

    The generalized k-fractional calculus operators have advantage that it generalizes Saigo's fractional integral and derivative operators, therefore, many authors called this a general operator. So, we conclude this paper by emphasizing that many other interesting image formulas can be derived as the specific cases of our leading results Theorems 2.1, 2.2, 3.1 and 3.2, involving familiar k-fractional integral and derivative operators as above said. Some special cases of k-fractional calculus involving k-Struve function have been explored in the literature by a authors [26] with different arguments. Therefore, results existing in this article are easily regenerate in terms of a comparable type of novel interesting integrals with diverse arguments after various suitable parametric replacements.

    The authors are thankful to the referee's for their valuable remarks and comments for the improvement of the paper.

    The authors declare no conflict of interest in this paper.



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