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ORIGINAL RESEARCH article

Front. Phys., 14 August 2018
Sec. Statistical and Computational Physics
This article is part of the Research Topic Fractional Calculus and its Applications in Physics View all 10 articles

A Remark on the Fractional Integral Operators and the Image Formulas of Generalized Lommel–Wright Function

  • 1Department of Mathematics, Malaviya National Institute of Technology, Jaipur, India
  • 2Department of Mathematics, Faculty of Applied Science, Madhav Institue of Technology & Sciences, Gwalior, India
  • 3Department of Mathematics, Texas A&M University, Kingsville, TX, United States
  • 4Department of Mathematics, Çankaya University, Ankara, Turkey
  • 5Institute of Space Science, Bucharest, Romania

In this paper, the operators of fractional integration introduced by Marichev-Saigo-Maeda involving Appell's function F3(·) are applied, and several new image formulas of generalized Lommel–Wright function are established. Also, by implementing some integral transforms on the resulting formulas, few more image formulas have been presented. We can conclude that all derived results in our work generalize numerous well-known results and are capable of yielding a number of applications in the theory of special functions.

Primary: 44A20 Transforms of special functions; 65R10 For numerical methods; 26A33 Fractional derivatives and integrals; Secondary: 33C20 Generalized hypergeometric series, pFq; 33E50 Special functions in characteristic p (gamma functions, etc.); 2010 AMS classification by MathSciNet

1. Introduction and Preliminaries

Fractional calculus (FC) represents a complex physical phenomenon in a more accurate and efficient way than classical calculus. In recent years, many researchers [17] have used fractional order integral models in real-world problems in various fields of science and technology. There exists several definitions of fractional order integrals in the literature that can be used to solve the fractional integral equations involving special functions. For an exhaustive literature review, one may refer to the paper by Srivastava and Saxena [8].

The generalized functions such as Bessel, Lommel, Struve, and Lommel–Wright functions have originated from concrete problems in applied fields of sciences viz mechanics, physics, engineering, astronomy, etc.

The generalized Lommel–Wright function Jω,ϑφ,m(z) is defined by de'Oteiza et al. [9] and is represented in the following manner:

Jω,ϑφ,m(z)=(z/2)ω+2ϑk=0(1)k(z/2)2k(Γ(ϑ+k+1))mΓ(ω+kφ+ϑ+1)             =(z/2)ω+2ϑ1ψm+1[(1,1);  (ϑ+1,1)mtimes,  (ω+ϑ+1,φ);z2/4]    z\(,0],  φ>0,  m,   ω,ϑ,    (1.1)

where pψq denotes the Fox–Wright generalized hypergeometric function which is defined as given in Srivastava and Karlsson [10, p. 21] and Kilbas et al. [11, P. 56]

pψq((a1,A1),...(ap,Ap);(b1,B1),...,(bq,Bq);z)=n=0Πj=1pΓ(aj+nAj)Πj=1qΓ(bj+nBj)znn!,    (1.2)

where ai, bj ∈ ℂ and Ai, Bj ∈ ℝ = (−∞, ∞); Ai, Bj ≠ 0, i = 1, 2, …, p, j = 1, 2, …, q, j=1qBj-j=1pAj>-1.

A useful generalization of the Lommel–Wright function and its special cases, Jωφ(z)Jω,ϑφ(z), depending on the arbitrary fractional parameter φ > 0 presents a fractional order extension of the Bessel function Jω(z).

Prieto et al. [12] studied some useful results in the theory of fractional calculus operators of generalized Lommel–Wright function. The convergence of series involving generalized Lommel–Wright function was studied by Konovska [13].

When m = 1, the following generalization of the Bessel function, introduced by Pathak [14] is obtained as a special case of generalized Lommel–Wright function (1.1) (see e.g., [15, p. 353]):

Jω,ϑφ(z)=Jω,ϑφ,1(z)             =(z2)ω+2ϑk=0(1)k( z2 )2kΓ(ϑ+k+1)Γ(ω+kφ+ϑ+1),          z\(,0],  φ>0,   ω,ϑ.    (1.3)

On taking m = 1, φ = 1, and ϑ=12 in (1.1), we obtain the Struve function Hω(·) (see e.g., [16, p. 28, Equation (1.170)])

Hω(z)=Jω,1/21,1          =(z2)ω+1k=0(1)k(z2)2kΓ(k+32)Γ(k+ω+32)z,ω.    (1.4)

If we take m = 1, φ = 1, and ϑ = 0 in (1.1), it gives the relationship with the Bessel function as follows (see e.g., [16, p. 27, Equation (1.161)]):

Jω(z)=Jω,01,1(z)=k=0(1)k(z/2)ω+2kΓ(ω+k+1)k!              z,ω,  z0,  (ω)>1.    (1.5)

A generalization of the hypergeometric fractional integrals, including the Saigo operators [17, 18] has been introduced by Marichev [19]. The details of these fractional operators have been found in Samko et al. [5, p. 194, Equation (10.47)] and later extended and studied by Saigo and Maeda [20, p. 393, Equation (4.12) and Equation (4.13)] in terms of complex order Appell function F3(·) of two variables (see [10, p. 23]) in the kernel

F3(ζ,ζ,ϱ,ϱ;η;x;y) =m,n=0(ζ)m(ζ)n(ϱ)m(ϱ)n(η)m+nxmm!ynn!,  (max{|x|,|y|}<1).    (1.6)

