Resurgence in the Transition Region: The Incomplete Gamma Function

Resurgence in the Transition Region:
The Incomplete Gamma Function††This paper is a contribution to the Special
Issue on Asymptotics and Applications of Special Functions in Memory of
Richard Paris. The full collection is available at
https://www.emis.de/journals/SIGMA/Paris.html

Gergő NEMES

G. Nemes

Department of Physics, Tokyo Metropolitan University,
1–1 Minami-osawa, Hachioji-shi, Tokyo, 192-0397, Japan<EMAIL_ADDRESS>
Received January 31, 2024, in final form March 24, 2024; Published online
March 31, 2024

We study the resurgence properties of the coefficients $C_{n}(\tau)$ appearing
in the asymptotic expansion of the incomplete gamma function within the
transition region. Our findings reveal that the asymptotic behaviour of
$C_{n}(\tau)$ as $n\to+\infty$ depends on the parity of $n$. Both
$C_{2n-1}(\tau)$ and $C_{2n}(\tau)$ exhibit behaviours characterised by a
leading term accompanied by an inverse factorial series, where the
coefficients are once again $C_{2k-1}(\tau)$ and $C_{2k}(\tau)$, respectively.
Our derivation employs elementary tools and relies on the known resurgence
properties of the asymptotic expansion of the gamma function and the uniform
asymptotic expansion of the incomplete gamma function. To the best of our
knowledge, prior to this paper, there has been no investigation in the
existing literature regarding the resurgence properties of asymptotic
expansions in transition regions.

asymptotic expansions; incomplete gamma function; resurgence; transition
regions

34E05; 33B20

##  1 Introduction and main result

Resurgence refers to the phenomenon wherein the “late” coefficients and
remainder terms in an asymptotic expansion can be re-expanded as generalised
asymptotic expansions, with their coefficients corresponding to the “early”
coefficients in the original expansion. Initially observed by Dingle [4] and
further developed by Écalle [6, 7, 8] in his theory of resurgent functions,
this phenomenon has been subsequently identified in a broad class of
asymptotic expansions. Examples include asymptotic expansions for integrals
with saddles [1, 2, 9], integrals with coalescing saddles [14], ordinary
differential equations with irregular singularities at infinity [10, 12],
nonlinear ordinary differential equations [16, 17], and second-order
difference equations [15].

In cases where a problem involves additional parameters alongside the
asymptotic variable, the coefficients of the asymptotic expansion can become
singular as these parameters approach specific critical values, termed
transition points. From a numerical perspective, the asymptotic expansion
becomes impractical in an entire region surrounding a transition point. This
region is known as a transition region, and its extent depends on the relative
magnitudes of the corresponding parameter and the asymptotic variable, as well
as the nature of the transition point. To address the computational challenge
posed by transition regions, a commonly employed approach is the utilisation
of uniform asymptotic expansions. Uniform asymptotic expansions are valid in
large domains, including the transition region. However, the coefficients of
such expansions are complicated functions with removable singularities, making
them difficult to compute. Another solution is offered by the lesser-known
transitional asymptotic expansions, providing an alternative approach. In
contrast to their uniform counterparts, these expansions hold validity within
smaller domains. However, they possess simple polynomial coefficients that are
easy to compute. The region of validity for a transitional expansion is large
enough to cover the transition region, making it a useful alternative to
uniform expansions.

As far as we are aware, there has been no investigation in the current
literature concerning the resurgence properties of transitional asymptotic
expansions. This paper marks the beginning of research in this direction, with
a specific emphasis on the transitional asymptotic expansion for the
(normalised) incomplete gamma function $Q(a,z)$ [18, Section 8.2 (i)]. In this
case, the role of the large asymptotic variable is played by $a$, while the
relevant parameter in the problem is $\lambda=z/a$. The transition region,
surrounding the transition point at $\lambda=1$, has a width of
$\mathcal{O}\big{(}|a|^{-1/2}\big{)}$, and within this region, the asymptotic
behaviour of $Q(a,z)$ undergoes an abrupt change. Specifically, $Q(a,z)$
demonstrates a sharp decay near the transition point $\lambda=1$, as it
approaches unity for $\lambda<1$ and progressively decreases algebraically to
zero for $\lambda>1$.

In the paper [11], it was established that the incomplete gamma function
admits the transitional asymptotic expansion

$Q\big{(}a,a+\tau
a^{\frac{1}{2}}\big{)}\sim\frac{1}{2}\operatorname{erfc}\big{(}2^{-\frac{1}{2}}\tau\big{)}+\frac{1}{\sqrt{2\pi
a}}\exp\left(-\frac{\tau^{2}}{2}\right)\sum_{n=0}^{\infty}\frac{C_{n}(\tau)}{a^{n/2}}$
(1.1)

as $a\to\infty$ in the sector $|\arg a|\leq\pi-\delta<\pi$, uniformly with
respect to bounded complex values of $\tau$. Here, $\operatorname{erfc}$
denotes the complementary error function [18, equation (2.2)]. The
coefficients $C_{n}(\tau)$ are polynomials in $\tau$ of degree $3n+2$ and
satisfy

$\displaystyle C_{0}(\tau)=\frac{1}{3}\tau^{2}-\frac{1}{3},$ $\displaystyle
C_{n}(\tau)+\tau
C^{\prime}_{n}(\tau)-C^{\prime\prime}_{n}(\tau)=\tau\big{(}\tau^{2}-2\big{)}C_{n-1}(\tau)-\big{(}2\tau^{2}-1\big{)}C^{\prime}_{n-1}(\tau)+\tau
C^{\prime\prime}_{n-1}(\tau)$ (1.2)

for $n\geq 1$. Additionally, the even- and odd-order polynomials are even and
odd functions, respectively. An algorithm for generating these coefficients is
discussed in Appendix B.

In this paper, we are interested in the asymptotic behaviour of the
coefficients $C_{n}(\tau)$ as $n\to+\infty$. Our main result can be summarised
as follows.

######  Theorem 1.1.

The coefficients $C_{n}(\tau)$ posses the inverse factorial series

$\displaystyle
C_{2n-1}(\tau)\sim\frac{\Gamma(n)}{(2\pi)^{n+1/2}}\sin\left(\frac{n}{2}\pi\right)\exp\left(\frac{\tau^{2}}{2}\right)\operatorname{erf}\big{(}2^{-\frac{1}{2}}\tau\big{)}$
$\displaystyle\phantom{C_{2n-1}(\tau)\sim}{}-\frac{1}{\pi}\sum_{k=1}^{\infty}C_{2k-1}(\tau)\sin\left(\frac{n-k}{2}\pi\right)\frac{\Gamma(n-k)}{(2\pi)^{n-k}},$
(1.3) $\displaystyle
C_{2n}(\tau)\sim-\frac{\Gamma\left(n+\frac{1}{2}\right)}{(2\pi)^{n+1}}\sin\left(\left(n+\frac{1}{2}\right)\frac{\pi}{2}\right)\exp\left(\frac{\tau^{2}}{2}\right)$
$\displaystyle\phantom{C_{2n}(\tau)\sim}{}-\frac{1}{\pi}\sum\limits_{k=0}^{\infty}C_{2k}(\tau)\sin\left(\frac{n-k}{2}\pi\right)\frac{\Gamma(n-k)}{(2\pi)^{n-k}}$
(1.4)

as $n\to+\infty$, uniformly with respect to bounded complex values of $\tau$.
Here, $\operatorname{erf}$ denotes the error function [18, equation (7.2.1)].
Moreover, under the condition that $\tau=o\big{(}n^{1/6}\big{)}$, these
expansions hold as generalised asymptotic expansions [18, Section 2.1 (v)]
when $n\to+\infty$.

