All the special functions are fractional differintegrals of elementary functions

Abstract

In this survey we discuss a unified approach to the generalized hypergeometric functions based on a generalized fractional calculus developed in the monography by Kiryakova. This generalization of the classical theory of the operators of integration and differentiation of fractional order deals with integral (differintegral) operators involving Meijer's G- and Fox's H-functions as kernel functions. Their theory is fully developed and illustrated by various special cases and applications in different areas of the applicable analysis. Usually, the special functions of mathematical physics are defined by means of power series representations. However, some alternative representations can be used as their definitions. Let us mention the well known Poisson integrals for the Bessel functions and the analytical continuation of the Gauss hypergeometric function via the Euler integral formula. The Rodrigues differential formulae, involving repeated or fractional differentiation are also used as definitions of the classical orthogonal polynomials and their generalizations. As to the other special functions (most of them being - and -functions), such representations are less popular and even unknown in the general case. There exist various integral and differential formulae, but, unfortunately, they are quite peculiar for each corresponding special function and scattered in the literature without any common idea to relate them. Here, all the generalized hypergeometric functions are proved to be generalized fractional integrals or derivatives of three basic elementary functions. On this base, they are classified in three specific classes and several new integral and differential representations are found.