The Marichev-Saigo-Maeda Fractional-Calculus Operators Involving the (p,q)-Extended Bessel and Bessel-Wright Functions

The goal of this article is to establish several new formulas and new results related to the Marichev-Saigo-Maeda fractional integral and fractional derivative operators which are applied on the (p, q)-extended Bessel function. The results are expressed as the Hadamard product of the (p, q)extended Gauss hypergeometric function Fp,q and the Fox-Wright function rΨs(z). Some special cases of our main results are considered. Furthermore, the (p, q)-extended Bessel-Wright function is introduced. Finally, a variety of formulas for the Marichev-Saigo-Maeda fractional integral and derivative operators involving the (p, q)-extended Bessel-Wright function is established.


Introduction
Many generalizations and extensions of special functions of mathematical physics have witnessed a significant evolution in recent years. This advancement in the theory of special functions serves as an analytic foundation for the majority of problems in mathematical physics and applied sciences, which have been solved exactly and which have found broad practical applications. Further, the importance of Bessel functions appears in many areas of applied mathematics, mathematical physics, astronomy, engineering, et cetera. The Bessel function was first introduced by and named after Friedrich Wilhelm Bessel (1784-1846) and it was subsequently developed by (among others) Euler, Lagrange, Bernoulli, and others. The Bessel function is a solution of a homogeneous second-order differential equation which is called the Bessel's differential equation and it is given by (see [1]) (−1) n B n + 1 2 , ν + 1 2 ; p, q n! Γ n + 1 2 B 1 2 , ν + 1 where min{ (p), (q)} 0, and (ν) > −1 when p = q = 0 and B(x, y; p, q) is the (p, q)-extended Beta function, which is defined as follows (see [3]): where min{ (x), (y)} > 0 and min{ (p), (q)} 0.
It should be remarked here that the existing literature on the subject contains much more general extensions of the classical Beta function, especially in the case when p = q (see, for example, [4,5]).
For p = q = 1, the (p, q)-extended Bessel function of the first kind J ν,p,q (z) and the (p, q)-extended Beta function B(x, y; p, q) reduce to the Bessel function J ν (z) of the first kind and the classical Beta function B(x, y), respectively.
Recently, Bessel functions have become widely used in fractional calculus and its applications (see, for example, [7,8]).
Let f and g be two functions having the following power-series representations: Then the familiar Hadamard product (or convolution) of the functions f and g, is given by The introduction of fractional calculus is a very important development in the field of calculus due to the fact that it has proven to be widely applicable in many fields of mathematical, physical and applied sciences. Initially, fractional calculus is the study of derivative and integral operators with a real or complex order, and thus it is a generalization of the traditional calculus. The fractional derivative was first discussed by l'Hôpital and Leibniz in the 16th century and attracted the attention of many mathematicians such as Euler, Laplace, Fourier, Abel, Liouville, Riemann, Grünwald, Letnikov, Weyl, Lévy, and Riesz. Due to its usefulness in different emerging branches of applied mathematics, physics, engineering, quantum mechanics, electrical engineering, telecommunications, digital image processing, robotics, system identification, chemistry, and biology (see, for example, [12][13][14][15][16]), it has been one of the most significant branches of applied mathematics. The development and study of fractional calculus opens the possibility of generalizations of formulas; furthermore, the generalized Marichev-Saigo-Maeda fractional integral was introduced by Marichev [17] as Mellin-type convolution operators with the Appell function F 3 in their kernel.) These operators were rediscovered and studied by Saigo [18] (and, subsequently, by Saigo and Maeda [19]) as generalizations of the Saigo fractional integral operators, which were first studied by Saigo [20] and then applied by Srivastava and Saigo [21] in their systematic investigation of several boundary-value problems involving the Euler-Darboux partial differential equation.
The study of fractional calculus provides many important tools for dealing with derivative and integral equations involving certain special functions and provides the generalized integrals and derivatives of arbitrary fractional order (see, for example, [21,26]). For several general results associated with the Marichev-Saigo-Maeda fractional integrals and derivatives, see the recent work by Srivastava et al. [27].
Motivated by these applications, in this work, we establish various formulas for the Marichev-Saigo-Maeda fractional derivative and integral operators involving the (p, q)-extended Bessel function of the first kind of order ν in terms of the Hadamard product of the Fox-Wright function and the (p, q)-extended Gauss hypergeometric function.
In the next section, we establish formulas for the Marichev-Saigo-Maeda fractional integrals and derivatives involving the (p, q)-extended Bessel function J ν,p,q (z) in terms of the Fox-Wright function r Ψ s and the (p, q)-extended Gauss hypergeometric function.

