I am trying to solve the closed formula for $x_n$ given the first order non-linear recurrence relation below. Unlike linear recurrences with direct solutions, I can't find any good reference for non-linear recurrences especially of this form. Any help is appreciated.
$x_n = \frac{\theta}{\beta x_{n-1}}-1$ for $n > 1 $ and $x_1 = 1$. Both $\theta$ and $\beta$ are constants.