Lots of results are known about the asymptotic ratio of two gamma functions when $z\to\infty$ and $\alpha,\beta$ are constants, the most basic one being:
$$\frac{\Gamma(z+\alpha)}{\Gamma(z+\beta)} \sim z^{\alpha - \beta}$$
What about the case where $\alpha,\beta \to \infty$ as well? Experimenting a bit, everything seems to work as expected provided that $\alpha$ and $\beta$ are $o(z)$, but the asymptotic is wrong if one is $O(z)$ and the other is not. Are there other regimes I need to worry about? Pointers to the literature would be appreciated.
$$\ln (\Gamma(t)) = t \ln(t) - t - \frac{\ln(t)}{2} + \frac{\ln(2\pi)}{2} + o(1)\ \text{as}\ t \to \infty$$ so if $z, \alpha, \beta$ are all reals with $z+\alpha$ and $z+\beta$ going to $+\infty$ $$ \ln\left(\frac{\Gamma(z+\alpha)}{\Gamma(z+\beta)}\right) = (z+\alpha)\ln(z+\alpha) - (z+\beta)\ln(z+\beta) - \alpha + \beta - \frac{\ln(z+\alpha)}{2} + \frac{\ln(z+\beta)}{2} + o(1)$$ Which of those terms are larger and which are smaller will depend on the relationships between $z, \alpha, \beta$.