Fractional Derivatives and Special Functions

Abstract

The fractional derivative operator is an extension of the familiar derivative operator $D^n $ to arbitrary (integer, rational, irrational, or complex) values of n. The most important representations which have been proposed for this concept are reviewed in this paper. In particular, those representations which appear to be of greatest interest for use in exploring the special functions, are presented in detail. A list of selected formulas and theorems on fractional differentiation is presented. Applications to the summation of series and the evaluation of definite integrals incorporating special functions are mentioned.