One-dimensional dynamical systems type delta over Integral Domains

TL;DR

This paper introduces a new framework for autonomous differential equations of order one over integral domains using delta operators, unifying differential and difference equations through algebraic structures.

Contribution

It defines delta operators on integral domains via Hurwitz expansion rings, generalizing differential and difference equations within an algebraic framework.

Findings

Defines delta operators on integral domains.

Unifies differential and difference equations.

Provides algebraic tools for analyzing such systems.

Abstract

In this paper, autonomous differential equations type delta of order one on integral domains are defined. For this we will use the autonomous ring defined on the Hurwitz expansion ring of exponential generating functions with coefficients in an integral domain. We will also use delta operators, which behave like derivatives when acting on polynomials, along with Umbral calculus. As a particular example of a delta operator we have the forward difference operator that defines the difference equations. Then the delta-type equations generalize to the ordinary equations and to the difference equations.