Modified Riemann–Liouville Fractional Operators via Probability Distributions

With the fractional logistic differential equation and its power-series solution, we show the problems when modeling with the Riemann–Liouville calculus. We propose a modification, in terms of the Beta probability distribution, that improves deficiencies of the Riemann–Liouville and Caputo operators. It is related to the L-fractional normalization. Fractional Euler numbers, connected with the famous Riemann zeta function, are computed. A Mittag-Leffler function is introduced.