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In this paper we consider a simple general form of a deterministic systemwith power-law memory whose state can be described by one variable andevolution by a generating function. A new value of the system's variable is atotal (a convolution) of the generating functions of all previous values of thevariable with weights, which are powers of the time passed. In discrete casesthese systems can be described by difference equations in which a fractionaldifference on the left hand side is equal to a total (also a convolution) ofthe generating functions of all previous values of the system's variable withfractional Eulerian number weights on the right hand side. In the continuouslimit the considered systems can be described by Gr$\ddot{u}$nvald-Letnikovfractional differential equations, which are equivalent to the Volterraintegral equations of the second kind. New properties of fractional Euleriannumbers and possible applications of the results are discussed.

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On Fractional Eulerian Numbers and Equivalence of Maps with Long Term  Power-Law Memory (Integral Volterra Equations of the Second Kind) to  Gr$\ddot{u}$nvald-Letnikov Fractional Difference (Differential) Equations