Hence, the coefficients $C_{n}(\tau)$ exhibit a resurgence property. Their
asymptotic behaviour expressed as an inverse factorial series, a
characteristic feature in resurgence theory, with the coefficients in these
series being once again the coefficients $C_{k}(\tau)$. As observed, the form
of the inverse factorial series for $C_{n}(\tau)$ depends on the parity of
$n$. Specifically, the expansion coefficients for odd $n$ comprise the
$C_{k}(\tau)$ with odd $k$, while those for even $n$ involve the $C_{k}(\tau)$
with even $k$. Additionally, it is worth noting that the leading term is
absent in the expansion of $C_{n}(\tau)$ when $n\equiv 3\bmod{4}$.

While the structure of the transitional expansion mirrors that of the
corresponding uniform expansion (compare (1.1) and (A.4)), the resurgence
properties of the former are notably simpler than those of the latter. The
asymptotics of the coefficients in the uniform asymptotic expansion of
$Q(a,z)$ are described in terms of the incomplete beta function, which is more
intricate than the error function appearing in the expansion of
$C_{2n-1}(\tau)$. We suspect that this phenomenon holds more generally, such
as when considering integrals with coalescing saddles. In such cases, the
asymptotic behaviour of the uniform expansion coefficients is no longer
represented by inverse factorial series but involves generalised asymptotic
expansions with complicated functions. Therefore, from a resurgent
perspective, transitional expansions seem to exhibit a much closer resemblance
to simple (non-uniform) asymptotic expansions than to uniform ones.

A primary tool for deriving asymptotic expansions of coefficients in
asymptotic series is the Borel transform. In the context of simple (non-
uniform) asymptotic expansions, the Borel transform is a well-established
technique that yields analytic functions with simple branch points. The
asymptotic behaviour of the coefficients is subsequently obtained through the
application of Darboux’s method. This paper takes a different approach,
employing elementary tools and drawing on the well-established resurgence
properties of the asymptotic expansion of the gamma function, as well as the
uniform asymptotic expansion of the incomplete gamma function. Details on
these expansions are given in Appendix A.

The remaining part of the paper is structured as follows. In Section 2, we
present the proof of the main result. Section 3 offers numerical examples that
demonstrate the applicability of the results. The paper concludes with a brief
discussion in Section 4.

##  2 Proof of the main result

The essence of proving our main result lies in relating the coefficients
$C_{n}(\tau)$ to some coefficients $D_{n}(\tau)$, whose asymptotic behaviour
is easier to analyse using the known resurgence properties of the asymptotic
expansion of the gamma function.

In the following discussion, we will assume that $|\tau|<|a|^{1/2}$ and $|\arg
a|<\pi$. Utilising the definitions of the incomplete gamma function $Q(a,z)$
and the scaled gamma function $\Gamma^{\ast}(a)$ (refer to (A.1)), we can
readily deduce that

$\displaystyle\frac{\partial Q\big{(}a,a+\tau
a^{\frac{1}{2}}\big{)}}{\partial\tau}$
$\displaystyle=-\frac{a^{\frac{1}{2}}\mathrm{e}^{-a}}{\Gamma(a)}\big{(}a+\tau
a^{\frac{1}{2}}\big{)}^{a-1}\mathrm{e}^{-\tau
a^{\frac{1}{2}}}=-\frac{1}{\sqrt{2\pi}}\frac{1}{\Gamma^{\ast}(a)}\big{(}1+\tau
a^{-\frac{1}{2}}\big{)}^{a-1}\mathrm{e}^{-\tau a^{\frac{1}{2}}}$
$\displaystyle=-\frac{1}{\sqrt{2\pi}}\exp\left(-\frac{\tau^{2}}{2}\right)\frac{1}{\Gamma^{\ast}(a)}\mathrm{e}^{F(a,\tau)},$
(2.1)

where

$F(a,\tau)=\frac{\tau^{2}}{2}+(a-1)\log\big{(}1+\tau
a^{-\frac{1}{2}}\big{)}-\tau a^{\frac{1}{2}}.$

Using the Maclaurin series of the logarithm, we derive the expansion

$F(a,\tau)=\sum_{n=1}^{\infty}\left(\frac{1}{n}-\frac{\tau^{2}}{n+2}\right)\frac{(-\tau)^{n}}{a^{n/2}}.$

By exponentiating both sides and expanding the right-hand side in negative
powers of $a^{-1/2}$, we obtain

$\mathrm{e}^{F(a,\tau)}=\sum_{n=0}^{\infty}\frac{p_{n}(\tau)}{a^{n/2}},$

where $p_{n}(\tau)$ is a polynomial in $\tau$ of degree $3n$. By substituting
this expansion into (2.1) and employing the asymptotic series (A.2) of the
reciprocal scaled gamma function, we deduce the asymptotic expansion

$\frac{\partial Q\big{(}a,a+\tau
a^{\frac{1}{2}}\big{)}}{\partial\tau}\sim-\frac{1}{\sqrt{2\pi}}\exp\left(-\frac{\tau^{2}}{2}\right)\left(1+\sum_{n=0}^{\infty}\frac{D_{n}(\tau)}{a^{(n+1)/2}}\right)$

as $a\to\infty$ in the sector $|\arg a|\leq\pi-\delta<\pi$, uniformly with
respect to bounded complex values of $\tau$. The coefficients $D_{n}(\tau)$
are expressed in terms of the Stirling coefficients $\gamma_{n}$ (refer to
Appendix A) and the polynomials $p_{n}(\tau)$ as follows:

$D_{2n-1}(\tau)=\sum_{k=0}^{n}\gamma_{n-k}p_{2k}(\tau),\qquad
D_{2n}(\tau)=\sum_{k=0}^{n}\gamma_{n-k}p_{2k+1}(\tau).$ (2.2)

We note that $D_{n}(\tau)$ is a polynomial in $\tau$ of degree $3n+3$. On the
other hand, differentiating each side of (1.1) with respect to $\tau$ yields
the asymptotic expansion

$\frac{\partial Q\big{(}a,a+\tau
a^{\frac{1}{2}}\big{)}}{\partial\tau}\sim-\frac{1}{\sqrt{2\pi}}\exp\left(-\frac{\tau^{2}}{2}\right)\left(1+\sum_{n=0}^{\infty}\frac{\tau
C_{n}(\tau)-C^{\prime}_{n}(\tau)}{a^{(n+1)/2}}\right)$

as $a\to\infty$ in the sector $|\arg a|\leq\pi-\delta<\pi$, uniformly with
respect to bounded complex values of $\tau$. Therefore, by invoking the
uniqueness property of the coefficients in an asymptotic expansion, we deduce
the following relation between the coefficients $C_{n}(\tau)$ and
$D_{n}(\tau)$: $\tau C_{n}(\tau)-C^{\prime}_{n}(\tau)=D_{n}(\tau)$. Solving
for $C_{n}(\tau)$ gives the following expressions:

$C_{2n-1}(\tau)=-\exp\left(\frac{\tau^{2}}{2}\right)\int_{0}^{\tau}\exp\left(-\frac{t^{2}}{2}\right)D_{2n-1}(t)\mathrm{d}t,$
(2.3)

and

$C_{2n}(\tau)=\exp\left(\frac{\tau^{2}}{2}\right)C_{2n}(0)-\exp\left(\frac{\tau^{2}}{2}\right)\int_{0}^{\tau}\exp\left(-\frac{t^{2}}{2}\right)D_{2n}(t)\mathrm{d}t.$
(2.4)

Here, we utilised the fact that $C_{2n-1}(\tau)$, being an odd function,
satisfies $C_{2n-1}(0)=0$.

