The characteristics of a higher-order rational difference equation (Q861136)

The authors study the semicycles, periodicity, global stability and boundedness of the solutions of the rational difference equation \[ x_{n+1}=\frac{\alpha+\beta x_{n-k+1} + \gamma x_{n-2k+1}}{A+Bx_{n-k+1}},\quad n=0,1,2,\dots \] where \(A,~B\) and \(\alpha,~\beta,~\gamma\) are positive, \(k\in \{1,2,3,\dots\}\), and the initial conditions \(x_{-2k+1},\dots,x_{-1},x_0\) are positive real numbers.