Consider a dynamical systems over complex numbers $$ z_{n+1}=\frac{\alpha}{z_{n}}+ \frac{\beta}{z_{n-1}},\qquad n=0,1,\ldots $$

where the parameters $\alpha, ~\beta$ are complex numbers, and the initial conditions $% z_{-1}$ and $z_{0}$ are arbitrary complex numbers.

For any initial values $z_{-1}$ and $z_{0}$, does the sequence $\{z_n\}$ bounded for all $n$? If so, what is the proof?

Does this sequence $\{z_n\}$ converge always? If so, how that be ensured?