Grouping the terms of $F(z)$ by the height reached, we get
$$F(z) = \frac{1}{(1 - z\gamma_0)} + \frac{z^2 \beta_1}{(1 - z\gamma_0)^2 (1 - z\gamma_1)} + \frac{z^4 \beta_1 \beta_2}{(1 - z\gamma_0)^2 (1 - z\gamma_1)^2 (1 - z\gamma_2)} + \cdots \\
$$
This has the form of Euler's continued fraction $$a_0 + a_0a_1 + a_0a_1a_2 + \cdots  = \cfrac{a_0}{1 - \cfrac{a_1}{1 + a_1 - \cfrac{a_2}{1 + a_2 - \ddots}}}$$
with $$a_0 = \frac{1}{1 - z\gamma_0} \\
a_1 = \frac{z^2 \beta_1}{(1 - z\gamma_0)(1 - z \gamma_1)} \\
a_2 = \frac{z^2 \beta_2}{(1 - z\gamma_1)(1 - z \gamma_2)} \\
\vdots
$$
It is perhaps more natural to drop the denominators to the next level: i.e. instead of $$\cfrac{\frac{1}{1 - z\gamma_0}}{1 - \cfrac{\frac{z^2 \beta_1}{(1 - z\gamma_0)(1 - z \gamma_1)}}{1 + \frac{z^2 \beta_1}{(1 - z\gamma_0)(1 - z \gamma_1)} - \cfrac{\frac{z^2 \beta_2}{(1 - z\gamma_1)(1 - z \gamma_2)}}{1 + \frac{z^2 \beta_2}{(1 - z\gamma_1)(1 - z \gamma_2)} - \ddots}}}$$ we could write $$\cfrac{1}{(1-z\gamma_0) - \cfrac{(1-z\gamma_0) z^2 \beta_1}{(1-z\gamma_0)(1-z\gamma_1) + z^2 \beta_1 - \cfrac{(1-z\gamma_0)(1-z\gamma_1) z^2 \beta_2}{(1-z\gamma_1)(1-z\gamma_2) + z^2 \beta_2 - \cfrac{(1-z\gamma_1)(1-z\gamma_2) z^2 \beta_3}{(1-z\gamma_2)(1-z\gamma_3) + z^2 \beta_3 - \ddots}}}} \\
$$