In this paper, based on the analytic method and the properties of Gauss sums, we study the computational problems of the fourth power mean value of one kind two-term exponential sums through the classification and estimation of Dirichlet characters and give it a calculation formula or asymptotic formula in different conditions.
Citation: Jinmin Yu, Renjie Yuan, Tingting Wang. The fourth power mean value of one kind two-term exponential sums[J]. AIMS Mathematics, 2022, 7(9): 17045-17060. doi: 10.3934/math.2022937
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In this paper, based on the analytic method and the properties of Gauss sums, we study the computational problems of the fourth power mean value of one kind two-term exponential sums through the classification and estimation of Dirichlet characters and give it a calculation formula or asymptotic formula in different conditions.
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 x∈R+, ω,ξ,γ∈C with ℜ(ω)>0,k>0, we have
(Iω,ξ,γ0+,kf)(x)=x−ω−ξkkΓk(ω)∫x0(x−t)ωk−1 |
×2F1,k((ω+ξ,k),(−γ,k);(ω,k);(1−tx))f(t)dt, | (1.1) |
(Iω,ξ,γ−,kf)(x)=1kΓk(ω)∫∞x(t−x)ωk−1t−ω−ξk |
×2F1,k((ω+ξ,k),(−γ,k);(ω,k);(1−xt))f(t)dt, | (1.2) |
(Dω,ξ,γ0+,kf)(x)=(ddx)r(I−ω+r,−ξ−r,ω+γ−r0+,kf)(x),r=[ℜ(ω)+1], | (1.3) |
=(ddx)rxω+ξkkΓk(−ω+r)∫x0(x−t)ωk+r−1 |
(×)2F1,k((−ω−ξ,k),(−γ−ω+r,k);(−ω+r,k);(1−tx))f(t)dt, |
(Dω,ξ,γ−,kf)(x)=(−ddx)r(I−ω+r,−ξ−r,ω+γ−,kf)(x),r=[ℜ(ω)+1], | (1.4) |
=(−ddx)r1kΓk(−ω+r)∫∞x(t−x)−ω+rk−1tω+ξk |
(×)2F1,k((−ω−ξ,k),(−γ−ω,k);(−ω+r,k);(1−xt))f(t)dt, |
where [ℜ(ω)] is the integer part of ℜ(ω) and 2F1,k((ω,k),(ξ,k);(γ,k);x) defined by [11] for x∈C,|x|<1, ℜ(γ)>ℜ(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!,k∈R+,ξ,γ∈C,ℜ(ξ)>0,ℜ(γ)>0. | (1.6) |
and its integral representation can be obtained as follows:
1F1((ξ,k);(γ,k);x)=Γk(γ)kΓk(ξ)Γk(γ−ξ)∫10tξk−1(1−t)γ−ξk−1extdt, | (1.7) |
Also, if ℜ(γ)>ℜ(ξ)>0,k>0,m≥0,m∈R+ and |x|<1, then
m+1Fm,k[(ω,k),(ξm,k),(ξ+km,k),⋯,(γ+(m−1)km,k);(γm,k),(γ+km,k),⋯,(γ+(m−1)km,k);x] |
=Γk(γ)kΓk(ξ)Γk(γ−ξ)∫10tξk−1(1−t)γ−ξk−1(1−kxt)−ωkdt. | (1.8) |
and if ℜ(γ)>ℜ(ξ)>0 and |x|<1, then
2F1,k((ω,k),(ξ,k);(γ,k);x)=Γk(γ)kΓk(ξ)Γk(γ−ξ)∫10tξk−1(1−t)γ−ξk−1(1−kxt)−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,ϑ∈C and ℜ(ω)>0,k∈R+(0,∞) such that ℜ(ϑ)>max [0,ℜ(ξ−γ)], then
(Iω,ξ,γ0+,ktϑk−1)(x)=∞∑r=0krΓk(ϑ)Γk(ϑ−ξ+γ)Γk(ϑ−ξ)Γk(ϑ+ω+γ)xϑ−ξk−1. | (1.10) |
