The special functions of fractional calculus as generalized fractional calculus operators of some basic functions

We propose a unified approach to the so-called Special Functions of Fractional Calculus (SFs of FC), recently enjoying increasing interest from both theoretical mathematicians and applied scientists. This is due to their role as solutions of fractional order differential and integral equations, as the better mathematical models of phenomena of various physical, engineering, automatization, chemical, biological, Earth science, economics etc. nature. Our approach is based on the use of Generalized Fractional Calculus (GFC) operators. Namely, we show that all the Wright generalized hypergeometric functions (W.ghf-s) p q.z/ can be represented as generalized fractional integrals, derivatives or differ-integrals of three basic simpler functions as cosq􀀀p.z/, exp.z/ and 1 0.z/ (reducible in particular to the elementary function z .1 􀀀 z/ , the Beta-distribution), depending on whether p < q, p D q or p D q C 1 and on the values of their indices and parameters. In this way, the SFs of FC can be separated into three classes with similar behaviour, and also new integral and differential formulas can be derived, useful for computational procedures.