The Marichev-Saigo-Maeda Fractional Calculus Operators Pertaining to the Generalized K-Struve Function

Seema Kabra 1 , Harish Nagar 2 , Kottakkaran Sooppy Nisar 3 , and D.L. Suthar 4
  • 1 Department of Mathematics, Sangam University, Bhilwara, India
  • 2 Department of Mathematics, Sangam University, Bhilwara, India
  • 3 Department of Mathematics, College of Arts & Sciences, Wadi Aldawaser, Prince Sattam bin Abdulaziz University, Saudi Arabia
  • 4 Department of Mathematics, Wollo University, Dessie, P.O. Box:1145, Amahara Region, Ethiopia
Seema Kabra, Harish Nagar, Kottakkaran Sooppy Nisar
  • Department of Mathematics, College of Arts & Sciences, Wadi Aldawaser, Prince Sattam bin Abdulaziz University, Saudi Arabia
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and D.L. Suthar
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  • Department of Mathematics, Wollo University, Dessie, P.O. Box:1145, Amahara Region, Ethiopia
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Abstract

In the present paper, we establish some compositions formulas for Marichev-Saigo-Maeda (MSM) fractional calculus operators with k-Struve function Sν,ck as of the kernel. The results are presented in terms of generalized k-Wright function pΨqk .

1 Introduction and Preliminaries

The Wright function play an important role in the partial differential equation of fractional order which is familiar and extensively treated in papers by a number of authors including Gorenflo et al. [6].

For ςi, τj ∈ ℝ\{0} and ai,bj ∈ ℂ,i = (1̅, p); j = (1̅,q) the generalized form of Wright function is defined by Wright ([13,14,15,16,17]) as following:
pΨq(z)=pΨq[(ai,ςi)1,p(bj,τj)1,q|z]=n=0i=1pΓ(ai+nςi)j=1qΓ(bj+nτj)znn!,z,
where Γ (z) is the well-known Euler gamma function [4].The condition for existence of (1.1) with its depiction in terms of Mellin-Barnes integral and the H-function were obtained by Kilbas et al. [10].
The generalized form of the above Wright function (1.1) was given by Gehlot and Prajapati [5], named as generalized K-Wright function which is defined as
pΨqk(z)=pΨqk[(ai,ςi)1,p(bj,τj)1,q|z]=n=0i=1pΓk(ai+nςi)j=1qΓk(bj+nτj)znn!,z,
where k ∈ ℝ+ and (ai + i), (bj + j) ∈ ℂ\k for all n ∈ ℕ0. The generalized k-gamma function [3] is defined as
Γk(z)=0etkktz1dt;((z)>0;k+)
and
Γk(z)=limnn!kn(nk)zk1(z)n,k,k+,z\k
Also
Γk(z)=kzk1Γ(zk),
where (z)n,k is the k-Pochammer symbol introduced by Diaz and Pariguan [3] defined for complex z ∈ ℂ and k ∈ ℝ as
(z)n,k={1ifn=0,z(z+k)(z+2k)(z+(n1)k)ifn.}
On taking k = 1, then the generalized K-Wright function (1.2) diminishes to the generalized Wright function (1.1).

1.1 Saigo fractional calculus operators

Saigo [18] defined the fractional integral and differential operators with the Gauss hyergeometric function as kernel, which are remarkable generalizations of the Riemann-Liouville (R-L) and Erdélyi-Kober fractional calculus operators (see; [11]).

