Using singularity analysis for probability at a threshold? In some urn model with parameter $p$, the generating function
$$
f_p(z) \;=\; \frac{1+p\,z}{1-(1-p)\,z\,(1+p\,z)}
$$
is such that $[z^n]f_p(z)$ is the probability that an $n$-urn configuration has a particular property.
It is known that this probability tends to zero if $p\gg n^{-1/2}$ and tends to one if $p\ll n^{-1/2}$, so $n^{-1/2}$ is a threshold for the property.
The following approach appears to give the correct probability for the property when $p=c\,n^{-1/2}$:
Let $\rho(p)$ be the dominant singularity of $f_p$ (a simple pole). Then,
$$
\lim_{n\to\infty}\rho\big(c\,n^{-1/2}\big)^{-n} \;=\; e^{-c^2} .
$$
Obviously, for fixed $p$, we have $[z^n]f_p(z)\sim c_p\,\rho(p)^{-n}$, for some (computable) constant $c_p$, but substituting $p=c\,n^{-1/2}$ and then taking limits seems rather dubious.
Question: Can singularity analysis be used in this way? If so, under what conditions, and where can I find the appropriate justification?
[Of course, the specific details of this urn model and property aren't relevant; it's the general method I'm asking about.]
 A: There are a few things working in this particular question that allow for that to occur.  This is a rational function and the denominator is a quadratic in $z$ meaning that we can write $$[z^n] f_p(z) = c_1 \rho_1^{-n} + c_2 \rho_2^{-n}$$ where $\rho_j$ are the roots of the denominator and $c_j$ are functions of $p$.
In the case at hand, we can substitute $p = c/\sqrt{n}$ and then see if the limit of this exact expression exists.  Yes, $\rho_2$ is larger than $\rho_1$,  but it could a priori be the case that $c_2 \rho_2^{-n}$ does not tend to $0$ as $n \to \infty$.  In this concrete example, you can compute that $c_2 \to 0$ and $\rho_2^{-n} \to 0$ for $p = c/\sqrt{n}$ and $n \to \infty$.
Similarly, we need to understand the limiting behavior of $c_1$.  A priori, it could be the case that $c_1 \to 0$ or something strange, but here it can be shown that $c_1 \to 1$ in the regime listed.
It could be possible that there are more general conditions that guarantee that $c_1 \not\to 0$ and similarly to deduce behavior on $c_2 \rho_2^{-n}$; nothing I've written---or you used in your post---takes into account that this is actually a bivariate rational function, just that it's rational in $z$.  Regardless, some info about $c_1$ has be taken into account, as we could just multiply the generating function by $2$ and alter the asymptotics by a constant factor.
