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

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 $4$.