In the following proposition, we present detailed information regarding the
asymptotic behaviour of $D_{n}(\tau)$ as $n$ becomes large, expressed as a
truncated inverse factorial series along with a remainder. This result, in
combination with (2.3) and (2.4), will be used to establish our main result.
In this paper, empty sums are interpreted as zero, and subscripts in the
$\mathcal{O}$ notations indicate the dependence of the implied constant on
certain parameters.

######  Proposition 2.1.

For any non-negative integer $N$, the coefficients $D_{n}(\tau)$ exhibit the
following asymptotic behaviours:

$\displaystyle D_{2n-1}(\tau)={}$
$\displaystyle-\frac{1}{\pi}\frac{\Gamma(n)}{(2\pi)^{n}}\sin\left(\frac{n}{2}\pi\right)-\frac{1}{\pi}\sum_{k=1}^{N-1}D_{2k-1}(\tau)\sin\left(\frac{n-k}{2}\pi\right)\frac{\Gamma(n-k)}{(2\pi)^{n-k}}$
$\displaystyle+\mathcal{O}_{N}(1)\bigl{(}\left|\tau\right|^{3}+1\bigr{)}^{2N}\exp\big{(}2\pi\bigl{(}\left|\tau\right|^{3}+1\bigr{)}^{2}\big{)}\frac{\Gamma(n-N)}{(2\pi)^{n-N}}$
(2.5)

and

$\displaystyle D_{2n}(\tau)={}$
$\displaystyle-\frac{1}{\pi}\sum_{k=0}^{N-1}D_{2k}(\tau)\sin\left(\frac{n-k}{2}\pi\right)\frac{\Gamma(n-k)}{(2\pi)^{n-k}}$
$\displaystyle+\mathcal{O}_{N}(1)\bigl{(}\left|\tau\right|^{3}+1\bigr{)}^{2N+1}\exp\big{(}2\pi\bigl{(}\left|\tau\right|^{3}+1\bigr{)}^{2}\big{)}\frac{\Gamma(n-N)}{(2\pi)^{n-N}}$
(2.6)

as $n\to+\infty$, provided $\tau=o\big{(}n^{1/6}\big{)}$.

In order to prove Proposition 2.1, we need to establish some lemmas.

######  Lemma 2.2.

For every non-negative integer $k$ and any complex number $\tau$, the
inequality

$\left|p_{k}(\tau)\right|\leq 4\bigl{(}\left|\tau\right|^{3}+1\bigr{)}^{k}$

is satisfied.

###### Proof.

Starting from the expansion defining the polynomials $p_{k}(\tau)$ and using
Cauchy’s formula, we find that

$p_{k}(\tau)=\frac{1}{2\pi\mathrm{i}}\oint_{\left|t\right|=r}\exp\bigg{(}\sum_{n=1}^{\infty}\left(\frac{1}{n}-\frac{\tau^{2}}{n+2}\right)(-\tau
t)^{n}\bigg{)}\frac{\mathrm{d}t}{t^{k+1}},$ (2.7)

where $r=\frac{1}{\left|\tau\right|^{3}+1}$. Assuming
$\left|t\right|=\frac{1}{\left|\tau\right|^{3}+1}$, we can bound the
exponential part as follows:

$\displaystyle\left|\exp\bigg{(}\sum_{n=1}^{\infty}\left(\frac{1}{n}-\frac{\tau^{2}}{n+2}\right)(-\tau
t)^{n}\bigg{)}\right|$
$\displaystyle\leq\exp\bigg{(}\bigg{(}\frac{\left|\tau\right|^{2}}{3}+1\bigg{)}\sum_{n=1}^{\infty}\left|\tau\right|^{n}\left|t\right|^{n}\bigg{)}$
$\displaystyle=\exp\bigg{(}\bigg{(}\frac{\left|\tau\right|^{2}}{3}+1\bigg{)}\sum_{n=1}^{\infty}\bigg{(}\frac{\left|\tau\right|}{\left|\tau\right|^{3}+1}\bigg{)}^{n}\bigg{)}$
$\displaystyle=\exp\bigg{(}\bigg{(}\frac{\left|\tau\right|^{2}}{3}+1\bigg{)}\frac{\left|\tau\right|}{\left|\tau\right|^{3}+1-\left|\tau\right|}\bigg{)}<4.$

Applying this bound in (2.7) and performing a straightforward estimation
yields the desired result. ∎

######  Lemma 2.3.

For every positive integer $n$, the inequality

$\left|\gamma_{n}\right|\leq\frac{\Gamma(n)}{(2\pi)^{n}}$

is satisfied.

###### Proof.

For the case $n=1$, the inequality holds trivially since

$\left|\gamma_{1}\right|=\frac{1}{12}<\frac{1}{2\pi}=\frac{\Gamma(1)}{2\pi}.$

Now, assuming $n\geq 2$, let $R_{n}$ denote the error incurred by truncating
the asymptotic expansion of the scaled gamma function after $n$ terms:

$\Gamma^{\ast}(a)=\sum_{k=0}^{n-1}(-1)^{k}\frac{\gamma_{k}}{a^{k}}+R_{n}(a).$

In the paper [3, equation (3.11)], it was established that for $n\geq 2$ and
$a>0$, the following inequality holds:

$\left|a^{n}R_{n}(a)\right|\leq\frac{1+\zeta(n)}{2\pi}\frac{\Gamma(n)}{(2\pi)^{n}}.$

Here, $\zeta$ denotes the Riemann zeta function [18, equation (25.2.1)].
Consequently,

$\left|\gamma_{n}\right|=\lim_{a\to+\infty}\left|a^{n}R_{n}(a)\right|\leq\frac{1+\zeta(n)}{2\pi}\frac{\Gamma(n)}{(2\pi)^{n}}\leq\frac{\Gamma(n)}{(2\pi)^{n}}$

provided $n\geq 2$. ∎

###### Proof of Proposition 2.1..

We will prove the statement for the coefficients $D_{2n-1}(\tau)$. The proof
for $D_{2n}(\tau)$ follows in a completely analogous manner. Let $N$ be any
non-negative integer, and express (2.2) in the following form:

$D_{2n-1}(\tau)=\sum_{k=0}^{N-1}\gamma_{n-k}p_{2k}(\tau)+\sum_{k=N}^{n}\gamma_{n-k}p_{2k}(\tau).$
(2.8)

We shall establish an asymptotic order estimate for the second sum as
$n\to+\infty$. To do so, we first observe that if $q$ and $r$ are non-negative
integers, then $q!r!\leq(q+r)!$. Thus,

$\Gamma(n-N-k)k!=(n-N-1-k)!k!\leq(n-N-1)!=\Gamma(n-N)$

for integers $n$, $N$, and $k$ satisfying $n\geq N+k+1$ and $k\geq 0$.
Employing this inequality in combination with those provided in Lemmas 2.2 and
2.3, we deduce that

