title: On the fractional Eulerian numbers and equivalence of maps with long term power-law memory (integral Volterra equations of the second kind) to Grünvald-Letnikov fractional difference (differential) equations.
abstract: 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 the fractional Eulerian number weights on the right hand side. In the continuous limit, the considered systems can be described by the Grünvald-Letnikov fractional differential equations, which are equivalent to the Volterra integral equations of the second kind. New properties of the fractional Eulerian numbers and possible applications of the results are discussed.
authors.affiliation: Department of Physics, Stern College at Yeshiva University, 245 Lexington Ave., New York, New York 10016, USA; Courant Institute of Mathematical Sciences, New York University, 251 Mercer St., New York, New York 10012, USA; and Department of Mathematics, BCC, CUNY, 2155 University Avenue, Bronx, New York 10453, USA.
authors.forename: Mark
authors.initials: M
authors.lastname: Edelman