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On the recursive sequence \(x_{n}=1+\sum _{i=1}^{k}\alpha_i x_{n - p_{i}}/\sum _{j=1}^{m} \beta _{j}x_{n - q_j}\). (English) Zbl 1180.39006

Summary: We give a complete picture regarding the behavior of positive solutions of the following important difference equation: \(x_n=1+ \sum_{i=1}^k\alpha_i x_{n-p_i}/ \sum_{j=1}^m\beta_jx_{n-q_j}\), \(n\in\mathbb N_0\), where \(\alpha_i\), \(i\in\{1,\dots,k\}\), and \(\beta_j\), \(j\in\{1,\dots,m\}\), are positive numbers such that \(\sum_{i=1}^k\alpha_i= \sum_{j=1}^m\beta_j=1\), and \(p_i\), \(i\in\{1,\dots,k\}\), and \(q_j\), \(j\in\{1,\dots,m\}\), are natural numbers such that \(p_1<p_2<\cdots<p_k\) and \(q_1<q_2<\dots<q_m\). The case when \(\gcd(p_1,\dots,p_k,q_1,\dots,q_m)=1\) is the most important. For the case we prove that if all \(p_i\), \(i\in\{1,\dots,k\}\), are even and all \(q_j\), \(j\in\{1,\dots,m\}\), are odd, then every positive solution of this equation converges to a periodic solution of period two, otherwise, every positive solution of the equation converges to a unique positive equilibrium.

MSC:

39A10 Additive difference equations
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