For ς, τ, γ ∈ ℂ and x ∈ ℝ+ with ℜ(ς) > 0, the left-hand and the right-hand sided generalized fractional integral operators connected with Gauss hypergeometric function are defined as below:
(I0+ς,τ,γf)(x)=xςτΓ(ς)0x(xt)ς12F1(ς+τ,γ;ς;1tx)f(t)dt
and
(Iς,τ,γf)(x)=1Γ(ς)x(tx)ς1tς+τ2F1(ς+τ,γ;ς;1xt)f(t)dt
respectively. Here, 2F1(ς, τ; γ; z) is the Gauss hypergeometric function [11] defined for z ∈ ℂ, |z| < 1 and ς, τ ∈ ℂ, γ\0 by
2F1(ς,τ;γ;z)=n=0(ς)n(τ)n(γ)nznn!,
where (z)n = (z)n,1. The corresponding fractional differential operators are
(D0+ς,τ,γf)(x)=(ddx)l(I0+ς+l,τl,ς+γlf)(x)
and
(Dς,τ,γf)(x)=(ddx)l(Iς+l,τl,ς+γf)(x)
where l = [ℜ (ς)] + 1 and [ℜ (ς)] is the integer part of ℜ(ς). Substituting τ = −ς and τ = 0 in equation (1.7) – (1.10), we get the corresponding R-L and Erdélyi-Kober fractional operators, respectively.

1.2 Marichev-Saigo-Maeda fractional operators

Marichev [13] was introduced and studied fractional calculus operators which are the generalization of the Saigo operators, later generalized by Saigo and Maeda [19]. For ς, ς′, τ, τ′, γ ∈ ℂ and x ∈ ℝ+ with ℜ(γ) > 0, the left-hand and right-hand sided MSM fractional integral and derivative operators associated with third Appell function F3 are defined as
(I0+ς,ς,τ,τ,γf)(x)=xςΓ(γ)0x(xt)γ1tςF3(ς,ς,τ,τ,γ,1tx,1xt)f(t)dt
and
(Iς,ς,τ,τ,γf)(x)=xςΓ(γ)x(tx)γ1tςF3(ς,ς,τ,τ,γ,1xt,1tx)f(t)dt
(D0+ς,ς,τ,τ,γf)(x)=(ddx)m(I0+ς,ς,τ+m,τ,γ+mf)(x)
and
(Dς,ς,τ,τ,γf)(x)=(ddx)m(Iς,ς,τ,τ+m,γ+mf)(x)
respectively, where m = [ℜ(γ)] + 1 and the third Appell function [17], is defined by
F3(ς,ς,τ,τ,γ;x,y)=m,n,=0(ς)m(ς)n(τ)m(τ)n(γ)m+nxmynm!n!,max{|x|,|y|}<1.

1.3 Generalized k-Struve function

The generalized k-Struve function was defined by Nisar et al. [14] as
Sν,ck(t)=n=0(c)nΓk(nk+ν+3k2)Γ(n+32)n!(t2)2n+νk+1(k+;c;ν>1)
taking k→ 1 and c = 1; (1.15) reduces to yield the well-known Struve function of order ν is defined by [1] as
Hν(t)=n=0(1)nΓ(n+ν+32)Γ(n+32)n!(t2)2n+ν+1
For more details about Struve functions, their generalizations and properties, the esteemed reader is invited to consider references [2, 7, 8, 14, 15, 20,21,22].

The following MSM integral operators are required here [19, p. 394] to obtain the MSM fractional integration of generalized k-Struve function.

Lemma 1

Let ς, ς′, τ, τ′, γ, ρ ∈ ℂ such that ℜ(ς) > 0

  1. (i)ℜ(ρ) > 0 max{0,ℜ(ς′ − τ′),ℜ(ς + ς′ + τγ)}, then
    (I0+ς,ς,τ,τ,γtρ1)(x)=Γ(ρ)Γ(ς+τ+ρ)Γ(ςςτ+γ+ρ)Γ(τ+ρ)Γ(ςς+γ+ρ)Γ(ςτ+γ+ρ)xςς+γ+ρ1
  2. (ii)If ℜ(ρ) > max{ℜ(τ),ℜ(−ςς′ + γ),ℜ(−ςτ′ + γ}, then
    (Iς,ς,τ,τ,γtρ)(x)=Γ(τ+ρ)Γ(ς+ςγ+ρ)Γ(ς+τγ+ρ)Γ(ρ)Γ(ςτ+ρ)Γ(ς+ς+τγ+ρ)xςς+γρ

