# Dingle’s final main rule, Berry’s transition,
and Howls’ conjecture

Gergő Nemes  Department of Analysis, Alfréd Rényi Institute of Mathematics,
Reáltanoda utca 13–15, Budapest H-1053, Hungary<EMAIL_ADDRESS>Department of Physics, Tokyo Metropolitan University, 1–1 Minami-osawa,
Hachioji-shi, Tokyo 192-0397, Japan<EMAIL_ADDRESS>Dedicated to Sir Michael
V. Berry on the occasion of his 80${}^{\text{th}}$ birthday.

###### Abstract.

The Stokes phenomenon is the apparent discontinuous change in the form of the
asymptotic expansion of a function across certain rays in the complex plane,
known as Stokes lines, as additional expansions, pre-factored by exponentially
small terms, appear in its representation. It was first observed by G. G.
Stokes while studying the asymptotic behaviour of the Airy function. R. B.
Dingle proposed a set of rules for locating Stokes lines and continuing
asymptotic expansions across them. Included among these rules is the “final
main rule” stating that half the discontinuity in form occurs on reaching the
Stokes line, and half on leaving it the other side. M. V. Berry demonstrated
that, if an asymptotic expansion is terminated just before its numerically
least term, the transition between two different asymptotic forms across a
Stokes line is effected smoothly and not discontinuously as in the
conventional interpretation of the Stokes phenomenon. On a Stokes line, in
accordance with Dingle’s final main rule, Berry’s law predicts a multiplier of
$\frac{1}{2}$ for the emerging small exponentials. In this paper, we consider
two closely related asymptotic expansions in which the multipliers of
exponentially small contributions may no longer obey Dingle’s rule: their
values can differ from $\frac{1}{2}$ on a Stokes line and can be non-zero only
on the line itself. This unusual behaviour of the multipliers is a result of a
sequence of higher-order Stokes phenomena. We show that these phenomena are
rapid but smooth transitions in the remainder terms of a series of optimally
truncated hyperasymptotic re-expansions. To this end, we verify a conjecture
due to C. J. Howls.

###### Key words and phrases:

asymptotic expansions, Stokes phenomenon, hyperasymptotics, gamma function

###### 2020 Mathematics Subject Classification:

41A60, 41A80, 33B15

##  1\. Introduction

During the course of a Stokes phenomenon, an asymptotic expansion of a
function can change its form near certain rays in the complex plane, known as
Stokes lines, through the seemingly discontinuous appearance of further series
with exponentially small pre-factors. It was first observed by G. G. Stokes
[27] while studying the asymptotic behaviour of the Airy function. Although
negligible at the place of their birth, the exponentially small contributions
introduced by the Stokes phenomenon can grow to significantly influence or
dominate the behaviour of the function in other regions. As noted in [13], due
to the omission of exponentially small terms in the subsequently adopted
definition of an asymptotic expansion formulated by H. Poincaré [25], the
Stokes phenomenon caused controversy, ambiguity and misunderstanding for over
a century.

During his work in solid-state physics, R. B. Dingle encountered mathematical
difficulties, associated with the evaluation of some integrals, leading to
divergent asymptotic series. Dingle realised the limitations of Poincaré’s
prescription and that existing techniques for interpreting asymptotic series
were often vague and severely limited in accuracy and range of applicability.
Building on previous works by T. J. Stieltjes, J. R. Airey and others, Dingle
developed a new and extensive formal, interpretative theory of asymptotic
series, and summarised his findings in his exhaustive and defining 1973 book,
_Asymptotic Expansions: Their Derivation and Interpretation_ [12]. In his
book, Dingle proposed a set of rules for locating Stokes lines and continuing
asymptotic expansions across them. To illustrate these rules in the simplest
case of a compound asymptotic expansion consisting of just two asymptotic
series, we consider the parabolic cylinder functions. Let $U(a,z)$ and
$V(a,z)$ denote those solutions of the parabolic cylinder equation

(1.1)
$\frac{\mathrm{d}^{2}w}{\mathrm{d}z^{2}}=\left(\tfrac{1}{4}z^{2}+a\right)w,\quad
a\in\mathbb{R},$

which are respectively exponentially decreasing and increasing when $z$ is
real and positive, and are real-valued for real $z$. The solution $U(a,z)$, up
to scalar multiplication, is uniquely determined by these properties, whereas
$V(a,z)$ may be defined uniquely through power series [21, §12.4]. By trial
substitution to the equation (1.1), the asymptotic expansions of these
solutions are found to be

