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Modified Riemann-Liouville derivative and fractional Taylor series of nondifferentiable. functions. Further results. (English) Zbl 1137.65001

Summary: The paper gives some results and improves the derivation of the fractional Taylor’s series of nondifferentiable functions obtained recently by the author [Appl. Math. Lett. 18, No. 7, 739–748 (2005; Zbl 1082.60029); ibid. 18, No. 7, 817–826 (2005; Zbl 1075.60068)] in the form $f\left(x+h\right)={E}_{\alpha }\left({h}^{\alpha }{D}_{x}^{\alpha }\right)f\left(x\right)$, $0<\alpha \le 1$, where ${E}_{\alpha }$ is the Mittag-Leffler 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 one’s.

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 Taylor’s series of multivariate functions. The relation with irreversibility of time and symmetry breaking is exhibited, and to some extent, this F-Taylor’s series generalizes the fractional mean value formula obtained by K. M. Kolwankar and A. D. Gangal [Phys. Rev. Lett. 80, No. 2, 214–217 (1998; Zbl 0945.82005)].

##### MSC:
 65B15 Euler-Maclaurin formula (numerical analysis) 26A33 Fractional derivatives and integrals (real functions) 40A05 Convergence and divergence of series and sequences