$\displaystyle\left|\sum_{k=N}^{n}\gamma_{n-k}p_{2k}(\tau)\right|$
$\displaystyle\qquad=\left|\sum_{k=N}^{n-1}\gamma_{n-k}p_{2k}(\tau)+p_{2n}(\tau)\right|\leq
4\sum_{k=N}^{n-1}\frac{\Gamma(n-k)}{(2\pi)^{n-k}}\bigl{(}\left|\tau\right|^{3}+1\bigr{)}^{2k}+4\bigl{(}\left|\tau\right|^{3}+1\bigr{)}^{2n}$
$\displaystyle\qquad=4\frac{\Gamma(n-N)}{(2\pi)^{n}}\left(\sum_{k=N}^{n-1}\frac{\Gamma(n-k)}{\Gamma(n-N)}\big{(}2\pi\bigl{(}\left|\tau\right|^{3}+1\bigr{)}^{2}\big{)}^{k}+\frac{\big{(}2\pi\bigl{(}\left|\tau\right|^{3}+1\bigr{)}^{2}\big{)}^{n}}{\Gamma(n-N)}\right)$
$\displaystyle\qquad=4\big{(}2\pi\bigl{(}\left|\tau\right|^{3}+1\bigr{)}^{2}\big{)}^{N}\frac{\Gamma(n-N)}{(2\pi)^{n}}$
$\displaystyle\phantom{\qquad=}{}\times\left(\sum_{k=0}^{n-N-1}\frac{\Gamma(n-N-k)}{\Gamma(n-N)}\big{(}2\pi\bigl{(}\left|\tau\right|^{3}+1\bigr{)}^{2}\big{)}^{k}+\frac{\big{(}2\pi\bigl{(}\left|\tau\right|^{3}+1\bigr{)}^{2}\big{)}^{n-N}}{\Gamma(n-N)}\right)$
$\displaystyle\qquad\leq
4\big{(}2\pi\bigl{(}\left|\tau\right|^{3}+1\bigr{)}^{2}\big{)}^{N}\frac{\Gamma(n-N)}{(2\pi)^{n}}\left(\sum_{k=0}^{n-N-1}\frac{\big{(}2\pi\bigl{(}\left|\tau\right|^{3}+1\bigr{)}^{2}\big{)}^{k}}{k!}+\frac{\big{(}2\pi\bigl{(}\left|\tau\right|^{3}+1\bigr{)}^{2}\big{)}^{n-N}}{\Gamma(n-N)}\right)$
$\displaystyle\qquad\leq
4\big{(}2\pi\bigl{(}\left|\tau\right|^{3}+1\bigr{)}^{2}\big{)}^{N}\frac{\Gamma(n-N)}{(2\pi)^{n}}\left(\exp\big{(}2\pi\bigl{(}\left|\tau\right|^{3}+1\bigr{)}^{2}\big{)}+\frac{\big{(}2\pi\bigl{(}\left|\tau\right|^{3}+1\bigr{)}^{2}\big{)}^{n-N}}{\Gamma(n-N)}\right)$
$\displaystyle\qquad=\mathcal{O}_{N}(1)\bigl{(}\left|\tau\right|^{3}+1\bigr{)}^{2N}\exp\big{(}2\pi\bigl{(}\left|\tau\right|^{3}+1\bigr{)}^{2}\big{)}\frac{\Gamma(n-N)}{(2\pi)^{n-N}}$

as $n\to+\infty$, provided $\tau=o\big{(}n^{1/6}\big{)}$. The condition
$\tau=o\big{(}n^{1/6}\big{)}$ on $\tau$ is imposed to ensure that

$\frac{\big{(}2\pi\bigl{(}\left|\tau\right|^{3}+1\bigr{)}^{2}\big{)}^{n-N}}{\Gamma(n-N)}=\mathcal{O}_{N}(1)\exp\big{(}2\pi\bigl{(}\left|\tau\right|^{3}+1\bigr{)}^{2}\big{)}.$

We have thus established that, for any fixed non-negative integer $N$,

$D_{2n-1}(\tau)=\sum_{k=0}^{N-1}\gamma_{n-k}p_{2k}(\tau)+\mathcal{O}_{N}(1)\bigl{(}\left|\tau\right|^{3}+1\bigr{)}^{2N}\exp\big{(}2\pi\bigl{(}\left|\tau\right|^{3}+1\bigr{)}^{2}\big{)}\frac{\Gamma(n-N)}{(2\pi)^{n-N}}$
(2.9)

as $n\to+\infty$, provided $\tau=o\big{(}n^{1/6}\big{)}$.

Continuing, our aim now is to derive a precise asymptotic expression for the
sum in (2.9) for large $n$, using the known inverse factorial series of the
Stirling coefficients. Referring to (A.3), we can assert that for each fixed
non-negative integer $N$:

$\gamma_{n}=-\frac{1}{\pi}\sum_{m=0}^{N-1}\gamma_{m}\sin\left(\frac{n-m}{2}\pi\right)\frac{\Gamma(n-m)}{(2\pi)^{n-m}}+\mathcal{O}_{N}(1)\frac{\Gamma(n-N)}{(2\pi)^{n-N}}$

as $n\to+\infty$. Applying this expression to the sum in (2.9), we infer that

$\displaystyle\sum_{k=0}^{N-1}\gamma_{n-k}p_{2k}(\tau)=$
$\displaystyle-\sum_{k=0}^{N-1}\frac{1}{\pi}\sum_{m=0}^{N-1}\gamma_{m}\sin\left(\frac{n-k-m}{2}\pi\right)\frac{\Gamma(n-k-m)}{(2\pi)^{n-k-m}}p_{2k}(\tau)$
$\displaystyle+\sum_{k=0}^{N-1}\mathcal{O}_{N}(1)\frac{\Gamma(n-k-N)}{(2\pi)^{n-k-N}}p_{2k}(\tau)$
$\displaystyle=$
$\displaystyle-\frac{1}{\pi}\sum_{k=0}^{N-1}\left(\sum_{j=0}^{k}\gamma_{k-j}p_{2j}(\tau)\right)\sin\left(\frac{n-k}{2}\pi\right)\frac{\Gamma(n-k)}{(2\pi)^{n-k}}$
$\displaystyle-\frac{1}{\pi}\sum_{k=N}^{2N-2}\left(\sum_{j=k-N+1}^{N-1}\gamma_{k-j}p_{2j}(\tau)\right)\sin\left(\frac{n-k}{2}\pi\right)\frac{\Gamma(n-k)}{(2\pi)^{n-k}}$
$\displaystyle+\sum_{k=0}^{N-1}\mathcal{O}_{N}(1)\frac{\Gamma(n-k-N)}{(2\pi)^{n-k-N}}p_{2k}(\tau)$

as $n\to+\infty$, with any fixed non-negative integer $N$. From (2.2), we can
affirm that

$\displaystyle-\frac{1}{\pi}\sum_{k=0}^{N-1}\left(\sum_{j=0}^{k}\gamma_{k-j}p_{2j}(\tau)\right)\sin\left(\frac{n-k}{2}\pi\right)\frac{\Gamma(n-k)}{(2\pi)^{n-k}}$
$\displaystyle\qquad=-\frac{1}{\pi}\frac{\Gamma(n)}{(2\pi)^{n}}\sin\left(\frac{n}{2}\pi\right)-\frac{1}{\pi}\sum_{k=1}^{N-1}D_{2k-1}(\tau)\sin\left(\frac{n-k}{2}\pi\right)\frac{\Gamma(n-k)}{(2\pi)^{n-k}}.$

