On Fractional Difference/Sum Operators and Standard Fractional Difference $\alpha$-Family of Maps

In this paper the author presents additional arguments in support of the Miller and Ross' definition of the fractional difference/sum operator and continues his work on the properties of fractional dynamical systems and systems with memory in general. Using the Standard Fractional Difference $\alpha$-Family of Maps as an example, the author shows that properties of systems with falling factorial-law memory are similar to properties of systems with power-law memory. The similarities (types of attractors, power-law convergence of trajectories, existence of cascade of bifurcations and intermittent cascade of bifurcations trajectories, and dependence of properties on the memory parameter $\alpha$) and differences in properties of falling factorial- and power-law memory maps are investigated.