(1.2)
$U(a,z)\sim\mathrm{e}^{-\frac{1}{4}z^{2}}z^{-a-\frac{1}{2}}\sum\limits_{n=0}^{\infty}(-1)^{n}\frac{\left(\frac{1}{2}+a\right)_{2n}}{n!(2z^{2})^{n}}$

and

(1.3)
$V(a,z)\sim\sqrt{\frac{2}{\pi}}\mathrm{e}^{\frac{1}{4}z^{2}}z^{a-\frac{1}{2}}\sum\limits_{n=0}^{\infty}\frac{\left(\frac{1}{2}-a\right)_{2n}}{n!(2z^{2})^{n}},$

for large positive $z$ and fixed $a$, with $(w)_{n}=\Gamma(w+n)/\Gamma(w)$.
The factor $\sqrt{2/\pi}$ in the expansion of $V(a,z)$ is inserted to match
the normalisation convention in the literature, e.g., [21, Ch. 12]. The rays
$\arg z=\pm\frac{\pi}{2}$ are Stokes lines for $U(a,z)$ where the first series
(1.2) maximally dominates the second series (1.3) (Dingle’s second rule). This
also follows from the like signs and phases of the late terms in (1.2)
(Dingle’s first rule). When the ray $\arg z=\frac{\pi}{2}$ is crossed the
Stokes phenomenon switches on a divergent series proportional to (1.3)
(Dingle’s fourth rule). When $\arg z=\pi$, the second series (1.3) is at peak
dominance relative to the first series, so has a Stokes line there (Dingle’s
second rule). This is also clear from the like signs and phases of the late
terms in (1.3) (Dingle’s first rule). Consequently, the asymptotic behaviour
of $U(a,z)$ in the sector $|\arg z|<\frac{\pi}{2}$ is described by the
expansion (1.2), while in the sector $\frac{\pi}{2}<\arg z<\pi$ we have the
compound expansion

(1.4)
$U(a,z)\sim\mathrm{e}^{-\frac{1}{4}z^{2}}z^{-a-\frac{1}{2}}\sum\limits_{n=0}^{\infty}(-1)^{n}\frac{\left(\frac{1}{2}+a\right)_{2n}}{n!(2z^{2})^{n}}+S\sqrt{\frac{2}{\pi}}\mathrm{e}^{\frac{1}{4}z^{2}}z^{a-\frac{1}{2}}\sum\limits_{n=0}^{\infty}\frac{\left(\frac{1}{2}-a\right)_{2n}}{n!(2z^{2})^{n}}.$

The magnitude of the constant $S$, which may depend on $a$ but is independent
of $z$, is unknown at this stage. Its value can be determined by continuing
the analysis for the next phase sector $\pi<\arg z<\frac{3\pi}{2}$ and
alluding to the fact that $U(a,z)$ is real when $z$ is real (cf. [12, pp.
9–10]), or by examining the asymptotic behaviour of the high-order
coefficients in (1.2) [19] (see also [29, §16.511] for yet another derivation
using power series). With either of these methods, it is found that

$S=\frac{\pi\mathrm{i}}{\Gamma\left(\frac{1}{2}+a\right)}\mathrm{e}^{-\pi\mathrm{i}a}.$

If $S$ is regarded not as a constant but as a function of $\theta=\arg z$, the
expansion (1.4) applies in the larger sector $-\frac{\pi}{2}<\arg z<\pi$
provided we set

(1.5) $S=S(\theta)=\begin{cases}0&\text{if
}\;|\theta|<\frac{\pi}{2},\\\\[2.5pt]
\dfrac{\pi\mathrm{i}}{2\Gamma\left(\frac{1}{2}+a\right)}\mathrm{e}^{-\pi\mathrm{i}a}&\text{if
}\;\theta=\frac{\pi}{2},\\\\[8.00003pt]
\dfrac{\pi\mathrm{i}}{\Gamma\left(\frac{1}{2}+a\right)}\mathrm{e}^{-\pi\mathrm{i}a}&\text{if
}\;\frac{\pi}{2}<\theta<\pi.\end{cases}$

The $S$ is called the Stokes multiplier. The value of $S$ on the Stokes line
$\theta=\frac{\pi}{2}$ was inferred from Dingle’s “final main rule”, namely
that half the discontinuity in form occurs on reaching the Stokes line, and
half on leaving it the other side. In Dingle’s terminology, the combination of
(1.4) and (1.5) constitutes the complete asymptotic expansion of $U(a,z)$ in
the sector $-\frac{\pi}{2}<\arg z<\pi$. A similar analysis may be carried out
for the conjugate sector $-\pi<\arg z<-\frac{\pi}{2}$ and for the expansion
(1.3) of $V(a,z)$.

