Analysis of the Fractional Integrodifferentiability of Power Functions and some Identities with Hypergeometric Functions

TL;DR

This paper derives formulas for fractional integrals and derivatives of power functions using Riemann-Liouville calculus, linking them to hypergeometric functions, and emphasizes the importance of proper limit choices based on the function's domain.

Contribution

It provides explicit expressions for fractional derivatives of power functions in terms of hypergeometric functions, expanding the analytical tools available for fractional calculus.

Findings

Derived valid formulas for fractional integrals and derivatives of power functions.

Expressed fractional derivatives in terms of hypergeometric functions.

Highlighted the importance of limit selection based on the function's domain.

Abstract

In this work we show that it is possible to calculate the fractional integrals and derivatives of order $\alpha$ (using the Riemann-Liouville formulation) of power functions $\left( t-\ast\right) ^{\beta}$ with $\beta$ being any real value, so long as one pays attention to the proper choosing of the lower and upper limits according to the original function's domain. We, therefore, obtain valid expressions that are described in terms of function series of the type $\left( t-\ast\right) ^{\pm\alpha+k}$ and we also show that they are related to the famous hypergeometric functions of the Mathematical-Physics.