A characterization of iterative equations by their coefficients

TL;DR

This paper provides a detailed characterization of iterative linear equations by deriving explicit formulas for their coefficients, establishing properties, and identifying conditions for iteration, especially for fourth-order equations.

Contribution

It introduces explicit coefficient formulas, operator forms, and a necessary and sufficient condition for an equation to be iterative, advancing the theoretical understanding of linear iterative equations.

Findings

Explicit coefficient expressions for iterative equations

Operator form for general order iterative equations

Necessary and sufficient condition for fourth-order iterative equations

Abstract

An expression for the coefficients of a linear iterative equation in terms of the parameters of the source equation is given both for equations in standard form and for equations in reduced normal form. The operator generating an iterative equation of a general order in reduced normal form is also obtained and some other properties of iterative equations are established. In particular, a simple necessary and sufficient condition for an equation to be iterative is given for the general fourth-order linear equation solely in terms of its coefficients.