Modified Riemann-Liouville derivative and fractional Taylor series of nondifferentiable functions further results

The paper gives some results and improves the derivation of the fractional Taylorʹs series of nondifferentiable functions obtained recently in the form f (χ + h) = Eα (hαDχα)f(χ), 0 α ≤ 1, where Eα is the Mittag-Leffier function. Here, one defines fractional derivative as the limit of fractional difference, and by this way one can circumvent the problem which arises with the definition of the fractional derivative of constant using Riemann-Liouville definition. As a result, a modified Riemann-Liouville definition is proposed, which is fully consistent with the fractional difference definition and avoids any reference to the derivative of order greater than the considered oneʹs. In order to support this F-Taylor series, one shows how its first term can be obtained directly in the form of a mean value formula. The fractional derivative of the Dirac delta function is obtained together with the fractional Taylorʹs series of multivariate functions. The relation with irreversibility of time and symmetry breaking is exhibited, and to some extent, this F-Taylorʹs series generalizes the fractional mean value formula obtained a few years ago by Kolwantar.