This is the conventional view of the Stokes phenomenon, where the multiplier
of the exponentially small contribution changes abruptly on crossing a Stokes
line. In his insightful paper [3], M. V. Berry adopted the view that the
Stokes multiplier should be regarded as a continuous function of $\arg z$,
rather than a discontinuous one. In our example above, this is achieved by
terminating the two series in (1.4): the first series is truncated optimally
(that is, just before its least term) after $N$ terms, where
$N=\lfloor\frac{1}{2}\left|z\right|^{2}-a\rfloor$, and the second series is
truncated at its first term. A modified Stokes multiplier $\mathcal{S}_{N}$
for the Stokes line $\arg z=\frac{\pi}{2}$ is then defined exactly by the
identity

$U(a,z)=\mathrm{e}^{-\frac{1}{4}z^{2}}z^{-a-\frac{1}{2}}\sum\limits_{n=0}^{N-1}(-1)^{n}\frac{\left(\frac{1}{2}+a\right)_{2n}}{n!(2z^{2})^{n}}+\mathcal{S}_{N}\frac{\sqrt{2\pi}}{\Gamma\left(\frac{1}{2}+a\right)}\mathrm{e}^{\frac{1}{4}z^{2}}(z\mathrm{e}^{-\pi\mathrm{i}})^{a-\frac{1}{2}}.$

The $\mathcal{S}_{N}=\mathcal{S}_{N}(\theta)$ is a piecewise continuous
function of $|z|$, and a continuous function of $\theta=\arg z$. Berry
demonstrated that if $|z|$ is large and held fixed, then $\mathcal{S}_{N}$
changes rapidly but smoothly from approximately $0$ to approximately $1$ as
$\theta$ increases continuously from values somewhat below $\frac{\pi}{2}$ to
values somewhat above $\frac{\pi}{2}$. This is consistent with (1.4) and (1.5)
above. In other words, Berry showed that the contribution of the exponentially
small term in the neighbourhood of the Stokes line $\arg z=\frac{\pi}{2}$ is
introduced smoothly, and not as an abrupt jump as was commonly supposed.
Moreover, Berry demonstrated that for a wide class of asymptotic expansions,
this smooth transition of the Stokes multiplier obeys a universal form given
approximately by an error function whose argument is an appropriate variable
describing the transition across a Stokes line. In our example, Berry’s
approximation takes the form

$\mathcal{S}_{N}\approx\frac{1}{2}+\frac{1}{2}\operatorname{erf}\left(\frac{1}{2}\frac{\operatorname{Im}(-z^{2})}{\sqrt{\operatorname{Re}(-z^{2})}}\right)=\frac{1}{2}\operatorname{erfc}\left(\frac{1}{2}\frac{\operatorname{Im}(z^{2})}{\sqrt{\operatorname{Re}(-z^{2})}}\right),$

where $\operatorname{erf}$ and $\operatorname{erfc}$ are the error function
and the complementary error function, respectively. Thus, $\mathcal{S}_{N}$
increases smoothly from approximately $0$ when
$\operatorname{Im}(z^{2})/\sqrt{\operatorname{Re}(-z^{2})}$ is large and
negative (i.e., $|z|$ is large and $\arg z$ is just below $\frac{\pi}{2}$) to
approximately $1$ when
$\operatorname{Im}(z^{2})/\sqrt{\operatorname{Re}(-z^{2})}$ is large and
positive (i.e., $|z|$ is large and $\arg z$ is just above $\frac{\pi}{2}$). On
the Stokes line $\arg z=\frac{\pi}{2}$,
$\operatorname{Im}(z^{2})/\sqrt{\operatorname{Re}(-z^{2})}=0$ and therefore
$\mathcal{S}_{N}\approx\frac{1}{2}$, in accordance with Dingle’s final main
rule.

Berry’s approximation for the Stokes multiplier $\mathcal{S}_{N}$ is derived
by application of Dingle’s theory of terminants. Dingle observed that,
although the early terms of an asymptotic series can rapidly get extremely
complicated, the high-order coefficients of a wide class of asymptotic
expansions display a universal and simple form. (This phenomenon may be seen
as a consequence of G. M. Darboux’s approximation of high derivatives in terms
of factorials [11].) By a clever application of É. Borel’s summation method to
the asymptotic form of the high orders, Dingle transformed the divergent tail
of an asymptotic expansion into a new series in terms of certain integrals
that he called “basic terminants”. The resulting series, Dingle’s
“interpretation” of the asymptotic expansion, can yield exceedingly accurate
approximations and automatically incorporates the Stokes phenomenon. He
envisaged in his book that “these terminant expansions can themselves be
closed with new terminants; and so on stage after stage”. Such re-summations
are now known as hyperasymptotics and have a well-established theory [2, 5, 6,
16].

