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