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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Asymptotic behavior of solutions of Poincaré difference equations

Author: William F. Trench
Journal: Proc. Amer. Math. Soc. 119 (1993), 431-438
MSC: Primary 39A10
MathSciNet review: 1184088
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Abstract: It is shown that if the zeros $ {\lambda _1},{\lambda _2}, \ldots ,{\lambda _n}$ of the polynomial

$\displaystyle q(\lambda ) = {\lambda ^n} + {a_1}{\lambda ^{n - 1}} + \cdots + {a_n}$

are distinct and $ r$ is an integer in $ \{ 1,2, \ldots ,n\} $ such that $ \vert{\lambda _s}\vert \ne \vert{\lambda _r}\vert$ if $ s \ne r$, then the Poincaré difference equation

$\displaystyle y(n + m) + ({a_1} + {p_1}(m))y(n + m - 1) + \cdots + ({a_n} + {p_n}(m))y(m) = 0$

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).

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Keywords: Poincaré difference equation, asymptotic behavior, Perron's theorem, conditional convergence
Article copyright: © Copyright 1993 American Mathematical Society