In this paper, we revisit two familiar, closely related asymptotic expansions,
namely those of the gamma function $\Gamma(z)$ and its reciprocal. These
asymptotic expansions exhibit a significantly different behaviour than that
discussed above: in the neighbourhood of the rays $\arg z=\pm\frac{\pi}{2}$,
not one but infinitely many exponentially small contributions appear, each
associated with its own Stokes multiplier. Furthermore, these contributions
may no longer obey Dingle’s rule: the value of their associated Stokes
multipliers can differ from $\frac{1}{2}$ on the Stokes line and may be non-
zero only on the line itself. This unusual behaviour of the multipliers is a
result of a sequence of higher-order Stokes phenomena. We will demonstrate
that if the asymptotic expansions are interpreted as a series of optimally
truncated hyperasymptotic re-expansions, the multiplier of each subdominant
exponential experiences a smooth, but rapid, change in value described by a
multivariate polynomial in complementary error functions. This result is a
natural extension of Berry’s theory to a particular, yet important, case of
the higher-order Stokes phenomenon. Our proof relies crucially on a non-
trivial identity between hyperterminants, which was originally conjectured by
C. J. Howls.

The remaining part of the paper is organised as follows. In Section 2, we
derive the complete asymptotic expansions of the gamma function and its
reciprocal and determine the Stokes multipliers in the discontinuous sense.
The smooth interpretation of the (higher-order) Stokes phenomena via
hyperasymptotics is described in Section 3. In Section 4, we settle Howls’
conjecture on the hyperterminants. Section 5 contains the main result of the
paper: the uniform asymptotic approximations for the hyperterminants in terms
of complementary error functions. The paper concludes with a short discussion
in Section 6.

##  2\. The Stokes phenomenon associated with the gamma function and its
reciprocal

A well-known representation of the logarithm of the gamma function is given by

(2.1) $\log\Gamma(z)=\left(z-\frac{1}{2}\right)\log
z-z+\frac{1}{2}\log(2\pi)+g(z),$

where $g(z)$ is an analytic function of $z$ on the principal sheet $|\arg
z|<\pi$ of the complex logarithm. Several explicit representations for $g(z)$
are available in the literature (see, e.g., [24, §2.1.1]). It is well known
that as $z\to\infty$ in $|\arg z|<\frac{\pi}{2}$, $g(z)$ possesses the
asymptotic expansion

(2.2)
$g(z)\sim\sum\limits_{n=1}^{\infty}\frac{B_{2n}}{2n(2n-1)z^{2n-1}}=\frac{1}{12z}-\frac{1}{360z^{3}}+\frac{1}{1260z^{5}}-\ldots$

where $B_{n}$ are the Bernoulli numbers (see [21, Eq. 5.11.1]). Since
successive even-order Bernoulli numbers have opposite signs, all terms in the
asymptotic expansion (2.2) have the same phase when $\arg z=\pm\frac{\pi}{2}$,
so these are Stokes lines for $\log\Gamma(z)$.

To obtain the expansion in the sectors $\frac{\pi}{2}<|\arg z|<\pi$, we can
use the reflection formula in the form

$\Gamma(-z)=-\frac{\pi}{z\sin(\pi z)}\frac{1}{\Gamma(z)}.$

Substitution into (2.1) shows that $g(z)$ must satisfy the functional relation

(2.3) $g(z)=-g(z\mathrm{e}^{\mp\pi\mathrm{i}})-\log(1-\mathrm{e}^{\pm
2\pi\mathrm{i}z})=-g(z\mathrm{e}^{\mp\pi\mathrm{i}})+\sum\limits_{k=1}^{\infty}\frac{\mathrm{e}^{\pm
2\pi\mathrm{i}kz}}{k}.$

The upper or lower sign is taken according as $z$ is in the upper or lower
left half-plane. Thus the analytic continuation of $g(z)$ has generated an
infinite string of exponential terms. These terms are subdominant compared to
the series (2.2) in the upper and lower half-planes and are maximally
subdominant on the Stokes lines $\arg z=\pm\frac{\pi}{2}$, respectively. As
$z$ approaches the negative real axis, however, these exponentials steadily
grow in magnitude and add to generate the poles of $\Gamma(z)$.

