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.

New Bounds for the Modified Bessel Function of the First Kind and Toader-Qi Mean

Let Ipx be the modified Bessel function of the first kind of order p. The upper and lower bounds in the form of simple rational functions about cosht and (sinht)/t for the function I0x are obtained. The corresponding inequalities for the Toader-Qi mean do not match those in the existing literature.

The Bessel function J 0 in fractional differential equations

Spherical Bessel functions have many important applications in engineering, optic and science. In this work, wich is a continuation of the error function in fractional differential equations, it is shown how solve the fractional differential equation d α y d x α = j 0 ( x ) , y ( k ) ( 0 ) = 0 , k = 0 , … m − 1 , with m − 1 < α ≤ m , m ∈ Ν , where the nonhomogenous part is the function Bessel spherical J 0(x).

An integral transform and its application in the propagation of Lorentz-Gaussian beams

The aim of the present note is to derive an integral transform I = ∫ 0 ∞ x s + 1 e - β x 2 + γ x M k , v ( 2 ζ x 2 ) J μ ( χ x ) d x , I = \int_0^\infty {{x^{s + 1}}{e^{ - \beta x}}^{2 + \gamma x}{M_{k,v}}} \left( {2\zeta {x^2}} \right)J\mu \left( {\chi x} \right)dx, involving the product of the Whittaker function Mk, ν and the Bessel function of the first kind Jµ of order µ. As a by-product, we also derive certain new integral transforms as particular cases for some special values of the parameters k and ν of the Whittaker function. Eventually, we show the application of the integral in the propagation of hollow higher-order circular Lorentz-cosh-Gaussian beams through an ABCD optical system (see, for details [13], [3]).

Fractional Hermite–Jensen–Mercer Integral Inequalities with respect to Another Function and Application

In this paper, authors prove new variants of Hermite–Jensen–Mercer type inequalities using ψ –Riemann–Liouville fractional integrals with respect to another function via convexity. We establish generalized identities involving ψ –Riemann–Liouville fractional integral pertaining first and twice differentiable convex function λ , and these will be used to derive novel estimates for some fractional Hermite–Jensen–Mercer type inequalities. Some known results are recaptured from our results as special cases. Finally, an application from our results using the modified Bessel function of the first kind is established as well.