Riesz potential operators and inverses via fractional centred derivatives

Fractional centred differences and derivatives definitions are proposed generalising to real orders the existing ones valid for even and odd positive integer orders. For each one, suitable integral formulations are obtained. The computations of the involved integrals lead to new generalisations of the Cauchy integral derivative. To compute this integral, a special two straight line path was used. With this the referred integrals lead to the well known Riesz potential operators and their inverses that emerge as true fractional centred derivatives, but that can be computed through summations similar to the well known Grünwald-Letnikov derivatives Mathematics Subject Classification: 26A33