Two definitions of fractional derivatives of power functions

TL;DR

This paper introduces two unified definitions of fractional derivatives for power functions, exploring their properties and generalizations, and establishing a framework that encompasses derivatives and integrals of various orders.

Contribution

It presents a novel unified approach to defining fractional derivatives and integrals for power functions, including properties like linearity and commutativity, and introduces new notions such as semi-equality.

Findings

Unified definitions of fractional derivatives and integrals.

Properties of linearity, commutativity, semi-equality, semi-linearity, and semi-commutativity.

Generalization of derivatives and integrals to fractional orders.

Abstract

We consider the set of power functions defined on the set of positive real number, and their linear combinations. After recalling some properties of the gamma function, we give two general definitions of derivatives of positive and negative integers, positive and negative fractional orders.Properties of linearity and commutativity are studied and the notions of semi-equality,semi-linearity and semi-commutativity are introduced. Our approach gives a unified definition of the common derivatives and integrals and their generalization.