Lemma 1.3. Let ω,ξ,γ,ϑ∈C and ℜ(ω)>0, k∈R+(0,∞) such that ℜ(ϑ)>max [ℜ(−ξ),ℜ(−γ)], then
(Iω,ξ,γ−,kt−ϑk)(x)=∞∑r=0krΓk(ϑ+ξ)Γk(ϑ+γ)Γk(ϑ)Γk(ϑ+ω+ξ+γ)x−ϑ−ξk. | (1.11) |
Lemma 1.4. Let ω,ξ,γ,ϑ∈C, r=(ℜ(ω))+1,k∈R+(0,∞) such that Re(ϑ)>max [0,ℜ(−ω−ξ−γ)], then
(Dω,ξ,γ0+,ktϑk−1)(x)=∞∑r=0Γk(ϑ)Γk(ϑ+ξ+γ+ω)Γk(ϑ+γ)Γk(ϑ+ξ+r−rk)xϑ+ξ+rk−r−1. | (1.12) |
Lemma 1.5. Let ω,ξ,γ,ϑ∈C, r=(ℜ(ω))+1,k∈R+(0,∞) such that Re(ϑ)>max [ℜ(−ω−γ),ℜ(ξ−rk+r)], then
(Dω,ξ,γ−,kt−ϑk)(x)=∞∑r=0Γk(ϑ−ξ−r+rk)Γk(ϑ+ω+γ)Γk(ϑ)Γk(ϑ−ξ+γ)x−ϑ+ξ+rk−r. | (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 k∈R+;v>−1 and c∈R and Γk(z) is the k-gamma function defined in Dˊiaz and Pariguan [14] as:
Γk(z)=∫∞0tz−1e−tkkdt,z∈C. | (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 k→1 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 k∈R+;z,ai,bj∈C,Ai,Bj∈R (Ai,Bj≠0) where i=1,2,⋯,p; j=1,2,⋯,q and (ai+Air),(bj+Bjr)∈C∖kZ−
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
q∑j=1Bjk−p∑i=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,ς∈C,ℜ(v)>−1, k∈R+ such that ℜ(ω)>0, ℜ(ϑ)>max[0,ℜ(ξ−γ)]. 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(tϑk−1Skv,c[tςk]))(x)=√kxϑ+ς−ξk+vςk2−1(1/2)vk+1 |
×3Ψk4[(ϑ+ς+vςk,2ς),(ϑ+ς+vςk−ξ+γ,2ς),(k,k)(v+3k2,k),(3k2,k),(ϑ+ς+vςk−ξ,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(tϑk−1∞∑r=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+vςkk−1), | (2.2) |
which upon Lemma (1.2), yields
=xϑ+ς−ξ+vςkk−1∞∑r=01Γk(v+3k2+rk)Γ(r+32)(12)vk+1 |
×Γk(ϑ+ς+vςk+2rς)Γk(ϑ+ς+vςk−ξ+γ+2rς)Γk(ϑ+ς+vςk−ξ+2rς)Γk(ϑ+ς+vςk+ω+γ+2rς)(−ckx2ςk4)r, | (2.3) |
On using Γ(r+1)=k−rΓk(rk+k), we get
=xϑ+ς−ξ+vςkk−1∞∑r=0k12Γk(rk+k)Γk(v+3k2+rk)Γk(3k2+rk)r!(12)vk+1 |
×Γk(ϑ+ς+vςk+2rς)Γk(ϑ+ς+vςk−ξ+γ+2rς)Γk(ϑ+ς+vςk−ξ+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,ς∈C,ℜ(v)>−1, k∈R+ such that ℜ(ω)>0, ℜ(ω+ϑ)> max[−ℜ(ξ),−ℜ(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(t−ω−ϑkSkv,c[t−ςk]))(x)=x−ω−ϑ−ς−ξk−vςk2−1(1/2)vk+1√k |
×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)|−ckx−2ςk4]. | (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,ς∈C,ℜ(v)>−1, such that ℜ(ω)>0 and ℜ(ϑ)>max[0,ℜ(ξ−γ)], then the subsequent result true:
(Iω,ξ,γ0+(tϑ−1Hv[tς]))(x)=xϑ+(v+1)ς−ξ−1(1/2)v+1 |
×3Ψ4[(ϑ+(v+1)ς,2ς),(ϑ+(v+1)ς−ξ+γ,2ς),(1,1)(v+32,1),(32,1),(ϑ+(v+1)ς−ξ,2ς),(ϑ+(v+1)ς+ω+γ,2ς)|−x2ς4]. | (2.6) |
Corollary 2.4. Let ω,ξ,γ,ϑ,ς∈C,ℜ(v)>−1, such that ℜ(ω)>0 and ℜ(ω+ϑ)> max[−ℜ(ξ),−ℜ(γ)], then the following result true:
