Ovcharenko O.V.

Application of the Theory of Fractional Calculus to Integral Operators with Generalized Hypergeometric Functions

The aim of paper is to study the properties of the integral operators with generalized hypergeometric functions in the kernels, in particular, to study the conditions of existence and their boundedness, the study of compositional relations with fractional integrals. In the study of common methods used by the theory of special functions, the theory of integral transforms and operators of fractional integro-differentiation. We introduce integral operators with (τ,β)-generalized hypergeometric functions in the kernels.

r-Hypergeometric Function and its Application

In this paper with the help of the -generalized confluent hypergeometric function the (τ,β) r-hypergeometric function is considered. The aim of it is to study the main properties of the r hypergeometric function, in particular, to study the relation of Erdelyi’ types, the Mellin transform, the composite relation with integral operator of Erdelyi–Kober’ type. In the study used common methods of the theory of special functions, the theory of integral transforms and operators ofvfractional integration.

Differential formulas for <em>q</em>-integral representation of (&tau;,&beta;) - generalized hypergeometric function

The key objective of this paper is to obtain differential formulas for q-integral representation of (τ,β)-generalized hypergeometric function. To this end, we consider new -generalized hypergeometric functions and Using the integral property of q-beta function for the function we obtain the q-integral representation.

The asymptotic expansions and new application of the generalized hypergeometric Gaussian functions

We consider the (τ,β)-generalization (according to Wright) of the functions of hypergeometric type. We also obtain the asymptotic expansions, describe new properties and consider application of these functions in the theory of fractional integration and differentiation.

The study of integral operators properties with generalized hypergeometric function in the kernel

We consider the integral operators with generalized functions of hypergeometric type in the kernel, notably (τ,β) - and τ - generalized hypergeometric Gaussian functions and (τ,β)-generalized Kummer function. We also obtain compositional relations for these operators with Liouville fractional integral and fractional derivative.