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. Maps with memory have been investigated for many decades. The field of fractional difference equations is a few decades old. In this paper, we show that maps with power-law memory are equivalent to the Grünvald-Letnikov fractional difference equations. The fractional Eulerian numbers, introduced by Butzer and Hauss in 1993 in a paper which was cited only once in 1995 by Jean-Louis Nicolas, play the key role in the connection between maps with power-law memory and fractional difference equations. In the continuous limit, the relationship between maps with power-law memory and fractional difference equations leads to the equivalence of fractional differential equations and the Volterra integral equations of the second kind. Systems with power-law memory can be used to investigate chaos in continuous fractional systems of less than three dimensions.