Variations on the Stirling's formula for $\Gamma(z)$ 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! 
 A: Let me take a stab at (Q1). I find it better working for $\log\Gamma(z)$ rather than $\Gamma(z)$, since $\Gamma(z) = \exp( \log\Gamma(z))$.
Stirling formula reads as follows:
$$
   \log\Gamma(z) \sim (z-\frac{1}{2}) \log(z) - z + \frac{1}{2} \log(2 \pi) + o(1)
$$
for $\vert z \vert \to \infty$ and $ \vert \arg(z) \vert < \pi - \epsilon$.
Notice that shifted formula is a simple consequence of the above:
$$
  \begin{eqnarray}
  \log\Gamma(z+a) &\sim& ( z+a -\frac{1}{2}) \log(z+a) - z - a + \frac{1}{2} \log(2 \pi) + o(1) \\ 
      &\sim& ( z+a -\frac{1}{2}) \log(z) + ( z+a -\frac{1}{2}) \log(1+\frac{a}{z}) - z - a + \frac{1}{2} \log(2 \pi) + o(1) \\
     &\sim& ( z+a -\frac{1}{2}) \log(z) +  z \log(1+\frac{a}{z}) - z - a + \frac{1}{2} \log(2 \pi) + o(1) \\
    &\sim& ( z+a -\frac{1}{2}) \log(z) - z + \frac{1}{2} \log(2 \pi) + o(1)
  \end{eqnarray}
$$
Now
$$
 \begin{eqnarray}
  \log\Gamma(a z + b) + \log\Gamma(c z + d)  & \sim &
       ( a z +b -\frac{1}{2}) \log(a z) - a z + \frac{1}{2} \log(2 \pi) + \\
      & & ( c z +d -\frac{1}{2}) \log(c z) - c z + \frac{1}{2} \log(2 \pi) + o(1)
 \end{eqnarray}
$$
Then it is a matter of recombining terms as $ ( \mathcal{A} + \mathcal{B} z) + (\kappa z + \rho -\frac{1}{2}) \log (\kappa z) - \kappa z + \frac{1}{2} \log(2 \pi) + o(1) $.
See the book of Paris and Kaminski to fill in the details.