The Appell function F3 reduces to the Gauss hypergeomatric function 2F1 and satisfies the system of two linear partial differential equations of the second order as follows (see [10, p. 301, Equation 9.4]):

F3(ζ,ηζ,ϱ,ηϱ;η;x;y)=  2F1(ζ,ϱ;η;x+yxy).    (1.7)

Further, it is easy to see that

F3(ζ,0,ϱ,ϱ,η;x,y)=  2F1(ζ,ϱ;η;x)    (1.8)

and

F3(0,ζ,ϱ,ϱ,η;x,y)= 2F1(ζ,ϱ;η;y).    (1.9)

In this paper, we develop and study the image formulas involving the generalized Lommel–Wright function using fractional calculus integral operators. We use the generalized Marichev-Saigo-Maeda fractional integral operators, involving the Appell function, defined as follows:

(I0+ζ,ζ,ϱ,ϱ,κf)(x)=xζΓκ0x(xt)κ1tζ                                 ×F3(ζ,ζ,ϱ,ϱ;κ;1tx,1xt)f(t)dt,                                  (κ)>0,ζ,ζ,ϱ,ϱ,κ,x>0    (1.10)

and

(I0ζ,ζ,ϱ,ϱ,κf)(x)     =xζΓκ x (tx)  κ1tζ F3 (ζ,ζ,ϱ,ϱ;κ;1xt,1tx) f(t)dt,            (κ)>0,ζ,ζ,ϱ,ϱ,κ,x>0.    (1.11)

respectively.

The power functions of left-hand sided and right-hand sided Marichev-Saigo-Maeda fractional integral operators as given in the Equations (1.10) and (1.11) (see Saigo et al. [6, 20]) are given by

(I0+ζ,ζ,ϱ,ϱ,κtχ1)(x)   =Γ(χ)Γ(χ+κζζϱ)Γ(χ+ϱζ)Γ(χ+ϱ)Γ(χ+κζζ)Γ(χ+κζϱ)xχ+κζζ1,    (1.12)

where ζ, ζ′, ϱ, ϱ′, κ ∈ ℂ, x>0 and if ℜ(κ) > 0, ℜ(χ)>max{0, ℜ(ζ + ζ′ + ϱ − κ), ℜ(ζ′ − ϱ′)}.

(I0ζ,ζ,ϱ,ϱ,κtχ1)(x) =Γ(1χκ+ζ+ζ)Γ(1χ+ζ+ϱκ)Γ(1χϱ)Γ(1χ)Γ(1χ+ζ+ζ+ϱ+ϱκ)Γ(1χ+ζϱ)      ×xχζζ+κ1,    (1.13)

where ζ, ζ′, ϱ, ϱ′, κ ∈ ℂ are such that ℜ(κ) > 0 and ℜ(χ) < 1 + min{ℜ(−ϱ), ℜ(ζ + ζ′ − κ), ℜ(ζ + ϱ′ − κ)}.

1.1. Relation Among the Operators

In this section, we recall some relationships between the fractional integral operators.

If we set ζ′ = 0 then in view of the formula (1.8), the relationship between Marichev-Saigo-Maeda and the Saigo fractional integral operators is found by Saxena and Saigo [6, p. 93, Equation (2.15)] as

(I0,xζ,0,ϱ,ϱ,ηf)(x)=(I0,xη,ζη,ϱf)(x), ((η)>0)    (1.14)

and

(Ix,ζ,0,ϱ,ϱ,ηf)(x)=(Ix,η,ζη,ϱf)(x), ((η)>0),    (1.15)

where the general operators I0,xζ,0,ϱ,ϱ,η and I0,xζ,0,ϱ,ϱ,η reduce, respectively, to the Saigo operators I0,xζ,ϱ,η and Ix,ζ,ϱ,η [17] defined as follows:

(I0,xζ,ϱ,ηf)(x)=xζϱΓ(ζ)0x(xt)2ζ1                        ×F1(ζ+ϱ,η;ζ;1tx)f(t)dt, ((ζ)>0)    (1.16)

and

(Ix,ζ,ϱ,ηf)(x)=x(tx)ζ1tζϱ2                         ×F1(ζ+ϱ,η;ζ;1xt)f(t)dt, ((ζ)>0)    (1.17)

where integrals in (1.16) and (1.17) exist.

Let ζ, ϱ, η, χ ∈ ℂ with ℜ(ζ) > 0. Then the following power function formulas involving the Saigo operators hold true:

(I0,xζ,ϱ,ηtχ1)(x)=Γ(χ)Γ(χ+ηϱ)Γ(χϱ)Γ(χ+η+ζ)xχϱ1,                        (χ)>max{0,(ϱη)}    (1.18)

and

(Ix,ζ,ϱ,ηtχ1)(x)=Γ(1χ+ϱ)Γ(1χ+η)Γ(1χ)Γ(1χ+ζ+ϱ+η)xχϱ1,                   ((χ)<1+min{(ϱ),(η)}).    (1.19)

On replacing ϱ = −ζ in the operators I0,xζ,ϱ,η(·) and Ix,ζ,ϱ,η(·), these reduce to the Riemann-Liouville and the Weyl fractional integral operators, respectively, by means of the following relationships (see Kilbas [11]):

(R0,xζf)(x)=(I0,xζ,ζ,ηf)(x)    (1.20)

and

(Wx,ζf)(x)=(Ix,ζ,ζ,ηf)(x).    (1.21)

The Riemann-Liouville fractional integral operator and the Weyl fractional integral operator are defined as follows (see e.g., [21]):

(R0,xζf)(x)=1Γ(ζ)0x(xt)ζ1f(t)dt, ((ζ)>0)    (1.22)

and

(Wx,ζf)(x)=1Γ(ζ)0x(tx)ζ1f(t)dt, ((ζ)>0),    (1.23)

provided both the integrals converge.

