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The present contribution is organized as follows. We shall firstly bear in mind the essential of modified Riemann-Liouville derivative and of fractional difference, and then we shall directly calculate the Z-transform of the later, and by this way we shall cross over modified Ztransform. Therefore we shall have some suggestions to get a modelling for anticipatory systems which exhibit long-range memory effects. After some background on our modelling of fractional Gaussian white noise, we shall derive its Z-transform considered as a Gaussian random variable with given mean value and given variance.

Mainly we shall improve its derivation and clarify the effects of non-differentiability in the result. And to conclude, we shall outline some prospects for the application of fractional derivative to generalized functions. This present theory is a theory for non-differentiable functions, like for instance those functions which could be considered as generated by random noises, and a prospect for future research would be to examine whether white noises could not be considered as elemental processes in this approach.

1) F( ) ( z )  j 0  F ( z, T )    z  j F ( z, T ) j j   (1) j 0    z  j , j j that is to say F( ) ( z )  (1  z 1 ) F ( z, T ) . 5) Further Remarks and Comments Remark 1. If we notice that one has the equality 1  z  1       Z  (1) t    , T  1 , t   we come across the convolution F( ) ( z )  1  z  1  F ( z,  ), T  1      Z  (1) t    f (t   )  , T  1. 6) Remark 2. 7) and at first glance this relation appears to be an excellent support to an alternative for defining fractional derivative via fractional difference.

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Advances in Mathematics Research, Volume 20

by Steven

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