On a Poincar\'e-Perron problem for high order differential equation

TL;DR

This paper develops asymptotic formulas for high order linear differential equations with almost constant coefficients, extending classical results and providing weaker versions of well-known theorems without diagonalization.

Contribution

It introduces a novel approach using Riccati-type equations and Bell's polynomials to generalize Poincaré-Perron asymptotics for equations of order five and higher.

Findings

Derived asymptotic formulas for high order equations

Extended classical theorems to orders n≥5

Provided weaker versions of Levinson, Hartman-Wintner, Harris-Lutz results

Abstract

We address asymptotic formulae for the classical Poincar\'e-Perron problem of linear differential equations with almost constant coefficients in a half line $[t_0,+\infty)$ for high order equation $n\ge 5$ and some $t_0\in\mathbb{R}$. By using a scalar nonlinear differential equation of Riccati type of order $n-1$, we recover Poincar\'e's and Perron's results and provide asymptotic formulae with the aid of Bell's polynomials. Furthermore, we obtain some weaker versions of Levinson, Hartman-Wintner and Harris-Lutz type Theorems without the usual diagonalization process. For an arbitrary $n\ge 5$, these are corresponding versions to known results for cases $n=2,3$ and $4$.