The operators I0,xζ,ϱ,η(·) and Ix,ζ,ϱ,η(·) reduce to Erdélyi–Kober fractional integral operators on setting ϱ = 0 as follows:

(E0,xζ,ηf)(x)=(I0,xζ,0,ηf)(x),    (1.24)

and

(Kx,ζ,ηf)(x)=(I0,xζ,0,ηf)(x),    (1.25)

where the Erdélyi–Kober type fractional integral operators are defined as follows (see [22]):

(E0,xζ,ηf)(x)=xζηΓ(ζ)0x(xt)ζ1tηf(t)dt, ((ζ)>0)    (1.26)

and

(Kx,ζ,ηf)(x)=xηΓ(ζ)x(tx)ζ1tζηf(t)dt, ((ζ)>0),    (1.27)

The function f(t) is constrained so that both the defining integrals (1.26) and (1.27) converge.

The Beta transform (see, e.g.[23]) of a complex valued function f(t) of a real variable t is defined as follows:

B{f(t):a,b}=01ta1(1t)b1f(t) dt,                           (t)>0, (a),(b)>0.    (1.28)

Beta transform of the power function tχ−1 is given by:

B{tχ1;a,b}=01ta+χ2(1t)b1dt                        =Γ(a+χ1)Γ(b)Γ(a+χ+b1), (t)>0, (a),(b)>0.    (1.29)

The Pδ - transform of a complex valued function f(t) of a real variable t denoted by Pδ[f(t); s] is a function F(s) of a complex variable s, valid under certain conditions on f(t), (given below is defined as (see Kumar [24])

Pδ[f(t);s]=F(s)=0[1+(δ1)s] tδ1f(t)dt, δ>1.    (1.30)

Here f(t) as a function is integrable over any finite interval (a, b), 0 < a < t < b; there exists a real number c such that

(i) if b > 0 is arbitrary, then bΥe-ctf(t)dz tends to a finite limit as Υ → ∞

(ii) for arbitrary a > 0, ωa|f(t)dt| tends to a finite limit as ω → 0+, then the Pδ-transform Pδ[f(t); s] exists for (ln[1+(δ-1)s]δ-1)>c for s ∈ ℂ.

Pδ - transform of the power function tχ−1 is given by

Pδ[zχ1;s]={δ1ln[1+(δ1)s]}χΓ(χ),                           χ,  (χ)>0,  δ>1.    (1.31)

Pδ-transform has found many applications. The pathway transforms are the paths going from the binomial form ln[1+(δ-1)s]-tδ-1 to the exponential from est. In Pδ-transform, the variable t is shifted from the binomial factor ln[1+(δ-1)s]-tδ-1 to the exponent, Hence, this form is more suitable for obtaining translation, convolution, etc. Recently, Agarwal et al. [25] found the solution of non-homogeneous time fractional heat equation and fractional Volterra integral equation using integral transform of pathway type. Also, Srivastava et al. [26] and [27] found some results involving generalized hypergeometric function and generalized incomplete gamma function by using Pδ-transform.

If we take δ → 1 in Equation (1.30), the Pδ-transform reduces to Laplace integral transform (Sneddon [23]):

L[f(t);s]=0ets f(t)dz;,  (s)>0.    (1.32)

The following relationship between the Pδ-transform is defined by (1.30) and the classical Laplace transform is defined by (1.32)

Pδ[f(t):s]=L[f(t):ln[1+(δ1)s]δ1],  (δ>1)    (1.33)

or, equivalently,

L[f(t):s]=Pδ[f(t):e(δ1)s1δ1],  (δ>1),    (1.34)

which can be applied to convert the table of Laplace transforms into the corresponding table of Pδ-transforms and vice versa.

The following integral formula involving the Whittaker function (see Mathai et al. [16, p. 56]) is used in finding the image formula:

0tτ1e 12Wσ,η(t) dt=Γ(τ+η+12)Γ(τη+12)Γ(τσ+12),                          (σ,  (τ±η)>1/2).    (1.35)

The Whittaker function (see e.g., Mathai et al. [16, p. 22]) is defined by

Wσ,η(z)=Γ(2η)Γ(12ση)Mσ,η(z)+Γ(2η)Γ(12σ+η)Mσ,η(z)             =Wσ,η(z),                             σ,  (1/2+η±σ)>0    (1.36)

where

Mσ,η(z)=zη+12ez21F1(12σ+η;2η+1;z),                      (1/2+η±σ)>0, |argz|<π.    (1.37)

2. Image Formula Associated With Fractional Integral Operators

Here, we establish image formulas for the generalized Lommel–Wright function involving Saigo-Maeda fractional integral operators (1.10) and (1.11), in terms of the Fox–Wright function.