Moreover, it is evident that, for any fixed non-negative integer $N$,

$-\frac{1}{\pi}\sum_{k=N}^{2N-2}\left(\sum_{j=k-N+1}^{N-1}\gamma_{k-j}p_{2j}(\tau)\right)\sin\left(\frac{n-k}{2}\pi\right)\frac{\Gamma(n-k)}{(2\pi)^{n-k}}=\mathcal{O}_{N}(1)\bigl{(}\left|\tau\right|^{3}+1\bigr{)}^{2N}\frac{\Gamma(n-N)}{(2\pi)^{n-N}}$

and

$\sum_{k=0}^{N-1}\mathcal{O}_{N}(1)\frac{\Gamma(n-k-N)}{(2\pi)^{n-k-N}}p_{2k}(\tau)=\mathcal{O}_{N}(1)\bigl{(}\left|\tau\right|^{3}+1\bigr{)}^{2N}\frac{\Gamma(n-N)}{(2\pi)^{n-N}}$

as $n\to+\infty$, uniformly with respect to $\tau$. Consequently, in summary,

$\displaystyle\sum_{k=0}^{N-1}\gamma_{n-k}p_{2k}(\tau)={}$
$\displaystyle-\frac{1}{\pi}\frac{\Gamma(n)}{(2\pi)^{n}}\sin\left(\frac{n}{2}\pi\right)-\frac{1}{\pi}\sum_{k=1}^{N-1}D_{2k-1}(\tau)\sin\left(\frac{n-k}{2}\pi\right)\frac{\Gamma(n-k)}{(2\pi)^{n-k}}$
$\displaystyle+\mathcal{O}_{N}(1)\bigl{(}\left|\tau\right|^{3}+1\bigr{)}^{2N}\frac{\Gamma(n-N)}{(2\pi)^{n-N}}$

as $n\to+\infty$, with any fixed non-negative integer $N$ and uniformly with
respect to $\tau$. Inserting this asymptotic expression into (2.9) yields the
desired result for $D_{2n-1}(\tau)$. ∎

We are now in a position to prove Theorem 1.1.

###### Proof of Theorem 1.1..

By employing (2.5) in (2.3) with $t$ in place of $\tau$, integrating term-by-
term, and utilising (2.3) for each term, we deduce

$\displaystyle C_{2n-1}(\tau)={}$
$\displaystyle\frac{\Gamma(n)}{(2\pi)^{n+1/2}}\sin\left(\frac{n}{2}\pi\right)\exp\left(\frac{\tau^{2}}{2}\right)\operatorname{erf}\big{(}2^{-\frac{1}{2}}\tau\big{)}$
$\displaystyle-\frac{1}{\pi}\sum_{k=1}^{N-1}C_{2k-1}(\tau)\sin\left(\frac{n-k}{2}\pi\right)\frac{\Gamma(n-k)}{(2\pi)^{n-k}}$
$\displaystyle-\exp\left(\frac{\tau^{2}}{2}\right)\int_{0}^{\tau}\mathcal{O}_{N}(1)\exp\left(-\frac{t^{2}}{2}\right)\bigl{(}\left|t\right|^{3}+1\bigr{)}^{2N}$
$\displaystyle\phantom{-}{}\times\exp\big{(}2\pi\bigl{(}\left|t\right|^{3}+1\bigr{)}^{2}\big{)}\mathrm{d}t\frac{\Gamma(n-N)}{(2\pi)^{n-N}}$

as $n\to+\infty$, provided that $N\geq 0$ is fixed and
$\tau=o\big{(}n^{1/6}\big{)}$. Choosing the integration path in the error term
to be a straight line segment connecting $0$ to $\tau$ and employing a
straightforward estimation, we obtain the following bound:

$\displaystyle\left|\exp\left(\frac{\tau^{2}}{2}\right)\int_{0}^{\tau}\mathcal{O}_{N}(1)\exp\left(-\frac{t^{2}}{2}\right)\bigl{(}\left|t\right|^{3}+1\bigr{)}^{2N}\exp\big{(}2\pi\bigl{(}\left|t\right|^{3}+1\bigr{)}^{2}\big{)}\mathrm{d}t\right|$
$\displaystyle\qquad\leq\mathcal{O}_{N}(1)\bigl{(}\left|\tau\right|^{3}+1\bigr{)}^{2N}\exp\big{(}2\pi\bigl{(}\left|\tau\right|^{3}+1\bigr{)}^{2}\big{)}\int_{0}^{\left|\tau\right|}\left|\exp\left(\frac{\tau^{2}-t^{2}}{2}\right)\right|\mathrm{d}\left|t\right|$
$\displaystyle\qquad=\mathcal{O}_{N}(1)\left|\tau\right|\bigl{(}\left|\tau\right|^{3}+1\bigr{)}^{2N}\exp\big{(}2\pi\bigl{(}\left|\tau\right|^{3}+1\bigr{)}^{2}\big{)}\int_{0}^{1}\exp\bigg{(}\frac{\left|\tau\right|^{2}}{2}(1-s^{2})\cos(2\arg\tau)\bigg{)}\mathrm{d}s$
$\displaystyle\qquad=\mathcal{O}_{N}(1)\left|\tau\right|\bigl{(}\left|\tau\right|^{3}+1\bigr{)}^{2N}\exp\big{(}2\pi\bigl{(}\left|\tau\right|^{3}+1\bigr{)}^{2}\big{)}\exp\bigg{(}\frac{\left|\tau\right|^{2}}{2}\bigg{)}$
$\displaystyle\qquad=\mathcal{O}_{N}(1)\left|\tau\right|\bigl{(}\left|\tau\right|^{3}+1\bigr{)}^{2N}\exp\big{(}7\bigl{(}\left|\tau\right|^{3}+1\bigr{)}^{2}\big{)}.$

Accordingly,

$\displaystyle C_{2n-1}(\tau)={}$
$\displaystyle\frac{\Gamma(n)}{(2\pi)^{n+1/2}}\sin\left(\frac{n}{2}\pi\right)\exp\left(\frac{\tau^{2}}{2}\right)\operatorname{erf}\big{(}2^{-\frac{1}{2}}\tau\big{)}$
$\displaystyle-\frac{1}{\pi}\sum_{k=1}^{N-1}C_{2k-1}(\tau)\sin\left(\frac{n-k}{2}\pi\right)\frac{\Gamma(n-k)}{(2\pi)^{n-k}}$
$\displaystyle+\mathcal{O}_{N}(1)\left|\tau\right|\bigl{(}\left|\tau\right|^{3}+1\bigr{)}^{2N}\exp\big{(}7\bigl{(}\left|\tau\right|^{3}+1\bigr{)}^{2}\big{)}\frac{\Gamma(n-N)}{(2\pi)^{n-N}}$
(2.10)

as $n\to+\infty$, under the condition that $N\geq 0$ is fixed and
$\tau=o\big{(}n^{1/6}\big{)}$. In a similar fashion, by employing (2.4) and
(2.6), we find that for each fixed non-negative integer $N$:

