## 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.