Theorem 2.1. Let ζ, ζ′, ϱ, ϱ′, κ, ϑ ∈ ℂ, m ∈ ℕ, φ > 0 and x > 0 be such that

                                            (κ)>0, (ω)>1,(χ+ω)>max{0,(ζ+ζ+ϱκ),(ζϱ)}    (2.1)

then there holds the formula

I0+ζ,ζ,ϱ,ϱ,κ[tχ1Jω,ϑφ,m(tz)](x)=xAζζ+κ1(z2)ω+2ϑ4ψ4+m[     (A,2),(A+κζζϱ,2),(A+ϱζ,2),(1,1)(A+ϱ,2),(A+κζζ,2),(A+κζϱ,2),(ω+ϑ+1,φ),(ϑ+1,1)mtimes|(zx)24]    (2.2)

where A = χ + ω + 2ϑ.

Proof: Under the conditions stated with the Theorem 2.1, by taking the fractional integral of (1.1) using the equation (1.10) therein and changing the order of integration and summation, we get

I0+ζ,ζ,ϱ,ϱ,κ[tχ1Jω,ϑφ,m(tz)](x)         =k=0(1)k(z2)ω+2ϑ+2kΓ(k+1)(Γ(ϑ+k+1))mΓ(ω+kφ+ϑ+1)k!               ×I0+ζ,ζ,ϱ,ϱ,κ(tω+2ϑ+2k+χ1)(x)    (2.3)

Further, applying the result (1.12) with χ replaced by χ + ω + 2ϑ + 2k, we obtain

I0+ζ,ζ,ϱ,ϱ,κ[tχ1Jω,ϑφ,m(tz)](x)=xAζζ+κ1(z2)ω+2ϑk=0(1)kΓ(A+2k)Γ(k+1)Γ(A+ϱ+2k)(Γ(ϑ+1+k))m     ×Γ(A+κζζϱ+2k)Γ(A+ϱζ+2k)Γ(A+κζϱ+2k)Γ(ω+ϑ+1+φk)Γ(A+κζζ+2k)     ×(zx)2k4kk!    (2.4)

Here A = χ + ω + 2ϑ.

Interpreting the right-hand side of the above equation, in view of the definition (1.2), we arrive at the result (2.2).

Theorem 2.2. Let ζ, ζ′, ϱ, ϱ′, κ, ϑ ∈ ℂ, m ∈ ℕ, φ > 0 and x > 0 be such that

                             (κ)>0,  (ω)>1,(χω)>1+min{(ϱ),(ζ+ζκ),(ζ+ϱκ)}    (2.5)

then there holds the formula

I0ζ,ζ,ϱ,ϱ,κ[tχ1Jω,ϑφ,m(z/t)](x)=xκζζA(z2)ω+2ϑ4ψ4+m[   (Aκ+ζ+ζ,2),(A+ζ+ϱκ,2),        (Aϱ,2),(1,1)(A,2)(A+ζ+ζ+ϱκ,2),(A+ζϱ,2),            (ω+ϑ+1,φ),(ϑ+1,1)mtimes|z24x2]     (2.6)

where A = 1 − χ + ω + 2ϑ.

Proof: Under the conditions stated with the Theorem 2.2, on making use of the definitions (1.11) and (1.1) and changing the order of integration and summation, we have

I0ζ,ζ,ϱ,ϱ,κ[tχ1Jω,ϑφ,m(z/t)](x)      =k=0(1)k(z2)ω+2ϑ+2kΓ(k+1)(Γ(ϑ+k+1))mΓ(ω+kφ+ϑ+1)k!           ×I0ζ,ζ,ϱ,ϱ,κ(tχω2ϑ2k1)(x)    (2.7)

Here, on applying the formula (1.13) with χ replaced by χ − ω − 2ϑ − 2k, we obtain

I0ζ,ζ,ϱ,ϱ,κ[tχ1Jω,ϑφ,m(z/t)](x)    =xκζζA(z2)ω+2ϑk=0(1)kΓ(Aϱ+2k)Γ(A+2k)(Γ(ϑ+k+1))m        ×Γ(k+1)Γ(Aκ+ζ+ζ+2k)Γ(A+ζ+ϱκ+2k)Γ(A+ζϱ+2k)Γ(ω+kφ+ϑ+1)Γ(A+ζ+ζ+ϱκ+2k)        ×(z)2k(4x2)kk!    (2.8)

where A = 1 − χ + ω + 2ϑ.

So in view of the definition of the generalized Lommel–Wright function given by (1.1), the Equation (2.8) leads to the result (2.6).

For m = 1 and in the light of Equation (1.3), Theorem 2.1 leads to the following corollaries:

Corollary 2.1. Under the conditions stated with the Equation (2.1), the following image formula

I0+ζ,ζ,ϱ,ϱ,κ[tχ1Jω,ϑφ,1(zt)](x)=xAζζ+κ1(z2)ω+2ϑ    ×4ψ5  [      (A,2),(A+κζζϱ,2),      (A+ϱζ,2),(1,1)(A+ϱ,2),(A+κζζ,2),(A+κζϱ,2),     (ω+ϑ+1,φ),(ϑ+1,1)|(zx)24]           (2.9)

A = χ + ω + 2ϑ, for generalized Bessel function Jω,ϑφ,1(zt) holds true.

Corollary 2.2. Under the conditions stated with the Equation (2.5), the image formula

I0ζ,ζ,ϱ,ϱ,κ[tχ1Jω,ϑφ,1(z/t)](x)    =xκζζA(z2)ω+2ϑ    ×4ψ5[(Aκ+ζ+ζ,2),(A+ζ+ϱκ,2),      (Aϱ,2),(1,1)(A,2)(A+ζ+ζ+ϱκ,2),(A+ζϱ,2),     (ω+ϑ+1,φ),(ϑ+1,1)|z24x2]    (2.10)

A = 1 − χ + ω + 2ϑ, for generalized Bessel function Jω,ϑφ,1(z/t) holds true.