$\displaystyle C_{2n}(\tau)={}$
$\displaystyle\exp\left(\frac{\tau^{2}}{2}\right)C_{2n}(0)-\frac{1}{\pi}\sum_{k=0}^{N-1}\left(C_{2k}(\tau)-\exp\left(\frac{\tau^{2}}{2}\right)C_{2k}(0)\right)\sin\left(\frac{n-k}{2}\pi\right)$
$\displaystyle\times\frac{\Gamma(n-k)}{(2\pi)^{n-k}}+\mathcal{O}_{N}(1)\left|\tau\right|\bigl{(}\left|\tau\right|^{3}+1\bigr{)}^{2N+1}\exp\big{(}7\bigl{(}\left|\tau\right|^{3}+1\bigr{)}^{2}\big{)}\frac{\Gamma(n-N)}{(2\pi)^{n-N}}$

as $n\to+\infty$, provided $\tau=o\big{(}n^{1/6}\big{)}$. To simplify this
expression, we can refer to the known inverse factorial series of the
coefficients $c_{n}(\eta)$ that appear in the uniform asymptotic expansion of
the incomplete gamma function $Q(a,z)$. Specifically, since
$C_{2n}(0)=c_{n}(0)$, we can deduce from (A.6) that for each fixed non-
negative integer $N$:

$\displaystyle C_{2n}(0)={}$
$\displaystyle-\frac{\Gamma\left(n+\frac{1}{2}\right)}{(2\pi)^{n+1}}\sin\left(\left(n+\frac{1}{2}\right)\frac{\pi}{2}\right)-\frac{1}{\pi}\sum_{k=0}^{N-1}C_{2k}(0)\sin\left(\frac{n-k}{2}\pi\right)\frac{\Gamma(n-k)}{(2\pi)^{n-k}}$
$\displaystyle+\mathcal{O}_{N}(1)\frac{\Gamma(n-N)}{(2\pi)^{n-N}}$

as $n\to+\infty$. As a result,

$\displaystyle C_{2n}(\tau)={}$
$\displaystyle-\frac{\Gamma\left(n+\frac{1}{2}\right)}{(2\pi)^{n+1}}\sin\left(\left(n+\frac{1}{2}\right)\frac{\pi}{2}\right)\exp\left(\frac{\tau^{2}}{2}\right)-\frac{1}{\pi}\sum\limits_{k=0}^{N-1}C_{2k}(\tau)\sin\left(\frac{n-k}{2}\pi\right)$
$\displaystyle\times\frac{\Gamma(n-k)}{(2\pi)^{n-k}}+\mathcal{O}_{N}(1)(\left|\tau\right|+1)\bigl{(}\left|\tau\right|^{3}+1\bigr{)}^{2N+1}\exp\big{(}7\bigl{(}\left|\tau\right|^{3}+1\bigr{)}^{2}\big{)}\frac{\Gamma(n-N)}{(2\pi)^{n-N}}$
(2.11)

as $n\to+\infty$, provided that $N\geq 0$ is fixed and
$\tau=o\big{(}n^{1/6}\big{)}$.

If $\tau$ is bounded, the error terms in (2.10) and (2.11) are of the same
order of magnitude as the first neglected terms in the sums. Consequently, the
inverse factorial series expansions (1.3) and (1.4) are valid. Under the
weaker assumption that $\tau=o\big{(}n^{1/6}\big{)}$, it can readily be
verified that the ratio of consecutive remainder terms asymptotically decays
in magnitude as $n\to+\infty$. This confirms that the expansions (1.3) and
(1.4) remain valid in the context of generalised asymptotic expansions. ∎

######  Remark 2.4.

Starting from (2.4), it is not difficult to show that

$C_{2n}(\tau)=\exp\left(\frac{\tau^{2}}{2}\right)\int_{\tau}^{+\infty}\exp\left(-\frac{t^{2}}{2}\right)D_{2n}(t)\mathrm{d}t$
(2.12)

for all $n\geq 0$. However, this formula cannot be employed to derive the
inverse factorial series (1.4) from (2.6) due to the divergence of the
improper integrals involving the error term in (2.6) against
$\exp\bigl{(}-\frac{t^{2}}{2}\bigr{)}$. An alternative approach involves
substituting (2.8) into (2.12) initially and then estimating the second sum.
Yet, obtaining an estimate resulting in an error term comparable to that in
(2.11) proves challenging. Consequently, it appears necessary to invoke the
inverse factorial series of the coefficients in the uniform asymptotic
expansion.

##  3 Numerical examples

In this section, we present numerical results that confirm the accuracy of the
inverse factorial expansions given in Theorem 1.1. Truncating the series (1.3)
and (1.4) after $N$ and $N+1$ terms, respectively, we consider the
approximations

$\displaystyle
C_{2n-1}(\tau)\approx\frac{\Gamma(n)}{(2\pi)^{n+1/2}}\sin\left(\frac{n}{2}\pi\right)\exp\left(\frac{\tau^{2}}{2}\right)\operatorname{erf}\big{(}2^{-\frac{1}{2}}\tau\big{)}$
$\displaystyle\phantom{C_{2n-1}(\tau)\approx}{}-\frac{1}{\pi}\sum_{k=1}^{N}C_{2k-1}(\tau)\sin\left(\frac{n-k}{2}\pi\right)\frac{\Gamma(n-k)}{(2\pi)^{n-k}},$
(3.1) $\displaystyle
C_{2n}(\tau)\approx-\frac{\Gamma\left(n+\frac{1}{2}\right)}{(2\pi)^{n+1}}\sin\left(\left(n+\frac{1}{2}\right)\frac{\pi}{2}\right)\exp\left(\frac{\tau^{2}}{2}\right)$
$\displaystyle\phantom{C_{2n}(\tau)\approx}{}-\frac{1}{\pi}\sum\limits_{k=0}^{N}C_{2k}(\tau)\sin\left(\frac{n-k}{2}\pi\right)\frac{\Gamma(n-k)}{(2\pi)^{n-k}}.$
(3.2)

In Table 1, we provide exact numerical values of $C_{2n-1}(\tau)$, along with
the approximate values derived from (3.1), and the corresponding errors for
various combinations of $n$, $\tau$, and the truncation index $N$.
Correspondingly, Table 2 displays exact numerical values of $C_{2n}(\tau)$,
the corresponding approximations from (3.2), and the associated errors for
different values of $n$, $\tau$, and $N$. The coefficients $C_{n}(\tau)$ were
computed using the algorithm outlined in Appendix B. It is observed,
particularly from Table 2, that these approximations are effective only when
the magnitude of $\tau$ is much smaller than that of $n$.