(Iω,ξ,γ−(t−ω−ϑHv[t−ς]))(x)=x−ω−ϑ−(v+1)ς−ξ−1(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)|−x−2ς4]. | (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,ς∈C,ℜ(v)>−1, k∈R+ such that ℜ(ω)>0, then the pursuing result true:
(Iω,0+,k(tϑk−1Skv,c[tςk]))(x)=√kxϑ+ς+ωk+vςk2−1(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,ς∈C,ℜ(v)>−1, k∈R+ such that ℜ(ω)>0, then the following result true:
(Iω−,k(t−ω−ϑkSkv,c[t−ςk]))(x)=x−ϑ−ςk−vςk2−1(1/2)vk+1√k |
×2Ψk3[(ϑ+ς+vςk,2ς),(k,k)(v+3k2,k),(3k2,k),(ϑ+ς+ω+γ+vςk,2ς)|−ckx−2ςk4]. | (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,ς∈C,ℜ(v)>−1, r=(ℜ(ω))+1, k∈R+ be such that ℜ(ω)>0, ℜ(ϑ)>max[0,ℜ(−ω−ξ−γ)], 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(tϑk−1Skv,c[tςk]))(x)=xϑ+ξ+ςk+vςk2−1(1/2)vk+1√k |
×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,2ς−k+1)|−cx2ς+1k−14]. | (3.1) |
Proof. By applying Eq. (1.14) in the left-side of (3.1), we get
=Dω,ξ,γ0+,k(tϑk−1∞∑r=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+vςkk−1), | (3.2) |
Using Lemma (2.5), in the above equation can be written as
=xϑ+ξ+ςk+vςk2−1∞∑r=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+r−rk)x2rς+rk−r, | (3.3) |
On using Γ(r+1)=k−rΓk(rk+k), we get
=xϑ+ξ+ςk+vςk2−1∞∑r=0(−c)rΓk(rk+v+3k2)Γk(rk+k)Γk(rk+3k2)(12)2r+vk+1√k |
×∞∑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+r−rk)x2rς+rk−r, | (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,ς∈C,ℜ(v)>−1, and k∈R+ be such that ℜ(ω)>0, ℜ(ϑ)>max[ℜ(−ω−γ),ℜ(ξ−rk+r)], where (r=[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(tω−ϑkwkv,c[at−ςk]))(x)=xω−ϑ−ς+ξ−vςkk−1(1/2)vk+1√k |
×3Ψk4[(ϑ+ς−ω−ξ+vςk,2ς+k−1),(ϑ+ς+vςk+γ,2ς),(k,k)(v+3k2,k),(3k2,k),(ϑ−ω+ς+vςk,2ς),(ϑ+ς−ω−ξ+γ+vςk,2ς)|−cx−2ς+1k−14]. | (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,ς∈C,ℜ(v)>−1, such that ℜ(ω)>0, and ℜ(ϑ)>max[0,ℜ(−ω−ξ−γ)], then the following result true:
(Dω,ξ,γ0+(tϑ−1Hv[tς]))(x)=xϑ+ξ+(v+1)ς−1(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,ς∈C,ℜ(v)>−1 such that ℜ(ω)>0, ℜ(ϑ)>max [ℜ(−ω−γ),ℜ(ξ−rk+r)], then the following result true:
(Dω,ξ,γ−(tω−ϑHv[at−ς]))(x)=xω−ϑ−(v+1)ς+ξ−1(1/2)v+1 |
×3Ψ4[(ϑ+(v+1)ς−ω−ξ,2ς),(ϑ+(v+1)ς+γ,2ς),(1,1)(v+32,1),(32,1),(ϑ+(v+1)ς−ω,2ς),(ϑ+(v+1)ς−ω−ξ+γ,2ς)|−x−2ς4]. | (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,ς∈C,ℜ(v)>−1, k∈R+ be such that ℜ(ω)>0, then leading result true:
(Dω0+,k(tϑk−1Skv,c[tςk]))(x)=xϑ−ω+ςk+vςk2−1(1/2)vk+1√k |
