Is there any way I can calculate this ratio more quickly than evaluating the (potentially very large) numerator and denominator?
$$ \frac{\Gamma(z+\alpha)}{\Gamma(z)} $$
I'm specifically interested in the case where $0 < \alpha < 1$
I derived a formula for the special case where $\alpha = \frac{1}{2}$:
$$ \Gamma(n + \frac{1}{2}) = \binom{n - \frac{1}{2}}{n}n!\sqrt{\pi} $$ $$ \frac{\Gamma(n + \frac{1}{2})}{\Gamma(n + 1)} = \binom{n - \frac{1}{2}}{n}\sqrt{\pi} $$ $$ \frac{\Gamma(n + 1)}{\Gamma(n + \frac{1}{2})} = \frac{1}{\binom{n - \frac{1}{2}}{n}\sqrt{\pi}} $$ $$ \frac{\Gamma(z + \frac{1}{2})}{\Gamma(z)} = \frac{1}{\binom{z - 1}{z - \frac{1}{2}}\sqrt{\pi}} $$
and the obvious case where $\alpha = 1$:
$$ \frac{\Gamma(z+1)}{\Gamma(z)} = z $$
Is there a more general formula that encompasses any $\alpha$?
There is no general form, since that would imply $\Gamma$ had a "closed form" for any values. It is possible to approximate the ratio as:
$$\frac{\Gamma(z+\alpha)}{\Gamma(z)}\underset{z\to\infty}\sim z^\alpha$$
which holds regardless of $\alpha$ and may be a direct consequence of the definition of $\Gamma$ depending on how you choose to define it.
Of course better asymptotics may be found from Stirling approximations.