values of $n$, $\tau$ and $N$ |  $n=50$, $\tau=\frac{1}{2}$, $N=10$ |  |
---|---|---|---
exact numerical value of $C_{2n-1}(\tau)\qquad$ | $-0.1605549419108870432185698$ | $\times$ | $10^{20}$
approximation (3.1) to $C_{2n-1}(\tau)$ | $-0.1605549417678948233999888$ | $\times$ | $10^{20}$
error | $-0.1429922198185809$ | $\times$ | $10^{11}$
values of $n$, $\tau$ and $N$ |  $n=50$, $\tau=\frac{1}{2}$, $N=20$ |  |
exact numerical value of $C_{2n-1}(\tau)$ | $-0.1605549419108870432185698$ | $\times$ | $10^{20}$
approximation (3.1) to $C_{2n-1}(\tau)$ | $-0.1605549419107926902148106$ | $\times$ | $10^{20}$
error | $-0.943530037592$ | $\times$ | $10^{7}$
values of $n$, $\tau$ and $N$ |  $n=100$, $\tau=1+\frac{3}{2}\mathrm{i}$, $N=20$ |  |
exact numerical value of $C_{2n-1}(\tau)$ | $0.6104582432674722845198873$ | $\times$ | $10^{75}$
| $-\mathrm{i}\,0.3947300517906354580061330$ | $\times$ | $10^{75}$
approximation (3.1) to $C_{2n-1}(\tau)$ | $0.6104582432678493030809359$ | $\times$ | $10^{75}$
| $-\mathrm{i}\,0.3947300517908665670802024$ | $\times$ | $10^{75}$
error | $-0.3770185610486$ | $\times$ | $10^{63}$
| $+\mathrm{i}\,0.2311090740694$ | $\times$ | $10^{63}$
values of $n$, $\tau$ and $N$ |  $n=100$, $\tau=1+\frac{3}{2}\mathrm{i}$, $N=40$ |  |
exact numerical value of $C_{2n-1}(\tau)$ | $0.6104582432674722845198873$ | $\times$ | $10^{75}$
| $-\mathrm{i}\,0.3947300517906354580061330$ | $\times$ | $10^{75}$
approximation (3.1) to $C_{2n-1}(\tau)$ | $0.6104582432674722845199135$ | $\times$ | $10^{75}$
| $-\mathrm{i}\,0.3947300517906354580061436$ | $\times$ | $10^{75}$
error | $-0.262$ | $\times$ | $10^{53}$
| $+\mathrm{i}\,0.106$ | $\times$ | $10^{53}$
Table 1: Approximations for $C_{2n-1}(\tau)$ with various $n$, $\tau$ and $N$, using (3.1). values of $n$, $\tau$ and $N$ |  $n=50$, $\tau=1$, $N=10$ |  |
---|---|---|---
exact numerical value of $C_{2n}(\tau)\qquad$ | $0.9780541202848348054115227$ | $\times$ | $10^{23}$
approximation (3.2) to $C_{2n}(\tau)$ | $0.9780541202841343977234399$ | $\times$ | $10^{23}$
error | $0.7004076880828$ | $\times$ | $10^{11}$
values of $n$, $\tau$ and $N$ |  $n=50$, $\tau=1$, $N=20$ |  |
exact numerical value of $C_{2n}(\tau)$ | $0.9780541202848348054115227$ | $\times$ | $10^{23}$
approximation (3.2) to $C_{2n}(\tau)$ | $0.9780541202848340460370285$ | $\times$ | $10^{23}$
error | $0.7593744942$ | $\times$ | $10^{8}$
values of $n$, $\tau$ and $N$ |  $n=100$, $\tau=2+\mathrm{i}$, $N=20$ |  |
exact numerical value of $C_{2n}(\tau)$ | $0.3119948787485535986155779$ | $\times$ | $10^{77}$
| $-\mathrm{i}\,0.6504249040471427943527241$ | $\times$ | $10^{77}$
approximation (3.2) to $C_{2n}(\tau)$ | $0.3119948786903918092779561$ | $\times$ | $10^{77}$
| $-\mathrm{i}\,0.6504249040964131036369408$ | $\times$ | $10^{77}$
error | $0.581617893376218$ | $\times$ | $10^{67}$
| $-\mathrm{i}\,0.492703092842167$ | $\times$ | $10^{67}$
values of $n$, $\tau$ and $N$ |  $n=100$, $\tau=2+\mathrm{i}$, $N=40$ |  |
exact numerical value of $C_{2n}(\tau)$ | $0.3119948787485535986155779$ | $\times$ | $10^{77}$
| $-\mathrm{i}\,0.6504249040471427943527241$ | $\times$ | $10^{77}$
approximation (3.2) to $C_{2n}(\tau)$ | $0.3119948786894443063970208$ | $\times$ | $10^{77}$
| $-\mathrm{i}\,0.6504249040965108927273969$ | $\times$ | $10^{77}$
error | $0.59109292218557109$ | $\times$ | $10^{67}$
| $-\mathrm{i}\,0.49368098374672863$ | $\times$ | $10^{67}$
Table 2: Approximations for $C_{2n}(\tau)$ with various $n$, $\tau$ and $N$,
using (3.2).

##  4 Concluding remarks

We studied the asymptotic behaviour of the coefficients $C_{n}(\tau)$ as
$n\to+\infty$ appearing in the asymptotic expansion of the incomplete gamma
function within the transition region. We established inverse factorial series
expansions for $C_{n}(\tau)$ that exhibit a resurgence property–meaning the
coefficients of these series are once again $C_{k}(\tau)$. To the best of our
knowledge, there has been no investigation into the resurgence properties of
asymptotic expansions in transition regions in the existing literature prior
to this paper.

The example of the incomplete gamma function is specific in a sense, as its
derivative can be expressed in terms of the gamma function, whose resurgence
properties are well understood. It would be interesting to develop a more
general method for studying the resurgence properties of transitional
asymptotic expansions that lack such special properties. Two important
examples include the transitional expansions of the Bessel functions [18,
Section 10.19 (iii)] and the Coulomb functions [18, Section 33.12 (i)], both
of which do not exhibit these special characteristics. We anticipate that the
Borel transform will play a crucial role in such investigations. Additionally,
exploring beyond the asymptotic study of coefficients and developing a
hyperasymptotic theory for transitional expansions, similar to those existing
for standard (non-uniform) asymptotic expansions [1, 12], would be of great
interest.

##  Appendix A Auxiliary results

In this appendix, we provide a concise overview of the resurgence properties
of the asymptotic expansion of the gamma function and the uniform asymptotic
expansion of the incomplete gamma function. Our focus is on the asymptotic
behaviour of the coefficients of these expansions, as they play a significant
role in proving our main result.

We define the scaled gamma function $\Gamma^{\ast}(a)$ through the relation

$\Gamma^{\ast}(a)=\frac{\Gamma(a)}{\sqrt{2\pi}a^{a-1/2}\mathrm{e}^{-a}}$ (A.1)

valid for $|\arg a|<\pi$. It is a well-established fact that both the scaled
gamma function and its reciprocal possess asymptotic expansions given by

$\Gamma^{\ast}(a)\sim\sum_{n=0}^{\infty}(-1)^{n}\frac{\gamma_{n}}{a^{n}},\qquad\frac{1}{\Gamma^{\ast}(a)}\sim\sum_{n=0}^{\infty}\frac{\gamma_{n}}{a^{n}}$
(A.2)

as $a\to\infty$ in the sector $|\arg a|\leq\pi-\delta<\pi$ (see, e.g., [21,
Section 3.6]). Here, the $\gamma_{n}$ are known as the Stirling coefficients,
with the initial values being

$\gamma_{0}=1,\qquad\gamma_{1}=-\frac{1}{12},\qquad\gamma_{2}=\frac{1}{288},\qquad\gamma_{3}=\frac{139}{51840},\qquad\gamma_{4}=-\frac{571}{2488320}.$

Note that the asymptotic expansion of the reciprocal scaled gamma function
involves the same coefficients as that of the scaled gamma function, but with
different signs of the coefficients with odd index. For an explanation of this
phenomenon, refer to, for instance, [21, p. 63]. The asymptotic behaviour of
the Stirling coefficients is described by the inverse factorial series

$\gamma_{n}\sim-\frac{1}{\pi}\sum_{k=0}^{\infty}\gamma_{k}\sin\left(\frac{n-k}{2}\pi\right)\frac{\Gamma(n-k)}{(2\pi)^{n-k}}$
(A.3)

as $n\to+\infty$. Notably, the coefficients within the inverse factorial
series are once again the Stirling coefficients, highlighting the resurgence
property of the gamma function. The expansion (A.3) was established by Boyd
[3, equation (3.34)], although it had previously appeared without proof in an
earlier paper by Rosser [19, equation (97)].