×2Ψk3[(ϑ+ς+vςk,2ς),(k,k)(v+3k2,k),(3k2,k),(ϑ+ς+vςk−ω,2ς−k+1)|−cx2ς+1k−14]. | (3.8) |
Corollary 3.6. Let ω,γ,ϑ,ς∈C,ℜ(v)>−1, and k∈R+be such that ℜ(ω)>0, then the succeeding formula holds true:
(Dω−,k(tω−ϑkwkv,c[at−ςk]))(x)=x−ϑ−ς−vςkk−1(1/2)vk+1√k |
×2Ψk3[(ϑ+ς+vςk,2ς+k−1),(k,k)(v+3k2,k),(3k2,k),(ϑ−ω+ς+vςk,2ς)|−cx−2ς+1k−14]. | (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)=1k∫10tgk−1(1−t)hk−1dt,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)=1k∫10tgk−1(1−t)hk−1f(t)dt,g>0,h>0. | (4.3) |
Theorem 4.1. Let ω,ξ,γ,ϑ,ς∈C,ℜ(v)>−1, k∈R+ such that ℜ(ω)>0, ℜ(ϑ)>max[0,ℜ(ξ−γ)], then the leading fractional order integral holds true:
Bk((Iω,ξ,γ0+,k(tϑk−1Skv,c[(zt)ςk]))(x);g,h)=xϑ+ς−ξk+vςk2−1(1/2)vk+1Γk(h)√k |
×4Ψk5[(ϑ+ς+vςk,2ς),(ϑ+ς+vςk−ξ+γ,2ς),(v+3k2,k),(3k2,k),(ϑ+ς+vςk−ξ,2ς), |
(g+ς+vςk,2ς),(k,k)(ϑ+ς+vςk+ω+γ,2ς),(g+h+ς+vςk,2ς)|−ckx2ςk4] | (4.4) |
Proof. Let ℓ be the left-hand side of (4.4) and using (4.3), we have
ℓ=1k∫10zgk−1(1−z)hk−1(Iω,ξ,γ0+,k(tϑk−1Skv,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
ℓ=∞∑r=0(−c)rΓk(rk+v+3k2)Γ(r+32)(12)2r+vk+1(Iω,ξ,γ0+,k(tϑ+ς+2rς+vς/kk−1))(x) |
×1k∫10zg+ς+2nς+vς/kk−1(1−z)hk−1dz, | (4.6) |
which upon Lemma (1.2) and Eq. (4.2) in (4.6), we get
ℓ=xϑ+ς−ξk+vςk2−1∞∑r=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,ς∈C,ℜ(v)>−1, k∈R+ such that ℜ(ω)>0, ℜ(ω+ϑ)> max[−ℜ(ξ),−ℜ(γ)], then the following fractional integral holds true:
Bk((Iω,ξ,γ−,k(t−ω−ϑkSkv,c[(z/t)ςk]))(x);g,h)=x−ω−ϑ−ς−ξk−vςk2(1/2)vk+1√kΓk(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)|−ckx−2ςk4]. | (4.8) |
Proof. The proof is similar of Theorem 4.1. Therefore we omit the details.
Theorem 4.3. Let ω,ξ,γ,ϑ,ς∈C,ℜ(v)>−1, k∈R+ be such that ℜ(ω)>0, ℜ(ϑ)>max[0,ℜ(−ω−ξ−γ)], then the following fractional derivative holds true:
Bk((Dω,ξ,γ0+,k(tϑk−1Skv,c[(zt)ςk]))(x);g,h)=xϑ+ς+ξk+vςk2−1(1/2)vk+1√kΓk(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,2ς−k+1),(g+h+ς+vςk,2ς)|−cx2ς+1k−14]. | (4.9) |
Proof. Let ℑ be the left-hand side of (4.9) and using the definition of Beta transform, we have
ℑ=1k∫10zgk−1(1−z)hk−1(Dω,ξ,γ0+,k(tϑk−1Skv,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
ℑ=∞∑r=0(−c)rΓk(rk+v+3k2)Γ(r+32)(12)2r+vk+1(Dω,ξ,γ0+,k(tϑ+ς+2rς+vς/kk−1))(x) |
×1k∫10zg+ς+2nς+vς/kk−1(1−z)hk−1dz, | (4.11) |
which upon Lemma (1.4) and Eq. (4.2) in (4.11), we get
ℑ=xϑ+ς+ξk+vςk2−1∞∑r=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+r−rk) |
