I am currently reading some material that makes heavy usage of Hypergeometric functions, and there is one particular point about applying Stirling's approximation to various terms consisting of Gamma-Functions that is not very clear to me.
We have the classical Stirling's approximation formula for the Gamma-Function in the form: $ \Gamma(z)=\sqrt{2\pi}e^{-z}z^{z-1/2}\left(1+O\left(\frac{1}{|z|}\right)\right) $
for $|\arg(z)|<\pi$ as $|z|\to\infty$. In absolute value: $ |\Gamma(z)|=\sqrt{2\pi}e^{-\Re(z)}|z|^{\Re(z)-1/2}e^{-\Im(z)\arg(z)}\left(1+O\left(\frac{1}{|z|}\right)\right) $
Now, there is also the shifted Stirling's approximation for the Gamma-Function due to C. Rowe, if I am not mistaken, that says: $ \Gamma(z+a)=\sqrt{2\pi}e^{-z}z^{z+a-1/2}\left(1+O\left(\frac{1}{|z|}\right)\right) $
uniformly for $|\arg(z)|\leq\pi-\varepsilon$, $a$ in a compact subset of $\mathbb{C}$ and some suitable fixed $\varepsilon>0$, as $|z|\to\infty$. In absolute value: $ |\Gamma(z+a)|=\sqrt{2\pi}e^{-\Re(z)}|z|^{\Re(z+a)-1/2}e^{-\Im(z+a)\arg(z)}\left(1+O\left(\frac{1}{|z|}\right)\right) $
My first question refers to terms of the type $ \Gamma(az+b)\Gamma(cz+d) $ for some complex numbers $a,b,c$ and $d$.
(Q1) Using the classical Stirling's approximation formula (i.e. not the shifted one), how can one obtain a meaningful aggregated asymptotics for the above expression?
I am asking this because there are several places that apply the non-shifted version of Stirling's formula to shifted Gamma factors without mentioning any details, which leaves the impression that this is a fairly standard or even trivial argument. Unfortunately, at this point I am unable to see its triviality. What bothers me here is the term $ (az+b)^{az+b-1/2}(cz+d)^{cz+d-1/2} $ as well as $\arg(az+b)$ and $\arg(cz+d)$, since the shifts by $c$ and $d$ break any easy manipulations.
I am naturally assuming that I am missing something very obvious here (as usual).
I have intentionally not specified anything about the parameters $a,b,c$ and $d$ because my question rather aims at the principle of applying the non-shifted Stirling's approximation to Gamma terms like the above one.
(Q2) Are there any other versions or forms of the Stirling's approximation for $\Gamma(z)$ that could be particularly useful for computing such kinds of asymptotics?
I will be extremely thankful if someone could give some insight in (principle of) the application of Stirling's approximatioin formula(s) to terms composed of Gamma factors!