Asymptotic behavior of solutions of Poincaré difference equations

Abstract:

It is shown that if the zeros ${\lambda _1},{\lambda _2}, \ldots ,{\lambda _n}$ of the polynomial \[ q(\lambda ) = {\lambda ^n} + {a_1}{\lambda ^{n - 1}} + \cdots + {a_n}\] are distinct and $r$ is an integer in $\{ 1,2, \ldots ,n\}$ such that $|{\lambda _s}| \ne |{\lambda _r}|$ if $s \ne r$, then the Poincaré difference equation \[ y(n + m) + ({a_1} + {p_1}(m))y(n + m - 1) + \cdots + ({a_n} + {p_n}(m))y(m) = 0\] has a solution ${y_r}$ such that (A) ${y_r}(m) = \lambda _r^m(1 + o(1))$ as $m \to \infty$, provided that the sums $\sum \nolimits _{j = m}^\infty {{p_i}(j)\;(1 \leqslant i \leqslant n)}$ converge sufficiently rapidly. Our results improve over previous results in that these series may converge conditionally, and we give sharper estimates of the $o(1)$ terms in (A).