Further, to obtain the MSM fractional differentiation of the generalized k-Struve function, following results will be used from [9] as below:

Lemma 2

Let ς, ς′, τ, τ′, γ, ρ ∈ ℂ, such that ℜ(ς) > 0;

  1. (i)If ℜ(ρ) > max{0,ℜ(−ς + τ),ℜ(−ςς′ − τ′ + γ)}
    (D0+ς,ς,τ,τ,γtρ1)(x)=Γ(ρ)Γ(τ+ς+ρ)Γ(ς+ς+τγ+ρ)Γ(τ+ρ)Γ(ς+ςγ+ρ)Γ(ς+τγ+ρ)xς+ςγ+ρ1
  2. (ii)If ℜ(ρ) > max{ℜ(−τ′),ℜ(ς′ + τγ),ℜ(ς + ς′ − γ) + [ℜ(γ)] + 1}, then
    (Dς,ς,τ,τ,γtρ)(x)=Γ(τ+ρ)Γ(ςς+γ+ρ)Γ(ςτ+γ+ρ)Γ(ρ)Γ(ς+τ+ρ)Γ(ςςτ+γ+ρ)xς+ςγρ

2 Fractional Calculus Approach

In this section, the following six theorems for k-Struve function concerning to MSM fractional integral and differential operators are established here as main results.

Theorem 1
Let ς, ς′, τ, tau′, γ, ρ ∈ ℂ and k ∈ ℝ+ be such that (γ)>0,(λk)>max{0,(ςτ),(ς+ς+τγ)} . Also let c ∈ ℝ; ν > −1, then for t > 0
(I0+ς,ς,τ,τ,γ(tλk1Sν,ck(t)))(x)=kγ+12xςς+γ+λk+νk2νk+1×3Ψ5k[(λ+ν+k,2k),(kς+kτ+λ+ν+k,2k),(kτ+λ+ν+k,2k),(kςkς+kγ+λ+ν+k,2k),(kςkςkτ+kγ+λ+ν+k,2k)(kςkτ+kγ+λ+ν+k,2k),(ν+3k2,k),(3k2,k)|cx2k4].
Proof
On using (1.16) and taking the left-hand sided MSM fractional integral operator inside the summation, the left-hand side of (2.1) becomes
=n=0(c)nΓk(nk+ν+3k2)Γ(n+32)n!22n+νk+1(I0+ς,ς,τ,τ,γ{tλk+νk+2n})(x),
Making use of (1.18), we obtain
=n=0(c)nxςς+γλk+νk+2nΓk(nk+ν+3k2)Γ(n+32)n!22n+νk+1Γ(λk+νk+2n+1)Γ(τ+λk+νk+2n+1)×Γ(ςςτ+γ+λk+νk+2n+1)Γ(ς+τ+λk+νk+2n+1)Γ(ςτ+γ+λk+νk+2n+1)Γ(ςς+γ+λkνk+2n+1),
Now, using equation (1.5) on above term, then we get
=xςς+γ+λk+νk2νk+1kγ12n=0Γk(λ+ν+k+2nk)Γk(kς+kτ+λ+ν+k+2nk)Γk(kτ+λ+ν+k+2nk)Γk(kςkς+kγ+λ+ν+k+2nk)×Γk(kςkςkτ+kγ+λ+ν+k+2nk)Γk(kςkτ+kγ+λ+ν+k+2nk)Γk(nk+ν+3k2)Γk(3k2+nk)n!(cx2k22)n.
Using the definition of (1.2) in the above term, we arrive at the result (2.1).

Next theorem gives the right-hand MSM fractional integration of Sν,ck(.) .

