This lemma is often referred to as Darboux’s method [31]. Informally, it
states that a comparison function $g(z)$ which shares the same dominant
singularity as $f(z)$ has asymptotically identical series coefficients. To
make use of this, one can compute the asymptotic series coefficients for a
singularity of a standard form, then use this theorem to transfer the results
to a more complex function. Consider the basic singularity

$(1-z/z_{0})^{\alpha}=\sum_{n=0}^{\infty}(-1)^{n}\binom{\alpha}{n}\left(\frac{z}{z_{0}}\right)^{n}.$
(206)

Here, the explicit form of the series coefficients is known, so one can easily
extract the asymptotics by applying the Stirling approximation:

$\displaystyle(-1)^{n}\binom{\alpha}{n}$
$\displaystyle=\frac{(-\alpha)_{n}}{n!}$ (207)
$\displaystyle\sim\frac{n^{-\alpha-1}}{\Gamma(-\alpha)}\left[1+\frac{\alpha(\alpha+1)}{2n}+\cdots\right].$
(208)

This gives the general result

$[z^{n}](1-z/z_{0})^{\alpha}\ \sim\
z_{0}^{-n}\frac{n^{-\alpha-1}}{\Gamma(-\alpha)};\qquad\alpha\notin\mathbb{Z}_{+}.$
(209)

For a derivation of this formula using only explicit contour integration, see
[70]. Note that this vanishes when the exponent $\alpha$ is a non-negative
integer. The reason is straightforward: the left hand side of (206) is a
polynomial in $z$ of degree $\alpha$, so its asymptotic series coefficients
are identically zero [68]. In this special case, the leading term in the limit
$n\to\infty$ is simply the term of highest degree.

#### Firewalk polynomial asymptotics

Having established the asymptotic approximation for the standard singularity,
we can turn to the generating function (158) for the firewalk polynomials
$W_{n}(x)$:

$\Phi(z,x)=(1-z/u)^{A}(1-z/v)^{B}=\sum_{n=0}^{\infty}W_{n}(x)z^{n}.$ (210)

This function has two singularities which may be coincident in norm depending
on the value of $x$; one has

$\displaystyle u=\frac{x(\beta+1)}{2}+\frac{D(x)}{2};\quad
v=\frac{x(\beta+1)}{2}-\frac{D(x)}{2};$ (211) $\displaystyle
D(x)=\sqrt{x^{2}(\beta+1)^{2}-4\beta}.$ (212)

Recall that for $x$ real and $|x|\leq I_{\beta}$, the roots $u,v$ are
coincident in norm with $u=\bar{v}$. For positive $x>I_{\beta}$, one has
$|v|<|u|$, whereas for negative $x<-I_{\beta}$ the inequality is reversed.
Take $x\notin\mathbb{R}$ and $\operatorname{Re}x>0$; this means that $v$ is
the dominant singularity of the generating function (210). One should
construct a comparison function $g(z)$ which shares the singularity at $z=v$.
To do so, expand the generating function $\Phi(z,x)$ around the point $z=v$,
taking only the leading term as $z\to v$:

$g(z)=(1-v/u)^{A}(1-z/v)^{B}.$ (213)

One can now simply apply the result (209) to determine the asymptotic
coefficients of $g(z)$; Darboux’s lemma ensures that these are also the
asymptotics of the $W_{n}(x)$. Therefore, one finds

$\lim_{n\rightarrow\infty}W_{n}(x)\sim(1-v/u)^{A}\frac{n^{-B-1}}{\Gamma(-B)}v^{-n},$
(214)

with the caveat that this formula vanishes for $B\in\mathbb{Z}_{+}$. For the
firewalk polynomials of the second kind $W^{*}_{n}(x)$, one has the generating
function

$\Phi^{*}(x,z)=(\beta+\gamma+1)(u-z)^{A}(v-z)^{B}\\\ \times\int_{0}^{z}dt\
(u-t)^{-A-1}(v-t)^{-B-1}.$ (215)

By the same arguments as above, for positive $x$, the root $v$ is the dominant
singularity of this function. The procedure is therefore essentially the same.
To define a comparison function, expand $\Phi^{*}$ around the point $z=v$:

$g(z)=(\beta+\gamma+1)D^{A}v^{B}(1-z/v)^{B}\\\ \times\int_{0}^{v}dt\
(u-t)^{-A-1}(v-t)^{-B-1},$ (216)

to which one can apply the result for the standard singularity. This gives the
asymptotic result

$\lim_{n\rightarrow\infty}W_{n}^{*}(x)\sim(\beta+\gamma+1)\frac{D^{A}}{\Gamma(-B)}v^{B-n}n^{-B-1}\\\
\times\int_{0}^{v}dt\ (u-t)^{-A-1}(v-t)^{-B-1}.$ (217)

This completes the asymptotics needed to compute the resolvent for the
firewalk polynomial measure.

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