If we take m = 1, φ = 1, and ϑ=12 in (2.2), then we obtain the corresponding results for the Struve function Hω(·) [16] as

Corollary 2.3. Under the conditions stated with the Equation (2.1), the following image formula

I0+ζ,ζ,ϱ,ϱ,κ[tχ1Hω(zt)](x)    =xAζζ+κ1(z2)ω+1    ×4ψ5[    (A,2),(A+κζζϱ,2),         (A+ϱζ,2),(1,1)   (A+ϱ,2),(A+κζζ,2),(A+κζϱ,2),(ω+32,1),(32,1)|(zx)24]    (2.11)

A = χ + ω + 1, for Struve function Hω(zt) holds true.

Corollary 2.4. Under the conditions stated with the Equation (2.5), the following image formula

I0ζ,ζ,ϱ,ϱ,κ[tχ1Hω(z/t)](x)    =xχωζζ+κ2(z2)ω+1    ×4ψ5[  (Aκ+ζ+ζ,2),(A+ζ+ϱκ,2),     (Aϱ,2),(1,1)(A,2)(A+ζ+ζ+ϱκ,2),(A+ζϱ,2),     (ω+32,1),(32,1)|z24x2]    (2.12)

where A = 2 − χ + ω, for Struve function Hω(z/t) holds true.

2.1. Special Cases

(1) On taking φ = 1, m = 1, ϑ = 0, and z = 1 in Theorem 2.1, we obtain the image formula for the Bessel function considered by Purohit et al. [28, Theorem 1].

Corollary 2.5. Under the conditions stated with the Equation (2.1), the following image formula

I0+ζ,ζ,ϱ,ϱ,κ[tχ1Jω(t)](x)     =xχ+ωζζ+κ12ω    ×3ψ4[(χ+ω,2),(χ+ω+κζζϱ,2),                    (χ+ω+ϱζ,2)(χ+ω+ϱ,2),(χ+ω+κζζ,2),   (χ+ω+κζϱ,2),(ω+1,1)|x24]    (2.13)

for Bessel function Jω(t) holds true.

(2) Further, on taking φ = 1, m = 1, and ϑ = 0 in Theorem 2.2, we arrive the right-sided image formula for the Bessel function considered by Purohit et al. [28, Theorem 2].

Corollary 2.6. Under the conditions stated with the Equation (2.5), the image formula

I0ζ,ζ,ϱ,ϱ,κ[tχ1Jω(1/t)](x)     =xκζζ1+χω2ω      ×3ψ4[ (1χ+ωκ+ζ+ζ,2),(1χ+ω+ζ+ϱκ,2),(1χ+ωϱ,2)(1χ+ω,2)(1χ+ω+ζ+ζ+ϱκ,2),(1χ+ω+ζϱ,2),(ω+1,1)|14x2]    (2.14)

for Bessel function Jω(1/t) holds true.

3. Image Formulas Associated With Integral Transforms

In this section, we obtain the theorem involving the results obtained in previous sections associated with the integral transforms such as Beta transform, pathway transform, Laplace transform, and Whittaker transform.

3.1. Image Formulas for Beta Transform

Theorem 3.1. Let ζ, ζ′, ϱ, ϱ′, κ, ϑ ∈ ℂ, m ∈ ℕ, φ > 0, and x > 0 be such that

   (l)>0, (n)>0 (κ)>0, (ω)>1,(χ+ω)>max{0,(ζ+ζ+ϱκ),(ζϱ)}    (3.1)

then the following Beta transform formula holds:

B[I0+ζ,ζ,ϱ,ϱ,κ(tχ1Jω,ϑφ,m(tz))(x):l,n]    =xAζζ+κ1Γ(n)2ω+2ϑ     5ψ5+m[                     (A,2),(A+κζζϱ,2),                     (A+ϱζ,2),(Cn,2)(1,1)(A+ϱ,2),(A+κζζ,2),(A+κζϱ,2),                    (ω+ϑ+1,φ),(C,2),(ϑ+1,1)mtimes|x24]    (3.2)

Here A = χ + ω + 2ϑ and C = l + ω + 2ϑ + n.

Proof: For our convenience, let the left-hand side of the formula (3.2) be denoted by ς. Applying (1.28) to Equation (3.2), we get

ς=01 zl1(1z)n1[I0+ζ,ζ,ϱ,ϱ,κ( tχ1Jω,ϑφ,m(tz) )(x)]  dz.

Here, applying Equation (2.2) to the integral, we obtain the following expression

ς=01zl1(1z)n1zω+2ϑxAζζ+κ12ω+2ϑ    ×k=0(1)kΓ(A+2k)Γ(k+1)Γ(A+ϱ+2k)Γ(A+κζζ+2k)    ×Γ(A+ϱζ+2k)Γ(A+κζζϱ+2k)Γ(A+κζϱ+2k)Γ(ω+ϑ+1+φk)(Γ(ϑ+1+k))m    ×(zx2)k4kk!dz

Here A = χ + ω + 2ϑ.