×Γk(g+ς+2rς+vς/k)Γk(h)Γk(g+h+ς+2rς+vς/k)(x)2nς+rk−r, | (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,ς∈C,ℜ(v)>−1, and k∈R+ be such that ℜ(ω)>0, ℜ(ϑ)>max[ℜ(−ω−γ),ℜ(ξ−rk+r)], then the following formula holds true:
Bk((Dω,ξ,γ−,k(tω−ϑkSkv,c[(z/t)−ςk]))(x);g,h)=xω−ϑ−ς−ξk−vςk2√k(1/2)vk+1Γk(h) |
×4Ψk5[(ϑ+ς−ω+vςk−ξ,2ς+k−1),(ϑ+ς+vςk+γ,2ς),(v+3k2,k),(3k2,k),(ϑ+ς−ω+vςk,2ς), |
(g+ς+vςk,2ς),(k,k)(ϑ+ς−ω+vςk−ξ+γ,2ς),(g+h+ς+vςk,2ς)|−cx−2ς+1k−14]. | (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,ς∈C,ℜ(v)>−1, such that ℜ(ϑ)>max [0,ℜ(ξ−γ)], ℜ(ω)>0; then
B((Iω,ξ,γ0+(tϑ−1Hv[(zt)ς]))(x);g,h)=xϑ−ξ+(v+1)ς−1(1/2)v+1Γ(h) |
×4Ψ5[(ϑ+(v+1)ς,2ς),(ϑ+(v+1)ς−ξ+γ,2ς),(v+32,1),(32,1),(ϑ+(v+1)ς−ξ,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,ς∈C,ℜ(v)>−1, such that ℜ(ω+ϑ)> max[−ℜ(ξ),−ℜ(γ)], ℜ(ω)>0; then
B((Iω,ξ,γ−,(t−ω−ϑHv[(z/t)−ς]))(x);g,h)=x−ω−ϑ−ξ−(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)|−x−2ς4]. | (4.15) |
Corollary 4.7. Let ω,ξ,γ,ϑ,ς∈C,ℜ(v)>−1, be such that ℜ(ϑ)>max[0,ℜ(−ω−ξ−γ)], ℜ(ω)>0; then
B((Dω,ξ,γ0+(tϑ−1Hv[(zt)ς]))(x);g,h)=xϑ+ξ+(v+1)ς−1(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,ς∈C,ℜ(v)>−1, be such that ℜ(ω)>0, ℜ(ϑ)>max[ℜ(−ω−γ),ℜ(ξ)], then the following formula holds true:
B((Dω,ξ,γ−(tω−ϑHv[(z/t)−ς]))(x);g,h)=xω−ϑ−ς−ξ−vς(1/2)v+1Γ(h) |
×4Ψ5[(ϑ+(v+1)ς−ω−ξ,2ς),(ϑ+(v+1)ς+γ,2ς),(v+32,1),(32,1),(ϑ+(v+1)ς−ω,2ς), |
(g+(v+1)ς,2ς),(1,1)(ϑ+(v+1)ς−ω−ξ+γ,2ς),(g+(v+1)ς+h,2ς)|−x−2ς4]. | (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,ς∈C,ℜ(v)>−1, k∈R+ such that ℜ(ω)>0, then
Bk((Iω0+,k(tϑk−1Skv,c[(zt)ςk]))(x);g,h)=xϑ+ς+ωk+vςk2−1(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,ς∈C,ℜ(v)>−1, k∈R+ such that ℜ(ω)>0, then
Bk((Iω−,k(t−ω−ϑkSkv,c[(z/t)−ςk]))(x);g,h)=x−ϑ−ςk−vςk2(1/2)vk+1√kΓk(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)|−ckx−2ςk4]. | (4.19) |
Corollary 4.11. Let ω,γ,ϑ,ς∈C,ℜ(v)>−1, k∈R+ be such that ℜ(ω)>0, then
Bk((Dω0+,k(tϑk−1Skv,c[(zt)ςk]))(x);g,h)=xϑ+ς−ωk+vςk2−1(1/2)vk+1√kΓk(h) |
×3Ψk4[(ϑ+ς+vςk,2ς),(g+ς+vςk,2ς),(k,k)(v+3k2,k),(3k2,k),(ϑ+ς+vςk+ξ,2ς−k+1),(g+h+ς+vςk,2ς)|−cx2ς+1k−14]. | (4.20) |
Corollary 4.12. Let ω,γ,ϑ,ς∈C,ℜ(v)>−1, and k∈R+be such that ℜ(ω)>0, then
Bk((Dω−,k(tω−ϑkSkv,c[(z/t)−ςk]))(x);g,h)=x2ω−ϑ−ςk−vςk2√k(1/2)vk+1Γk(h) |
×3Ψk4[(ϑ+ς+vςk,2ς+k−1),(g+ς+vςk,2ς),(k,k)(v+3k2,k),(3k2,k),(ϑ+ς−ω+vςk,2ς),(g+h+ς+vςk,2ς)|−cx−2ς+1k−14]. | (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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