Dynamics of a \(k\)th order rational difference equation (Q856126)

This paper is concerned with the solutions, stability character and semicycle behavior of the rational difference equation \[ x_{n+1}=\frac{x_{n-k}}{A+x_{n-k}x_{n}}, \] where \(x_{-k}, x_{-k+1}, \dots, x_{0}>0\), and \(A>0\). The authors generalize the existing results in the reference. For the special case \(k=1\), they give an explicit solution of the difference equation \[ x_{n+1}=\frac{x_{n-1}}{A+x_{n-1}x_{n}}. \]