Moving on to the incomplete gamma function $Q(a,z)$, define

$\lambda=\frac{z}{a},\qquad\eta=(2(\lambda-1-\log\lambda))^{\frac{1}{2}},$

where the branch of the square root is continuous and satisfies
$\eta(\lambda)\sim\lambda-1$ as $\lambda\to 1$. Temme [20] established that
the incomplete gamma function admits the uniform asymptotic expansion

$Q(a,z)\sim\frac{1}{2}\operatorname{erfc}\bigg{(}\eta\sqrt{\frac{a}{2}}\bigg{)}+\frac{1}{\sqrt{2\pi
a}}\exp\left(-\eta^{2}\frac{a}{2}\right)\sum_{n=0}^{\infty}\frac{c_{n}(\eta)}{a^{n}},$
(A.4)

as $a\to\infty$ in the sector $|\arg a|\leq\pi-\delta<\pi$, uniformly with
respect to $\lambda$ in the sector $|\arg\lambda|\leq 2\pi-\delta<2\pi$. The
coefficients $c_{n}(\eta)$ are complicated functions of their argument with
removable singularities at $\eta=0$. The resurgence properties of this
expansion were studied by Olde Daalhuis [13]. He demonstrated that the
coefficients $c_{n}(\eta)$ possess inverse factorial series of the following
form as $n\to+\infty$, for $\eta\in\mathcal{N}$:

$\displaystyle\begin{split}c_{n}(\eta)\sim\;&\frac{(-1)^{n+1}}{2\sqrt{2\pi}}\Gamma\left(n+\frac{1}{2}\right)\left(\frac{1}{2}\eta^{2}+2\pi\mathrm{i}\right)^{-n-\frac{1}{2}}I_{1+\frac{\eta^{2}}{4\pi\mathrm{i}}}\left(n+\frac{1}{2},\frac{1}{2}\right)\\\
&+\frac{(-1)^{n+1}}{2\sqrt{2\pi}}\Gamma\left(n+\frac{1}{2}\right)\left(\frac{1}{2}\eta^{2}-2\pi\mathrm{i}\right)^{-n-\frac{1}{2}}I_{1-\frac{\eta^{2}}{4\pi\mathrm{i}}}\left(n+\frac{1}{2},\frac{1}{2}\right)\\\
&-\frac{1}{\pi}\sum_{k=0}^{\infty}c_{k}(\eta)\sin\left(\frac{n-k}{2}\pi\right)\frac{\Gamma(n-k)}{(2\pi)^{n-k}},\end{split}$
(A.5)

where

$\mathcal{N}=\left\\{\eta\in\mathbb{C}\right\\}\setminus\big{\\{}\eta=\alpha+\mathrm{i}\beta\mid\alpha\beta=\pm
2\pi,\;\alpha\leq-\sqrt{2\pi}\big{\\}}$

is the image of the sector $|\arg\lambda|<2\pi$ under the mapping
$\eta(\lambda)$. In (A.5), $I_{x}(a,b)$ denotes the incomplete beta function
[18, Section 8.17]. In the special case that $\eta=0$, the expansion (A.5)
simplifies to the inverse factorial series

$c_{n}(0)\sim-\frac{\Gamma\left(n+\frac{1}{2}\right)}{(2\pi)^{n+1}}\sin\left(\left(n+\frac{1}{2}\right)\frac{\pi}{2}\right)-\frac{1}{\pi}\sum_{k=0}^{\infty}c_{k}(0)\sin\left(\frac{n-k}{2}\pi\right)\frac{\Gamma(n-k)}{(2\pi)^{n-k}}$
(A.6)

as $n\to+\infty$.

An alternative asymptotic expansion for $c_{n}(\eta)$ was established in the
paper [5], but it does not exhibit the resurgence property.

##  Appendix B The computation of the coefficients $\bm{C_{n}(\tau)}$

In this appendix, we present an efficient method for computing the
coefficients $C_{n}(\tau)$. We substitute the polynomial expansion

$C_{n}(\tau)=\sum_{k=0}^{3n+2}c_{n,k}\tau^{k}$ (B.1)

into (1.2). As a result, this substitution yields the following recurrence
relation for the coefficients $c_{n,k}$:

$c_{n,k}=(k+2)c_{n,k+2}+(k+1)c_{n-1,k+1}-\frac{2k}{k+1}c_{n-1,k-1}+\frac{1}{k+1}c_{n-1,k-3}.$
(B.2)

Given that $C_{0}(\tau)=\frac{1}{3}\tau^{2}-\frac{1}{3}$, we deduce that

$c_{n,3n+2}=\frac{1}{3^{n+1}(n+1)!},\qquad c_{n,3n+1}=0$ (B.3)

holds for $n=0$. Using the recurrence relation (B.2), we can demonstrate that
(B.3) remains valid for $n\geq 1$. Initiating with the first equation in (B.3)
and subsequently setting $k=3n,3n-2,\allowbreak 3n-4,\dots$, enables the
computation of non-zero coefficients in the expansion (B.1). Moreover, we
establish that $c_{n,k}=0$ for $k=3n+1,3n-1,3n-3,\dots$. The following Wolfram
Mathematica [22] code implements the recursive method:

$\displaystyle\bm{c[0,2]=1/3;}$ $\displaystyle\bm{c[0,0]=-1/3;}$
$\displaystyle\bm{c[0,4]=0;}$ $\displaystyle\bm{c[0,-2]=0;}$
$\displaystyle\bm{\textbf{NN}=10;}$
$\displaystyle\bm{\textbf{For}[n=1,n\leq\textbf{NN},n\textbf{++},}$
$\displaystyle\quad\bm{c[n,3*n+2]=1/3{}^{\wedge}(n+1)/(n+1)!;}$
$\displaystyle\quad\bm{c[n,3*n+4]=0;}$
$\displaystyle\quad\bm{\textbf{For}[k=3*n,k\geq 0,k-=2,}$
$\displaystyle\quad\quad\bm{c[n,k]=(k+2)*c[n,k+2]+(1/(k+1))*c[n-1,k-3]}$
$\displaystyle\quad\quad\quad\bm{-(2k/(k+1))*c[n-1,k-1]+(k+1)*c[n-1,k+1];];}$
$\displaystyle\quad\bm{c[n,k]=0;}$ $\displaystyle\quad\bm{c[n,k-2]=0;];}$
$\displaystyle\bm{\textbf{Ccoefficient}[\textbf{n${}_{-}$},\textbf{t${}_{-}$}]\,\textbf{:=}\,\textbf{Sum}[c[n,3*n+2-2*k]*t{}^{\wedge}(3*n+2-2*k),}$
$\displaystyle\hphantom{\textbf{Ccoefficient}[\textbf{n${}_{-}$},\textbf{t${}_{-}$}]\,\textbf{:=}\,}\bm{\\{k,0,3/2*n+1\\}];}$
$\displaystyle\bm{\textbf{Table}[\textbf{Ccoefficient}[n,t],\\{n,0,\textbf{NN}\\}]}$

This code generates a list of coefficients $C_{n}(t)$ for $0\leq n\leq 10$.
Adjusting the value of the variable NN allows for computing a different number
of coefficients.

### Acknowledgements

The author’s research was supported by the JSPS KAKENHI Grants No. JP21F21020
and No. 22H01146. The author is grateful to the referees for their
comprehensive evaluation of the manuscript.

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