The combination of (2.1), (2.2) and (2.3), and Dingle’s final main rule yields
the complete asymptotic expansion of $\log\Gamma(z)$ in the sector $|\arg
z|<\pi$:

(2.4) $\log\Gamma(z)\sim\left(z-\frac{1}{2}\right)\log
z-z+\frac{1}{2}\log(2\pi)+\sum\limits_{n=1}^{\infty}\frac{B_{2n}}{2n(2n-1)z^{2n-1}}+\sum\limits_{k=1}^{\infty}S^{(k)}\frac{\mathrm{e}^{\pm
2\pi\mathrm{i}kz}}{k},$

where the Stokes multipliers $S^{(k)}$ are

$S^{(k)}=S^{(k)}(\theta)=\begin{cases}0&\text{if
}\;|\theta|<\frac{\pi}{2},\\\\[2.5pt] \frac{1}{2}&\text{if
}\;\theta=\pm\frac{\pi}{2},\\\\[2.5pt] 1&\text{if
}\;\frac{\pi}{2}<|\theta|<\pi,\end{cases}$

with $\theta=\arg z$. R. B. Paris and A. D. Wood [23] gave an alternative
derivation of the expansions on the Stokes lines by invoking the specific form
of $\left|\Gamma(\mathrm{i}y)\right|$, $y>0$.

It is expected that the small exponentials emerge from the remainder of the
optimally truncated divergent series (2.2) in a smooth manner, according to
the error function law, as $z$ crosses the rays $\arg z=\pm\frac{\pi}{2}$. The
demonstration of the smooth transition of the leading subdominant exponentials
($k=1$ in (2.4)) was first given by Paris and Wood [23]. Berry [4] proved by a
different method, involving a sequence of increasingly delicate subtractions
of optimally-truncated asymptotic series, that all the smaller exponentials
appear in this way. His result was later reproduced by Paris and D. Kaminski
using Mellin–Barnes integrals [24, §6.4.2] (see also [22]).

Our primary interest in this paper is concerned with the asymptotics, and the
related Stokes phenomena, of the gamma function itself rather than its
logarithm. With regard to this case, it is convenient to introduce the scaled
gamma function $\Gamma^{\ast}(z)$ defined by

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

The function $\Gamma^{\ast}(z)$ can be analytically continued as a meromorphic
function on the Riemann surface $\widehat{\mathbb{C}}$ associated with the
complex logarithm. Exponentiation of (2.2) produces the familiar asymptotic
expansion

(2.5)
$\Gamma^{\ast}(z)\sim\sum\limits_{n=0}^{\infty}(-1)^{n}\frac{\gamma_{n}}{z^{n}}=1+\frac{1}{12z}+\frac{1}{288z^{2}}-\frac{139}{51840z^{3}}-\ldots,$

as $z\to\infty$ in the sector $|\arg z|<\frac{\pi}{2}$, where $\gamma_{n}$ are
the so-called Stirling coefficients (see, e.g., [24, §2.1.2]). The
corresponding expansion for the reciprocal function is given by

(2.6)
$\frac{1}{\Gamma^{\ast}(z)}\sim\sum\limits_{n=0}^{\infty}\frac{\gamma_{n}}{z^{n}}=1-\frac{1}{12z}+\frac{1}{288z^{2}}+\frac{139}{51840z^{3}}-\ldots,$

as $z\to\infty$ in $|\arg z|<\frac{\pi}{2}$. Observe that (2.6) involves the
same set of coefficients as (2.5) but with different signs of the coefficients
with odd index. This may be explained by noting that the series (2.2) has odd
powers only [28, p. 63]. We remark that the reciprocal function is analytic on
$\widehat{\mathbb{C}}$.

The Stokes phenomena associated with the gamma function and its reciprocal are
directly inherited from that of the function $\log\Gamma(z)$. In particular,
the positive and negative imaginary axes are Stokes lines. There are however
significant differences which only become apparent when studying complete
asymptotic expansions and the values of Stokes multipliers on the rays $\arg
z=\pm\frac{\pi}{2}$. Indeed, exponentiating (2.4) and using the standard
Maclaurin series of the functions $\exp(\mp\log(1-w))=(1-w)^{\mp 1}$ and
$\exp\big{(}\mp\frac{1}{2}\log(1-w)\big{)}=(1-w)^{\mp 1/2}$ yields the
complete asymptotic expansions of $\Gamma^{\ast}(z)$ and its reciprocal in the
sector $|\arg z|<\pi$:

(2.7)
$\Gamma^{\ast}(z)\sim\sum\limits_{n=0}^{\infty}(-1)^{n}\frac{\gamma_{n}}{z^{n}}+\sum\limits_{k=1}^{\infty}\mathbf{S}^{(k)}\mathrm{e}^{\pm
2\pi\mathrm{i}kz}\sum\limits_{n=0}^{\infty}(-1)^{n}\frac{\gamma_{n}}{z^{n}}$

and

(2.8)
$\frac{1}{\Gamma^{\ast}(z)}\sim\sum\limits_{n=0}^{\infty}\frac{\gamma_{n}}{z^{n}}-\sum\limits_{k=1}^{\infty}\widetilde{\mathbf{S}}^{(k)}\mathrm{e}^{\pm
2\pi\mathrm{i}kz}\sum\limits_{n=0}^{\infty}\frac{\gamma_{n}}{z^{n}},$

where the Stokes multipliers $\mathbf{S}^{(k)}$ and
$\widetilde{\mathbf{S}}^{(k)}$ are

(2.9) $\mathbf{S}^{(k)}=\mathbf{S}^{(k)}(\theta)=\begin{cases}0&\text{if
}\;|\theta|<\frac{\pi}{2},\\\\[2.5pt]
\frac{1}{k!}\left(\frac{1}{2}\right)_{k}&\text{if
}\;\theta=\pm\frac{\pi}{2},\\\\[2.5pt] 1&\text{if
}\;\frac{\pi}{2}<|\theta|<\pi,\end{cases}$

for $k\geq 1$, and

(2.10)
$\widetilde{\mathbf{S}}^{(1)}=\widetilde{\mathbf{S}}^{(1)}(\theta)=\begin{cases}0&\text{if
}\;|\theta|<\frac{\pi}{2},\\\\[2.5pt] \frac{1}{2}&\text{if
}\;\theta=\pm\frac{\pi}{2},\\\\[2.5pt] 1&\text{if
}\;\frac{\pi}{2}<|\theta|<\pi,\end{cases}\quad\widetilde{\mathbf{S}}^{(k)}=\widetilde{\mathbf{S}}^{(k)}(\theta)=\begin{cases}0&\text{if
}\;|\theta|<\frac{\pi}{2},\\\\[2.5pt]
\frac{-1}{k!}\left(-\frac{1}{2}\right)_{k}&\text{if
}\;\theta=\pm\frac{\pi}{2},\\\\[2.5pt] 0&\text{if
}\;\frac{\pi}{2}<|\theta|<\pi,\end{cases}$

for $k\geq 2$, and with $\theta=\arg z$ (cf. [14, §3.2]). The upper or lower
signs are taken in (2.7) and (2.8) according as $z$ is in the upper or lower
left half-plane. Expansions of type (2.7) and (2.8) are commonly referred to
as transseries expansions. It is seen from (2.7) that when either the positive
or the negative imaginary axis is crossed the Stokes phenomenon switches on an
infinite copy of the expansion (2.5), pre-factored by increasingly subdominant
exponentially small terms. But the situation is different from the case of
$\log\Gamma(z)$, in that the Stokes multipliers of the sub-subdominant
contributions (corresponding to $k\geq 2$) do not follow Dingle’s final main
rule. The complete asymptotic expansion (2.8) of the reciprocal function
exhibits an even more striking behaviour: an infinite number of exponentially
small contributions of decreasing magnitude appear on each of the Stokes lines
$\arg z=\pm\frac{\pi}{2}$ from which only the leading one ($k=1$ in (2.8))
survives on the respective other side. (The disappearance of the exponentials
is essential since the reciprocal gamma function is analytic along the
negative real axis.) This unconventional behaviour of the Stokes multipliers
is a consequence of a series of higher-order Stokes phenomena taking place
across the lines $\arg z=\pm\frac{\pi}{2}$. These phenomena are not present in
the expansion (2.2) of $\log\Gamma(z)$. At a higher-order Stokes phenomenon, a
Stokes multiplier itself can change value, and in particular the potential for
a Stokes phenomenon to occur is changed (for a general account see [13]). This
is indeed reflected in the expressions for $\mathbf{S}^{(k)}$ and
$\widetilde{\mathbf{S}}^{(k)}$, $k\geq 2$.