Interchanging the order of integration and summation, we have

ς=xAζζ+κ12ω+2ϑk=0Γ(A+2k)Γ(A+κζζϱ+2k)Γ(A+κζζ+2k)Γ(A+κζϱ+2k)    ×Γ(A+ϱζ+2k)Γ(k+1)(1)kΓ(A+ϱ+2k)Γ(ω+ϑ+1+φk)(Γ(ϑ+1+k))m(x2)k4kk!×01zl+ω+2ϑ+2k1(1z)n1dz  =xAζζ+κ12ω+2ϑk=0Γ(l+ω+2ϑ+2k)Γ(n)Γ(A+2k)Γ(A+κζζϱ+2k)Γ(l+ω+2ϑ+2k+n)Γ(A+ϱ+2k)Γ(A+κζζ+2k) ×Γ(A+ϱζ+2k)Γ(k+1)Γ(A+κζϱ+2k)Γ(ω+ϑ+1+φk)(Γ(ϑ+1+k))m×(x2)k4kk!    (3.3)

Interpreting the right-hand side of the above equation, in the view of the definition (1.2), we arrive at the required result (3.2).

Theorem 3.2. Let ζ, ζ′, ϱ, ϱ′, κ, ϑ, ω ∈ ℂ, m ∈ ℕ, φ > 0, and x > 0 be such that

         (κ)>0, (ω)>1, (l)>0,  (n)>0,(χω)>1+min{(ϱ),(ζ+ζκ),(ζ+ϱκ)}    (3.4)

then the following Beta transform formula holds:

B[I0ζ,ζ,ϱ,ϱ,κ(tχ1Jω,ϑφ,m(z/t))(x):l,n}=xκζζAΓ(n)2ω+2ϑ×5ψ5+m[(Aκ+ζ+ζ,2),(A+ζ+ϱκ,2),                    (Aϱ,2),(Cn,2),(1,1)(A,2)(A+ζ+ζ+ϱκ,2),(A+ζϱ,2),                (ω+ϑ+1,φ),(C,2),(ϑ+1,1)mtimes|14x2]    (3.5)

where A = 1 − χ + ω + 2ϑ and C = l + ω + 2ϑ + n.

Proof: The proof of the fractional integral formula (3.5) is similar to the proof of the formula (3.2) given in Theorem 3.1.

Remark 3.1.

(1) For m = 1, Theorem 3.1 and Theorem 3.2 leads to the corresponding results for fractional integral of generalized Bessel function defined by (1.3).

(2) If we take m = 1, φ = 1, and ϑ=12 in (3.2) and (3.5), we get the corresponding results for fractional integral of Struve function defined in (1.4).

(3) On taking m = 1, φ = 1, and ϑ = 0, in (3.2) and (3.5), we get the results for fractional integral of Bessel function defined in (1.5).

3.2. Image Formulas for Pδ-Transform

Theorem 3.3. Let ζ, ζ′, ϱ, ϱ′, κ, χ, ϑ ∈ ℂ, m ∈ ℕ, φ > 0, ℜ(χ) > 0, ℜ(s) > 0, δ > 1, and x > 0 be such that

                            (κ)>0, (ω)>1, (s)>0,(χ+ω)>max{0,(ζ+ζ+ϱκ),(ζϱ)}    (3.6)

then the following Pδ-transform formula holds:

 Pδ[zl1(I0+ζ,ζ,ϱ,ϱ,κtχ1Jω,ϑφ,m(tz))(x):s]=(Λ(δ;s))l+ω+2ϑ xAζζ+κ12ω+2ϑ×5ψ4+m[      (A,2),(A+κζζϱ,2),(A+ϱζ,2),(l+ω+2ϑ,2),(1,1)    (A+ϱ,2),(A+κζζ,2),              (A+κζϱ,2),           (ω+ϑ+1,φ),(ϑ+1,1)mtimes|(Λ(δ;s)x)24]    (3.7)

where A = χ + ω + 2ϑ and Λ(δ;s)=(δ-1ln[1+(δ-1)s]).

Proof: For our convenience, we let the left-hand side of the formula (3.7) be denoted as Ξ. Applying (1.30) to Equation (3.2) we get,

Ξ=0[1+(δ1)s]zδ1zl1I0+ζ,ζ,ϱ,ϱ,κ(tχ1Jω,ϑφ,m(tz))(x)]dz

Here, applying Equation (2.4) to the integral, we obtain the following expression:

Ξ=xAζζ+κ12ω+2ϑk=0(1)kΓ(A+2k)Γ(A+κζζϱ+2k)Γ(A+ϱ+2k)Γ(A+κζζ+2k)Γ(A+κζϱ+2k)Γ(A+ϱζ+2k)Γ(k+1)Γ(ω+ϑ+1+φk)(Γ(ϑ+1+k))m(x)2k4kk!×0[1+(δ1)s]zδ1zω+2ϑ+2k+l1dz

Here making use of the result (1.31) and interchanging the order of integration and summation, we obtain,

Ξ=(Λ(δ;s))l+ω+2ϑ xAζζ+κ12ω+2ϑk=0Γ(A+2k)Γ(A+κζζϱ+2k)Γ(A+ϱ+2k)Γ(A+κζζ+2k)         ×Γ(ω+2ϑ+2k+l)Γ(A+ϱζ+2k)Γ(k+1)(1)kΓ(A+κζϱ+2k)Γ(ω+ϑ+1+φk)(Γ(ϑ+1+k))m{Λ(δ;s)x}2k4kk!    (3.8)

where A = χ + ω + 2ϑ and Λ(δ;s)=(δ-1ln[1+(δ-1)s]).

