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

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.