Marichev-Saigo-Maeda Fractional Integral of the Function J ν,p,q (z)
Now, we establish the Marichev-Saigo-Maeda fractional integral formulas involving the (p, q)-extended Bessel function of the first kind of order ν; the results are expressed as the Hadamard product of the Fox-Wright function and the (p, q)-extended Gauss hypergeometric function.
, which is given by Equation (8) on the function J ν,p,q (t), which is given by Equation (2), we have Now, taking advantage of Equation (24) in Lemma 1 and Equation (29), which is satisfied under the conditions of Theorem 1, we get Therefore, by expressing the above Equation (30) as the Hadamard product of the Fox-Wright function r Ψ s , which is given by Equation (5) and the (p, q)-extended Gauss hypergeometric function F p,q , which is given by Equation (6), we obtain the right-hand side of Equation (28).

Proof. Benefiting from Equations
Thus, by using Equations (26) and (41), which are satisfied under the conditions stated with Theorem 3, we get D α,α ,β,β ,γ 0+ Therefore, by expressing the above equation (42) as the Hadamard product of the Fox-Wright function r Ψ s given by Equation (5) and the (p, q)-extended Gauss hypergeometric function F p,q given by Equation (6), we obtain the right-hand side of Equation (40).

Theorem 4. The Marichev-Saigo-Maeda fractional derivative D
α,α ,β,β ,γ − (α, α , β, β , γ, σ, ν ∈ C) of the function J ν,p,q 1 t is given by Proof. In view of Equations (11), (2) and the left-hand side of Equation (43), we have Thus, making use of Equation (27) in Lemma 1 and Equation (44), which is satisfied under the conditions stated with Theorem 4, we get Therefore, by expressing the above Equation (45) as the Hadamard product of the Fox-Wright function r Ψ s , given by Equation (5) and the (p, q)-extended Gauss hypergeometric function F p,q given by Equation (6), we obtain the right-hand side of Equation (43).
The proofs of Theorems 5, 6, 7 and 8 are similar to those that we have already fully described for Theorems 1, 2, 3 and 4, respectively. We, therefore, choose to omit the details involved.

Conclusions
Motivated by the demonstrated usages and the potential for applications of the various operators of fractional calculus (that is, fractional integral and fractional derivative) and also of the considerably large spectrum of special functions and higher transcendental functions in mathematical, physical, engineering, biological and statistical sciences, we have established here several new formulas and new results for the Marichev-Saigo-Maeda fractional integral and fractional derivative operators, which are applied on the (p, q)extended Bessel function J ν,p,q (z). Our results have been expressed as the Hadamard product of the (p, q)-extended Gauss hypergeometric function F p,q (a, b; c; z) and the Fox-Wright function r Ψ s (z). Some special cases of our main results have also been considered. Furthermore, we have introduced and investigated the (p, q)-extended Bessel-Wright function J µ ν,p,q (z). Finally, we have proved several new formulas for the Marichev-Saigo-Maeda fractional integral and fractional derivative operators involving the (p, q)-extended Bessel-Wright function J µ ν,p,q (z). In concluding this investigation, we choose to indicate the possibility of further researches involving basic or quantum (or q-) extensions of the results which we have presented in this paper. At the same time, in order not to encourage the current trend of some amateurish-type publications, the authors should refer the interested reader to the well-demonstrated observations in [24] (pp. 1511-1512) that this trend of trivially and inconsequentially translating known q-results into the corresponding (p, q)-results leads to no more than a straightforward and shallow variation of the known q-results by means of a forced-in redundant (or superfluous) parameter p. Data Availability Statement: This study did not report any data.