# Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

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

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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by Ch. G. Philos and Y. G. Sficas
Proc. Amer. Math. Soc. 108 (1990), 107-115 Request permission

## Abstract:

Consider the difference equation $({\text {E}})\quad {( - 1)^{m + 1}}{\Delta ^m}{A_n} + \sum \limits _{k = 0}^\infty {{p_k}{A_{n - {l_k}}} = 0,}$ where $m$ is a positive integer, ${({p_k})_{k \geq 0}}$ is a sequence of positive real numbers and ${({l_k})_{k \geq 0}}$ is a sequence of integers with $0 \leq {l_0} < {l_1} < {l_2} < \cdots$. The characteristic equation of (E) is $( * )\quad - {(1 - \lambda )^m} + \sum \limits _{k = 0}^\infty {{p_k}{\lambda ^{ - {l_k}}} = 0.}$ We prove the following theorem. Theorem. (i) For $m$ even, (E) has a positive solution ${({A_n})_{n \in Z}}$ with $\lim {\text {su}}{{\text {p}}_{n \to \infty }}{A_n} < \infty$ if and only if (*) has a root in $(0,1)$. (ii) For $m$ odd, (E) has a positive solution ${({A_n})_{n \in Z}}$ if and only if (*) has a root in $(0,1)$.
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