Generalizations of Fractional Calculus, Special Functions and Integral Transforms, and Their Mutual Relationships

摘要

In this survey talk we aim to clarify the close relationships between the operators of the generalized fractional calculus (GFC), some classes of generalized hypergeometric functions and generalizations of the classical integral transforms. The GFC developed in [13] is based on the essential use of the Special Functions (SF). The generalized (multiple) fractional integrals and derivatives are defined by single (differ-)integrals with Meijer's G-and Fox H-functions as their kernels, but represent also products of classical Erdélyi-Kober operators. This theory is widely illustrated by numerous special cases and applications in different areas of Applied Analysis: SF, Integral Transforms (IT), operational calculus, differential and integral equations, hyper-Bessel and Gelfond-Leontiev operators, geometric function theory, etc. Here we focus our attention to the first two topics of applications, SF and IT. First of all, the GFC operators are defined by means of IT whose kernels are SF. Next, we provide a new sight on the SF (meant as all generalized hypergeometric functions pFq and pΨq) as operators of GFC of 3 simplest elementary functions, depending on either p