Theorem 2
Let ς, ς′, τ, τ′, γ, ρ ∈ ℂ and k ∈ ℝ+ be such that ℜ(γ) > 0, (γ)>0,(λk)>max{(τ),(ςς+γ),(ςτ+γ)} . Also let c ∈ ℝ; ν > −1, then for t > 0
(Iς,ς,τ,τ,γ(tλk1Sν,ck(t)))(x)=kγ+12xςς+γ+λk+νk2νk+1×3Ψ5k[(kτλν,2k),(kς+kςkγnu,2k),(λν,2k),(kςkτλν,2k),(kς+kτkγλν,2k)(kς+kς+kτkγλν,2k),(ν+3k2,k),(3k2,k)|cx2k4].
Proof
On using (1.16) and taking the right-hand sided MSM fractional integral operator inside the summation, the left hand side of (2.2) becomes
=n=0(c)nΓk(nk+ν+3k2)Γ(n+32)n!22n+νk+1(Iς,ς,τ,τ,γ{tλk+νk+2n})(x)
On using (1.19), we get
=n=0(c)nxςς+γ+λk+νk+2nΓk(nk+ν+3k2)Γ(n+32)n!22n+νk+1Γ(τλkνk2n)Γ(λkνk2n)×Γ(ς+τγλkνk2n)Γ(ς+ςγλkνk2n)Γ(ς+ς+τγλkνk2n)Γ(ςτλkνk2n)=n=0(cx2)nxςς+γ+λk+νkΓk(nk+ν+3k2)n!22n+νk+1Γ(τλkνk2n)Γ(ς+ςγλkνk2n)Γ(n+32)Γ(λkνk2n)Γ(ςτλkνk2n)×Γ(ς+τγλkνk2n)Γ(ς+ς+τγλkνk2n)=xςς+γ+λk+νk2νk+1kγ12n=0(ckx24)n1n!Γk(kτλν2nk)Γk(λν2nk)Γk(kςkτλν2nk)×Γk(kς+kςkγλν2nk)Γk(kς+kτkγλν2nk)Γk(kς+kς+kτkγλν2nk)Γk(nk+ν+3k2)Γk(3k2+nk)
and the result follows on making use of (1.5) and definition of generalized k-Wright function.
Theorem 3
Let ς, ς′, τ, τ′, γ, ρ ∈ ℂ and k ∈ ℝ+ be such that ℜ(γ) > 0, (γ)>0,(λk)>max{(τ),(ςς+γ),(ςτ+γ)} . Also let c ∈ ℝ; ν > −1, then for t > 0
(Iς,ς,τ,τ,γ(tλkSν,ck(t)))(x)=kγ12xςς+γ+νkλk+12νk+1×3Ψ5k[(kτ+λν,2k),(kς+kςkγ+λnuk,2k),(λνk,2k),(kςkτ+λνk,2k),(kς+kτkγ+λνk,2k)(kς+kς+kτkγ+λνk,2k),(ν+3k2,k),(3k2,k)|cx2k4].
Proof
On using (1.16) and taking the right-hand sided MSM fractional integral operator inside the summation, the left hand side of (2.3) becomes
=n=0(c)nΓk(nk+ν+3k2)Γ(n+32)n!22n+νk+1(Iς,ς,τ,τ,γ{tνλk+2n+1})
On using (1.19), we obtain
=(c)ntςς+γ+νkλk+2n+1Γk(nk+ν+3k2)Γ(n+32)n!22n+νk+1Γ(τ+λνk2n1)Γ(λnuk2n1)×Γ(ς+ςγ+λνk2n1)Γ(ς+τγ+λνk2n1)γ(ςτ+λνk2n1)Γ(ς+ς+τγ+λνk2n1)
Making use of (1.5), we get
=xςς+γ+νkλk+1kγ+122νk+1n=0(ckx2)n4nn!Γk(kτ+λνk2nk)Γk(λνk2nk)Γk(kςkτ+λνk2nk)×Γk(kς+kςkγ+λνk2nk)Γk(kς+kτkγ+λνk2nk)Γk(kς+kς+kτkγ+λνk2nk)Γk(nk+ν+3k2)Γk(nk+3k2)
This on expressing in terms of k-Wright function pΨqk using (1.2) leads to the right-hand side of (2.3). This completes the proof of theorem.

The next theorem obtains the left-hand sided MSM fractional differentiation of k-Struve function.

