On fractional powers of the Bessel operator on a semiaxis

In this paper we study fractional powers of the Bessel differential operator defined on a semiaxis. Some important properties of such fractional powers of the Bessel differential operator are proved. They include connections with Legendre functions for kernel representations, fractional integral operators of Liouville and Saigo, Mellin transform and index laws. Possible applications are indicated to differential equations with fractional powers of the Bessel differential operator.Comment: English version (pp. 1--8) and Russian version (pp. 9--18

Functional inequalities for the Fox-Wright functions

In this paper,our aim is to establish some mean value inequalities for the Fox–Wright functions,suchasTurán-type inequalities, Lazarevic and Wilker-type inequalities. As applications we derive some new type inequalities for hypergeometric functions and thefour-parametric Mittag–Lefflerfunctions.Furthermore,weprovethemonotonicity of ratios for sections of series of Fox–Wright function

Signal approximations by shifted Gaussians: a direct approach by finite linear systems

In this paper we consider interpolation problem connected with series by integer shifts of Gaussians. Known approaches for these problems met numerical difficulties. Due to it another method is considered based on finite-rank approximations by linear systems. The main result for this approach is to establish correctness of the finite-rank linear system under consideration. And the main result of the paper is to prove correctness of the finite-rank linear system approximation. For that an explicit formula for the main determinant of the linear system is derived to demonstrate that it is non-zer

On one integral equation in the theory of transform operators

Integral representations of solutions of one differential equation with singularities in the coefficients, containing the Bessel operator perturbed by some potential, are considered. The existence of integral representations of a certain type for such solutions is proved by the method of successive approximations using transform operator

Some problems in the modern theory of transmutations

Methods of transmutation theory now form an important part of modern mathematics. They have many applications to theoretical and applied problem