On some analytic properties of tempered fractional calculus

TL;DR

This paper explores the analytic properties of tempered fractional calculus, revealing connections with classical fractional calculus, and introduces new theoretical results including special functions, Taylor's theorem analogue, and integral inequalities.

Contribution

It provides new insights into the analytic structure of tempered fractional calculus and extends classical results to this generalized framework.

Findings

Connections with Riemann-Liouville fractional calculus established

Derivation of special functions like hypergeometric and Appell's functions

Proved an analogue of Taylor's theorem and integral inequalities

Abstract

We consider the integral and derivative operators of tempered fractional calculus, and examine their analytic properties. We discover connections with the classical Riemann-Liouville fractional calculus and demonstrate how the operators may be used to obtain special functions such as hypergeometric and Appell's functions. We also prove an analogue of Taylor's theorem and some integral inequalities to enrich the mathematical theory of tempered fractional calculus.