Theorem 4
Let ς, ς′, τ, τ′, γ, ρ ∈ ℂ and k ∈ ℝ+ be such that (λk)>max{0,(ς+τ),(ςςτ+γ)} . Also let c ∈ ℝ; ν > −1, then for t > 0
(D0+ς,ς,τ,τ,γ(tλk1Sν,ck(t)))(x)=kγ+12xς+ςγ+λk+νk2νk+1×3Ψ5k[(λ+ν+k,2k),(kτ+kς+λ+ν+k,2k),(kτ+λ+ν+k,2k),(kς+kςkγ+λ+ν+k,2k),(kς+kς+kτkγ+λ+ν+k,2k)(kς+kτkγ+λ+ν+k,2k),(ν+3k2,k),(3k2,k)|cx2k4].
Proof
On using (1.16) and taking the left-hand sided MSM fractional derivative inside the summation, the left-hand side of (2.4) becomes
=n=0(c)nΓk(nk+ν+3k2)Γ(n+32)n!2νk+2n+1(D0+ς,ς,τ,τ,γ(tλk+νk+2n))
Using (1.20) in above term, we obtain
=n=0(c)nΓ(λk+νk+2n+1)Γ(τ+ς+λk+νk+2n+1)Γk(nk+ν+3k2)Γ(n+32)n!2νk+2n+1Γ(τ+λk+νk+2n+1)×Γ(ς+ς+τγ+λk+νk+2n+1)Γ(ς+ςγ+λk+νk+2n+1)Γ(ς+τγ+λk+νk+2n+1)xς+ςγ+λk+νk+2n=xς+ςγλk+νk2νk+1n=0(cx2)nn!4nΓk(nk+ν+3k2)Γ(λk+νk+2n+1)Γ(n+32)Γ(τ+λk+νk+2n+1)×Γ(τ+ς+λk+νk+2n+1)Γ(ς+ς+τγ+λk+νk+2n+1)Γ(ς+ςγ+λk+νk+2n+1)Γ(ς+τγ+λk+νk+2n+1)=kγ+12xς+ςγ+λk+νk2νk+1n=0(ckx2)nn!4n×Γk(λ+ν+k+2nk)Γk(kτ+kς+λ+ν+k+2nk)Γk(nk+ν+3k2)Γk(nk+3k2)Γk(kτ+λ+ν+k+2nk)×Γk(kς+kς+kτkγ+λ+ν+k+2nk)Γk(kς+kςkγ+λ+ν+k+2nk)Γk(kς+kτkγ+λ+ν+k+2nk)
In above term, we use equation (1.5), and the result follows by using (1.2), then we arrive at (2.4).

The next theorem gives the right-hand sided MSM fractional derivative of k-Struve function.