In view of the definition (1.2), we arrive at the required result (3.7).

Theorem 3.4. Let ζ, ζ′, ϱ, ϱ′, κ, ϑ ∈ ℂ, m ∈ ℕ, φ > 0ℜ(χ) > 0, ℜ(s) > 0, δ > 1, and x > 0 be such that

              (κ)>0,  (ω)>1,  (s)>0,  (χω)>1+min{(ϱ),(ζ+ζκ),(ζ+ϱκ)}    (3.9)

then the following Pδ- transform formula holds:

Pδ(zl1[I0ζ,ζ,ϱ,ϱ,κtχ1Jω,ϑφ,m(z/t)](x):s)=(Λ(δ;s))l+ω+2ϑ xχω2ϑζζ+κ12ω+2ϑ×5ψ4+m[    (Aκ+ζ+ζ,2),(A+ζ+ϱκ,2),          (Aϱ,2),(l+ω+2ϑ,2),(1,1)(A,2)(A+ζ+ζ+ϱκ,2),(A+ζϱ,2),              (ω+ϑ+1,φ),(ϑ+1,1)mtimes |{Λ(δ;s)}24x2]    (3.10)

where A = 1 − χ + ω + 2ϑ and Λ(δ;s)={δ-1ln[1+(δ-1)s]}.

Proof: Our demonstration of the Pδ-transform of generalized Lommel–Wright function (3.10) is based upon the known result (2.6).

A limit case of the Theorems 3.3 and 3.4 when δ → 1 yields the following corollaries for the Laplace transform in view of the (1.32).

Corollary 3.1. Under the conditions stated with the Equation (3.6), the following Laplace transform formula holds true:

Pδ(zl1(I0+ζ,ζ,ϱ,ϱ,κtχ1Jω,ϑφ,m(tz))(x):s)=xAζζ+κ1sl 2ω+2ϑ×5ψ4+m[(A,2),(A+κζζϱ,2),(A+ϱζ,2),                           (l+ω+2ϑ,2),(1,1)(A+ϱ,2),(A+κζζ,2),(A+κζϱ,2),                          (ω+ϑ+1,φ),(ϑ+1,1)mtimes|x2s2l4]    (3.11)

where A = χ + ω + 2ϑ.

Corollary 3.2. Under the conditions stated with the Equation (3.9), the following Laplace transform formula holds true:

Pδ(zl1[I0ζ,ζ,ϱ,ϱ,κtχ1Jω,ϑφ,m(z/t)](x):s)=xχω2ϑζζ+κ1sl 2ω+2ϑ×5ψ4+m[  (Aκ+ζ+ζ,2),(A+ζ+ϱκ,2),        (Aϱ,2),(l+ω+2ϑ,2),(1,1)(A,2)(A+ζ+ζ+ϱκ,2),(A+ζϱ,2),                 (ω+ϑ+1,φ),(ϑ+1,1)mtimes |1s2l4x2]    (3.12)

where A = 1 − χ + ω + 2ϑ.

Remark 3.2.

(1) On taking m = 1, Theorems 3.3 and 3.4 lead to the Pδ-transform formulas for fractional integrals of generalized Bessel function.

(2) A limit case of the Theorems 3.3 and 3.4, when δ → 1 and m = 1, yields the Laplace transform formulas for fractional integrals of generalized Bessel function.

(3) On taking m = 1, φ = 1, and ϑ=12, Theorems 3.3 and 3.4 yield the Pδ-transform formulas for fractional integrals of Struve function.

(4) A limit case of Theorem 3.3 and 3.4, when δ → 1 and m = 1, φ = 1, and ϑ=12, yield the Laplace transform formulas for fractional integrals of Struve function.

(5) On taking m = 1, φ = 1, and ϑ = 0, Theorem 3.3 and 3.4 yield the corresponding results for fractional integrals of Bessel function.

(6) A limit case of Theorem 3.3 when δ → 1 and m = 1, φ = 1, and ϑ = 0 yield the corresponding Laplace transform formulas for fractional integrals of Bessel function.

3.3. Image Formulas for Whittaker Transform

Theorem 3.5. Let ζ, ζ′, ϱ, ϱ′, κ, ϑ, η, σ ∈ ℂ, m ∈ ℕ, φ > 0, and x > 0 be such that

         (κ)>0,(ω)>1,  (τ±η)>1/2, (χ+ω)>max{0,(ζ+ζ+ϱκ),(ζϱ)    (3.13)

then the following Whittaker transform formula holds:

0zσ1ez/2[Wσ,ηI0+ζ,ζ,ϱ,ϱ,κ(tχ1Jω,ϑφ,m(zt))(x)]dz     =xAζζ+κ12ω+2ϑ6ψ5+m[(A,2),(A+κζζϱ,2),(A+ϱζ,2),                      (E+η,2),(Eη,2),(1,1)(A+ϱ,2),(A+κζζ,2),(A+κζϱ,2),                (ω+ϑ+1,φ),(Eσ,2),(ϑ+1,1)mtimes  |x24]    (3.14)

where A = χ + ω + 2ϑ and E = τ + ω + 2ϑ + 1/2.