The smooth interpretation of the Stokes phenomenon associated with the leading
subdominant exponentials $\mathrm{e}^{\pm 2\pi\mathrm{i}z}$ in (2.7) and (2.8)
was studied by W. G. C. Boyd [8] and the present author [14]. They verified
that these small exponentials are born from the remainders of the optimally
truncated expansions (2.5) and (2.6), and obey Berry’s error function
smoothing law. As Berry [4] remarks, the smoothings of the smaller
exponentials which are apparent with $\log\Gamma(z)$ are obscured if one
considers $\Gamma(z)$ instead, and their recovery might require
hyperasymptotics. In the subsequent section we shall show this indeed to be
the case: the smaller exponentials are born from the remainder terms of
successive hyperasymptotic re-expansions. However, there are two major
differences compared with $\log\Gamma(z)$. First, the switching-on of
successively smaller exponential contributions can only be revealed one at a
time, and second, the appearance of the sub-subdominant expansions is no
longer described by a single error function. (We remark that an example of the
non-universality of the error function smoothing law was also found by S. J.
Chapman [10] in the solution of an inhomogeneous delay equation.)

##  3\. Hyperasymptotics and Stokes smoothings

In this section, we discuss the hyperasymptotics process and the smoothings of
the (higher-order) Stokes phenomena for the gamma function and its reciprocal.
Briefly, hyperasymptotics is the systematic re-expansion of successive
remainder terms in an asymptotic expansion that automatically incorporates the
Stokes phenomenon. Each step of the process reduces the size of the error term
by an exponentially small factor.

For any positive integer $N$, we denote by $R_{N}(z)$ and
$\widetilde{R}_{N}(z)$ the remainders of the series (2.5) and (2.6) truncated
to $N$ terms:

(3.1)
$\Gamma^{\ast}(z)=\sum\limits_{n=0}^{N-1}(-1)^{n}\frac{\gamma_{n}}{z^{n}}+R_{N}(z)\quad\text{
and
}\quad\frac{1}{\Gamma^{\ast}(z)}=\sum\limits_{n=0}^{N-1}\frac{\gamma_{n}}{z^{n}}+\widetilde{R}_{N}(z).$

Note that for each $N$, $R_{N}(z)$ and $\widetilde{R}_{N}(z)$ are analytic
functions of $z$ in the sector $|\arg z|<\pi$. The basis of hyperasymptotics
will be the exact resurgence formulae ([8, p. 616], [14, p. 576])

(3.2)
$R_{N}(z)=\frac{1}{2\pi\mathrm{i}}\frac{1}{z^{N}}\int_{0}^{[\pi/2^{-}]}\frac{\mathrm{e}^{2\pi\mathrm{i}t}t^{N-1}}{1-t/z}\Gamma^{\ast}(t)\mathrm{d}t-\frac{1}{2\pi\mathrm{i}}\frac{1}{z^{N}}\int_{0}^{[-\pi/2^{+}]}\frac{\mathrm{e}^{-2\pi\mathrm{i}t}t^{N-1}}{1-t/z}\Gamma^{\ast}(t)\mathrm{d}t$

and

(3.3)
$\widetilde{R}_{N}(z)=-\frac{1}{2\pi\mathrm{i}}\frac{1}{z^{N}}\int_{0}^{[\pi/2^{-}]}\frac{\mathrm{e}^{2\pi\mathrm{i}t}t^{N-1}}{1-t/z}\Gamma^{\ast}(t\mathrm{e}^{-\pi\mathrm{i}})\mathrm{d}t+\frac{1}{2\pi\mathrm{i}}\frac{1}{z^{N}}\int_{0}^{[-\pi/2^{+}]}\frac{\mathrm{e}^{-2\pi\mathrm{i}t}t^{N-1}}{1-t/z}\Gamma^{\ast}(t\mathrm{e}^{\pi\mathrm{i}})\mathrm{d}t.$

Here and throughout the paper we use the notation
$\int_{0}^{\left[\eta\right]}=\int_{0}^{+\infty\mathrm{e}^{\mathrm{i}\eta}}$.
These formulae are valid under the assumption that $|\arg z|<\frac{\pi}{2}$,
and they both incorporate the Stokes phenomenon. For example as $\arg z$
increases beyond $\frac{\pi}{2}$, the pole at
$t=|z|\mathrm{e}^{\pi\mathrm{i}/2}$ in the first integral of (3.3) is
entrapped: consequently for $\frac{\pi}{2}<\arg z<\frac{3\pi}{2}$, the term
$-\mathrm{e}^{2\pi\mathrm{i}z}\Gamma^{\ast}(z\mathrm{e}^{-\pi\mathrm{i}})$ has
to be added to the right-hand side of (3.3). Expanding
$\Gamma^{\ast}(z\mathrm{e}^{-\pi\mathrm{i}})$ via (2.6) then reproduces (2.8).

