Kneading determinants of infinite order linear recurrences

TL;DR

This paper introduces kneading matrices and determinants for infinite order linear recurrences, extending classical concepts and deriving asymptotic formulas through analysis of generating functions.

Contribution

It defines kneading matrices and determinants in a linear algebraic framework, extending classical finite recurrence notions to infinite order cases.

Findings

Defined kneading matrices and determinants for infinite recurrences

Extended Frobenius companion matrix concepts to infinite order

Derived asymptotic Binet formulas for solutions

Abstract

Infinite order linear recurrences are studied via kneading matrices and kneading determinants. The concepts of kneading matrix and kneading determinant of an infinite order linear recurrence, introduced in this work, are defined in a purely linear algebraic context. These concepts extend the classical notions of Frobenius companion matrix to infinite order linear recurrences and to the associated discriminant of finite order linear recurrences. Asymptotic Binet formulas are deduced for general classes of infinite order linear recurrences as a consequence of the analytical properties of the generating functions obtained for the solutions of these infinite order linear recurrences.