Proof: For simplicity, let ϖ be the left-hand side of the formula (3.14). Applying (1.35) to Equation (3.14), we have

ϖ=0zσ1ez/2Wσ,η [I0+ζ,ζ,ϱ,ϱ,κ(tχ1Jω,ϑφ,m(zt))(x)] dz.    (3.15)

Here, applying Equation (2.2) to the integral, we obtain the following expression:

ϖ=0zσ+ω+2ϑ1ez/2Wσ,η        [xAζζ+κ12ω+2ϑk=0Γ(A+2k)Γ(A+κζζϱ+2k)Γ(A+ϱ+2k)Γ(A+κζζ+2k)              ××Γ(A+ϱζ+2k)Γ(k+1)(1)kΓ(A+κζϱ+2k)Γ(ω+ϑ+1+φk)(Γ(ϑ+1+k))m        ×(zx)2k4kk!]   dz

where A = χ + ω + 2ϑ. Interchanging the order of integration and summation, we have

ϖ=xAζζ+κ12ω+2ϑk=0Γ(E+η+2k)Γ(Eη+2k)Γ(A+κζζϱ+2k)Γ(Eσ+2k)Γ(A+κζϱ+2k)      ×(1)kΓ(A+2k)Γ(A+ϱζ+2k)Γ(k+1)Γ(A+ϱ+2k)Γ(A+κζζ+2k)Γ(ω+ϑ+1+φk)(Γ(ϑ+1+k))mx2k4kk!    (3.16)

where A = χ + ω + 2ϑ and E = τ + ω + 2ϑ + 1/2.

Interpreting the right-hand side of the above equation, in view of the definition (1.2), we arrive at the required result (3.14).

Theorem 3.6. Let ζ, ζ′, ϱ, ϱ′, κ, ϑ, η, σ ∈ ℂ, m ∈ ℕ, φ > 0, and x > 0 be such that

              (κ)>0,  (ω)>1,  (τ±n)>1/2,(χω)>1+min{(ϱ),(ζ+ζκ),(ζ+ϱκ)}    (3.17)

then there holds the formula

 0zσ1ez/2Wσ,η[(I0ζ,ζ,ϱ,ϱ,κtχ1Jω,ϑφ,m(zt))(x)]dz=xχω2ϑζζ+κ12ω+2ϑ 6ψ5+m[     (Aκ+ζ+ζ,2),(A+ζ+ϱκ,2),         (Aϱ,2),(E+η,2),(Eη,2),(1,1)(A,2)(A+ζ+ζ+ϱκ,2),(A+ζϱ,2),            (ω+ϑ+1,φ),(Eσ,2),(ϑ+1)mtimes|14x2]    (3.18)

where A = 1 − χ + ω + 2ϑ and E = τ + ω + 2ϑ + 1/2.

Proof: We can establish the result given in Theorem 3.6 similar to the proof of Theorem 3.5.

Remark 3.3.

(1) For m = 1, Theorems 3.5 and 3.6 lead to the corresponding results for fractional integral of generalized Bessel function defined in (1.3).

(2) If we take m = 1, φ = 1, and ϑ=12, Theorems 3.5 and 3.6 yield the corresponding results for fractional integral of Struve function defined in (1.4).

(3) On taking m = 1, φ = 1, and ϑ = 0, Theorems 3.5 and 3.6 yield the corresponding results for fractional integral of Bessel function defined in (1.5).

4. Special Cases and Concluding Remarks

In this section, we consider some special cases of our main results involved in Theorems 2.1–3.6 which can be obtained by setting ζ′ = 0. These interesting corollaries of our results involve the Saigo fractional integral operators I0,xζ,ϱ,η and Ix,ζ,ϱ,η and can be deduced from the Theorems 2.1–3.6 by appropriately applying the relationships given in the definitions (1.16) and (1.17). If we set ϱ = −ζ in the Theorems 2.1–3.6, then from the relationships (1.20) and (1.21) we obtain the corresponding results for the Riemann–Liouville and the Weyl fractional integral operators, respectively. Again, if we put ϱ = 0 in the Theorems 2.1–3.6, then from the relationships (1.24) and (1.25) we obtain the analogous results for Erdélyi-Kober type fractional integral operators.

In our present investigation, we establish the relationship between well-known fractional integral operators with novel integral transforms. The results obtained here are useful in deriving at various image formulas. The results presented here are very generic and can be specialized to give further potentially interesting and useful formulas involving fractional integral operators.

Author Contributions

RPA devised the problem and supervised the manuscript by adding various results to it. RA and SJ worked on the mathematics in the manuscript. DB provided guidance, checked all calculations, and suggested language modifications to the article paper.

Conflict of Interest Statement

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

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Keywords: Marichev-Saigo-Maeda fractional integral operators, generalized Lommel–Wright function, generalized Fox–Wright function, generalized hypergeometric series, integral transform

Citation: Agarwal R, Jain S, Agarwal RP and Baleanu D (2018) A Remark on the Fractional Integral Operators and the Image Formulas of Generalized Lommel–Wright Function. Front. Phys. 6:79. doi: 10.3389/fphy.2018.00079

Received: 23 May 2018; Accepted: 05 July 2018;
Published: 14 August 2018.

Edited by:

Carla M. A. Pinto, Instituto Superior de Engenharia do Porto (ISEP), Portugal

Reviewed by:

Bruce J. West, United States Army Research Laboratory, United States
Carlo Cattani, Università degli Studi della Tuscia, Italy

Copyright © 2018 Agarwal, Jain, Agarwal and Baleanu. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

*Correspondence: Ritu Agarwal, ragarwal.maths@mnit.ac.in

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