Global analysis of solutions of \(x_{n+1}= \frac{\beta x_n + \delta x_{n-2}}{A+Bx_n+Cx_{n-1}}\) (Q819737)

The author investigates the convergence to the equilibrium solution(s), the existence of a prime period three solution, and the boundedness of the positive solutions of the rational difference equation \[ x_{n+1}=\frac{\beta x_n+\delta x_{n-2}}{A+Bx_n+Cx_{n-1}},\quad n=0,1,2,... \] where the parameters \(\beta,\delta,A\) are nonnegative, \(\beta+\delta>0,~B,C>0\), the initial conditions \(x_{-2},x_{-1},x_0\) are nonnegative real numbers and the denominator is always positive. He also poses open problems and formulates conjectures regarding the global behavior of the solutions of the above difference equation.