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

Abstract

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 [1–7] 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 is defined by de'Oteiza et al. [9] and is represented in the following manner:

where _pψ_q 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]

where a_i, b_j ∈ ℂ and A_i, B_j ∈ ℝ = (−∞, ∞); A_i, B_j ≠ 0, i = 1, 2, …, p, j = 1, 2, …, q, .

A useful generalization of the Lommel–Wright function and its special cases, depending on the arbitrary fractional parameter φ > 0 presents a fractional order extension of the Bessel function J_ω(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]):

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

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)]):

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

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]):

Further, it is easy to see that

and

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:

and

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

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′)}.

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

and

where the general operators and reduce, respectively, to the Saigo operators and [17] defined as follows:

and

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:

and

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

and

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

and

provided both the integrals converge.

The operators and reduce to Erdélyi–Kober fractional integral operators on setting ϱ = 0 as follows:

and

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

and

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:

Beta transform of the power function t^χ−1 is given by:

The P_δ - transform of a complex valued function f(t) of a real variable t denoted by P_δ[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])

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

• if b > 0 is arbitrary, then tends to a finite limit as Υ → ∞

• for arbitrary a > 0, tends to a finite limit as ω → 0+, then the P_δ-transform P_δ[f(t); s] exists for for s ∈ ℂ.

P_δ - transform of the power function t^χ−1 is given by

P_δ-transform has found many applications. The pathway transforms are the paths going from the binomial form to the exponential from e^−st. In P_δ-transform, the variable t is shifted from the binomial factor 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 P_δ-transform.

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

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

or, equivalently,

which can be applied to convert the table of Laplace transforms into the corresponding table of P_δ-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:

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

where

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 andx > 0 be such that

then there holds the formula

whereA = 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

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

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 andx > 0 be such that

then there holds the formula

whereA = 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

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

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

A = χ + ω + 2ϑ, for generalized Bessel functionholds true.

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

A = 1 − χ + ω + 2ϑ, for generalized Bessel functionholds true.

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

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

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

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

whereA = 2 − χ + ω, for Struve functionH_ω(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

for Bessel functionJ_ω(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

for Bessel functionJ_ω(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, andx > 0 be such that

then the following Beta transform formula holds:

HereA = χ + ω + 2ϑ andC = 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

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

Here A = χ + ω + 2ϑ.

Interchanging the order of integration and summation, we have

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, andx > 0 be such that

then the following Beta transform formula holds:

whereA = 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) Form = 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 takem = 1, φ = 1, andin (3.2) and (3.5), we get the corresponding results for fractional integral of Struve function defined in (1.4).

• (3) On takingm = 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 P_δ-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, andx > 0 be such that

then the followingP_δ-transform formula holds:

whereA = χ + ω + 2ϑ and

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,

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

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

where A = χ + ω + 2ϑ and

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, andx > 0 be such that

then the followingP_δ- transform formula holds:

whereA = 1 − χ + ω + 2ϑ and.

Proof: Our demonstration of the P_δ-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:

whereA = χ + ω + 2ϑ.

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

whereA = 1 − χ + ω + 2ϑ.

Remark 3.2.

• On takingm = 1, Theorems 3.3 and 3.4 lead to theP_δ-transform formulas for fractional integrals of generalized Bessel function.

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

• On takingm = 1, φ = 1, and, Theorems 3.3 and 3.4 yield theP_δ-transform formulas for fractional integrals of Struve function.

• A limit case of Theorem 3.3 and 3.4, when δ → 1 andm = 1, φ = 1, and, yield the Laplace transform formulas for fractional integrals of Struve function.

• On takingm = 1, φ = 1, and ϑ = 0, Theorem 3.3 and 3.4 yield the corresponding results for fractional integrals of Bessel function.

• A limit case of Theorem 3.3 when δ → 1 andm = 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, andx > 0 be such that

then the following Whittaker transform formula holds:

whereA = χ + ω + 2ϑ andE = 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

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

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

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, andx > 0 be such that

then there holds the formula

whereA = 1 − χ + ω + 2ϑ andE = 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.

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

• If we takem = 1, φ = 1, and, Theorems 3.5 and 3.6 yield the corresponding results for fractional integral of Struve function defined in (1.4).

• On takingm = 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 and 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.

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