Mittag-Leffler Functions and the Truncated V-fractional Derivative

In this paper, we introduce a new type of fractional derivative, which we called truncated V-fractional derivative, for a-differentiable functions, by means of the six-parameter truncated Mittag-Leffler function. One remarkable characteristic of this new derivative is that it generalizes several different fractional derivatives, recently introduced: conformable fractional derivative, alternative fractional derivative, truncated alternative fractional derivative, M-fractional derivative and truncated M-fractional derivative. This new truncated V-fractional derivative satisfies several important properties of the classical derivatives of integer order calculus: linearity, product rule, quotient rule, function composition and the chain rule. Also, as in the case of the Caputo derivative, the derivative of a constant is zero. Since the six parameters Mittag-Leffler function is a generalization of Mittag-Leffler functions of one, two, three, four and five parameters, we were able to extend some of the classical results of the integer-order calculus, namely: Rolle's theorem, the mean value theorem and its extension. In addition, we present a theorem on the law of exponents for derivatives and as an application we calculate the truncated V-fractional derivative of the two-parameter Mittag-Leffler function. Finally, we present the V-fractional integral from which, as a natural consequence, new results appear as applications. Specifically, we generalize the inverse property, the fundamental theorem of calculus, a theorem associated with classical integration by parts, and the mean value theorem for integrals. We also calculate the V-fractional integral of the two-parameter Mittag-Leffler function. Further, we were able to establish the relation between the truncated V-fractional derivative and the truncated V-fractional integral and the fractional derivative and fractional integral in the Riemann-Liouville sense when the order parameter alpha lies between 0 and 1 (0 < alpha < 1).