Abstract
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.
Highlights
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
Motivated by the demonstrated usages and the potential for applications of the various operators of fractional calculus and 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 Jν,p,q (z)
Summary
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. 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]) d2 u du z2 2 + z
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