Theorem 5
Let ς, ς′, τ, τ′, γ, ρ ∈ ℂ and k ∈ ℝ+ be such that (λk)>max{(τ),(ς+τγ),(ς+ςγ)+[(γ)]+1} . Also let c ∈ ℝ; ν > −1, then for t > 0
(Dς,ς,τ,τ,γ(tλk1Sν,ck(t)))(x)=kγ+12xς+ςγ+λk+νk2νk+1×3Ψ5k[(kτλν,2k),(kςkς+kγλν,2k),(λν,2k),(kς+kτλν,2k),(kς+kτ+kγλν,2k)(kςkςkτ+kγλν,2k),(ν+3k2,k),(3k2,k)|cx2k4].
Proof
On using (1.16) and taking the left-hand sided MSM fractional derivative inside the summation, the left-hand side of (2.5) becomes
=n=0(c)nΓk(nk+ν+3k2)Γ(n+32)n!2νk+2n+1(Dς,ς,τ,τ,γ(tλk+νk+2n))
Using (1.21) in above term, we obtain
=n=0(c)nΓ(τλkνk2n)Γk(nk+ν+3k2)Γ(n+32)n!2νk+2n+1Γ(λkνk2n)×Γ(ςς+γλkνk2n)Γ(ςτ+γλkνk2n)Γ(ςςτ+γλkνk2n)Γ(ς+τλkνk2n)xς+ςγ+λk+νk+2n=xς+ςγλk+νk2νk+1n=0(cx2)nn!4nΓk(nk+ν+3k2)Γ(τλkνk2n)Γ(λkνk2n)×Γ(ςς+γλkνk2n)Γ(ςτ+γλkνk2n)Γ(ς+τλkνk2n)Γ(ςςτ+γλkνk2n)=kγ+12xς+ςγ+λk+νk2νk+1n=0(ckx2)nn!4n×Γk(kτλν2nk)Γk(kςkς+kγλν2nk)Γk(λν2nk)Γk(kς+kτλν2nk)×Γk(kςkτ+kγλν2nk)Γk(kςkςkτ+kγλν2nk)tΓk(ν+3k2+nk)Γk(3k2+nk)
Thus, in accordance with (1.2), we get the required result (2.5).
Theorem 6
Let ς, ς′, τ, τ′, γ, ρ ∈ ℂ and k ∈ ℝ+ be such that (λk)>max{(τ),(ς+τγ),(ς+ςγ)+[(γ)]+1} . Also let c ∈ ℝ; ν > −1, then for t > 0
(Dς,ς,τ,τ,γ(tλkSν,ck(t)))(x)=kγ+12xς+ςγ+λk+νk+12νk+1×3Ψ5k[(kτ+λνk,2k),(kςkς+kγλνk,2k),(λνk,2k),(kς+kτ+λνk,2k),(kςkτ+kγ+λνk,2k)(kςkςkτ+kγ+λνk,2k),(ν+3k2,k),(3k2,k)|cx2k4].
Proof
On using (1.16) and taking the right-hand sided MSM fractional derivative inside the summation, the left-hand side of (2.6) becomes
=n=0(c)nΓk(nk+ν+3k2)Γ(n+32)n!2νk+2n+1(Dς,ς,τ,τ,γ(tνkλk+2n+1))
Using (1.21), we have
=n=0(c)nΓk(nk+ν+3k2)Γ(n+32)n!2νk+2n+1Γ(τ+λνk2n1)Γ(λνk2n1)×Γ(ςς+γ+λνk2n1)Γ(ςτ+γ+λνk2n1)Γ(ς+τ+λνk2n1)Γ(ςςτ+γ+λνk2n1)xς+ςγ+νkλk+2n+1
Making use of (1.5), we obtain
=tς+ςγ+νkλk+12νk+1n=0(ckt2)nn!4nkγ12×Γk(kτ+λνk2nk)Γk(λνk2nk)Γk(kς+kτ+λνk2nk)×Γk(kςkς+kγ+λνk2nk)Γk(kςkτ+kγ+λνk2nk)Γk(kςkςkτ+kγ+λνk2nk)Γk(nk+ν+3k2)Γk(nk+3k2)
This on expressing in terms of k-Wright function pΨqk using (1.2) leads to the right-hand side of (2.6). This completes the proof.

3 Concluding Remark

MSM fractional calculus operators have more advantage due to the generalize of Riemann-Liouville, Weyl, Erdélyi-Kober, and Saigo's fractional calculus operators; therefore, many authors are called as general operator. Now we are going to conclude of this paper by emphasizing that our leading results (Theorems 1 – 6) can be derived as the specific cases involving familiar fractional calculus operators as above said. On other hand, the k Struve function defined in (1.16) possesses the lead that a number of special functions occur to be the particular cases. Some of special cases respect to the integrals relating with k Struve function have been discovered in the earlier research works by various authors with not the same arguments.

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    Suthar D.L., Purohit S.D. and Nisar K.S., Integral transforms of the Galue type Struve function. TWMS J. Appl. Eng. Math., 8(1), (2018), 114–121.

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    Yagmur N. and Orhan H., Starlikeness and convexity of generalized Struve functions, Abstr. Appl. Anal., 2013, Art. ID 954513, pp. 6.

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