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

TL;DR

This paper explores the connection between systems with power-law memory, fractional difference equations, and fractional differential equations, revealing new properties of fractional Eulerian numbers and their applications.

Contribution

It establishes the equivalence of integral Volterra equations of the second kind to Gr"unvald-Letnikov fractional difference and differential equations, introducing new properties of fractional Eulerian numbers.

Findings

Fractional Eulerian numbers have novel properties.

Discrete systems with power-law memory relate to fractional difference equations.

Continuous limits lead to Gr"unvald-Letnikov fractional 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 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.