On the recursive sequence \(x_{n+1}=\frac{\alpha+\beta x_{n-k}}{f(x_n,\dots,x_{n-k+1})}\) (Q2493634)

The paper discusses qualitative properties for the solutions of the difference equation \[ x_{n+1}= {{(\alpha+\beta x_{n-k})}\over{f(x_n,\ldots,x_{n-k+1})}} \] with \(\alpha\geq 0\;,\;\beta\geq 0\) and \(f:\mathbb R_+^k\to\mathbb R_+\) continuous and non-decreasing in each argument such that \(f(0,0,\ldots,0)>0\). Only nonnegative solutions are considered. Several cases are tackled: \(\beta<1\); \(\beta>1\); \(\beta=1, \alpha>0\). An open problem is finally stated.