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Indication on the proof. The proof can be found in [9, 11], but for the convenience of the reader, we recall it here. 1), then it is easy to verify that it satisfies the equality Dh FW (h) f ( x)  FW (h) Dx f ( x) . 4) where denotes the Mittag-Leffler function defined by the expression E ( x ) :  xk  (1  k ) . 5) k 0 We then have the equality   f ( x  h)  E h Dx f ( x) . 2). For further details see [9, 10, 11, 12]. Remark. Clearly, the fractional Taylor’s series is obtained by using the definition via fractional difference (local definition), and thus applies also to the modified RiemannLiouville derivative (global definition).

Nevertheless, it is a wide area for future research, primarily with respect to the mathematical aspects. References              F. Aurenhammer, Voronoi diagrams-A survey of a fundamental geometric data structure, ACM Computing Surveys, V. 23 (1991), no. 3, 345-405. K. Bagi, Discussion of the paper “Tensorial form definitions of discrete mechanical quantities for granular assemblies” [M. Satake, Int. J. Solids and Structures 2004, 41(21), pp. 5775–5791] 2006, International Journal of Solids and Structures, V.

4. On the Modelling of Particle Dynamics by Using Complex-Valued Velocity The fact that the fractional derivative on the left may be imaginary suggests some remarks, like the following one, for instance. First remark. Time increases, and never decreases, and as a result the velocity on the left x(t  dt)  x(t )  dt v  (t )  strictly speaking, is meaningless on a practical standpoint, in such a manner that it could be considered as an imaginary parameter which characterizes a virtual dynamics.