We now define certain multiple integrals, the so-called “hyperterminants”,
which form the basis of the hyperasymptotics scheme. These hyperterminants are
multidimensional generalisations of Dingle’s basic terminants. For $m$ a non-
negative integer, we define

(3.4) $F^{(0)}(z)=1,\qquad
F^{(1)}\\!\left(z;\\!\begin{array}[]{c}N_{1}\\!\\!\\\ \sigma_{1}\\!\\!\\\
\end{array}\right)=\int_{0}^{[\pi-\arg\sigma_{1}]}\frac{\mathrm{e}^{\sigma_{1}t_{1}}t_{1}^{N_{1}-1}}{z-t_{1}}\mathrm{d}t_{1},$
(3.5) $F^{(m)}\\!\left(z;\\!\begin{array}[]{c}N_{1},\,\ldots,\\!\\!\\!\\!\\\
\,\sigma_{1},\,\,\ldots,\\!\\!\\!\\!\\\
\end{array}\begin{array}[]{c}N_{m}\\!\\!\\\ \sigma_{m}\\!\\!\\\
\end{array}\right)=\int_{0}^{[\pi-\arg\sigma_{1}]}\frac{\mathrm{e}^{\sigma_{1}t_{1}}t_{1}^{N_{1}-1}}{z-t_{1}}\prod_{k=2}^{m}\int_{0}^{[\pi-\arg\sigma_{k}]}\frac{\mathrm{e}^{\sigma_{k}t_{k}}t_{k}^{N_{k}-1}}{t_{k-1}-t_{k}}\mathrm{d}t_{k}\mathrm{d}t_{1},$

for an arbitrary set of complex numbers $N_{1},\ldots,N_{m}$ such that
$\operatorname{Re}(N_{k})>1$ with $k=1,\ldots,m$, and for an arbitrary set of
elements $\sigma_{1},\ldots,\sigma_{m}$ of $\widehat{\mathbb{C}}$ such that
$\arg\sigma_{k}\neq\arg\sigma_{k+1}\pmod{2\pi}$ with $k=1,\ldots,m$. These
multiple integrals converge provided $|\arg(\sigma_{1}z)|<\pi$, and can be
extended by analytic continuation to other values of $\arg z$. The $F^{(m)}$
is termed an $m$th-level hyperterminant. We will refer to the quantities
$\sigma_{k}$ as “singulants”, a term originating from the work of Dingle. We
remark that

(3.6) $F^{(1)}\\!\left(z;\\!\begin{array}[]{c}N\\!\\\ \sigma\\!\\\
\end{array}\right)=\mathrm{e}^{\pi\mathrm{i}N}\mathrm{e}^{\sigma
z}z^{N-1}\Gamma(N)\Gamma(1-N,\sigma z),$

where $\Gamma(a,w)$ is the incomplete gamma function. In the case that
$\arg\sigma_{k}=\arg\sigma_{k+1}\pmod{2\pi}$ for some $k$, we have to specify
whether the $t_{k}$-contour is indented to the left or right of the pole of
$(t_{k-1}-t_{k})^{-1}$. We make the choice by defining

(3.7) $F^{(m)}\\!\left(z;\\!\begin{array}[]{c}N_{1},\,\ldots,\\!\\!\\!\\!\\\
\,\sigma_{1},\,\,\ldots,\\!\\!\\!\\!\\\
\end{array}\begin{array}[]{c}N_{m}\\!\\!\\\ \sigma_{m}\\!\\!\\\
\end{array}\right)=\mathop{\lim}\limits_{\varepsilon\to
0^{+}}F^{(m)}\\!\left(z;\\!\begin{array}[]{c}N_{1},\\!\\!\\!\\!\\\
\sigma_{1}\mathrm{e}^{(m-1)\varepsilon\mathrm{i}},\\!\\!\\!\\!\\\
\end{array}\begin{array}[]{c}N_{2},\quad\quad\quad\,\ldots,\\!\\!\\!\\!\\\
\sigma_{2}\mathrm{e}^{(m-2)\varepsilon\mathrm{i}},\,\ldots,\\!\\!\\!\\!\\\
\end{array}\begin{array}[]{c}N_{m}\\!\\!\\\ \sigma_{m}\\!\\!\\\
\end{array}\right).$

Thus, we require that the pole of $(t_{k-1}-t_{k})^{-1}$ is on the left-hand
side of the $t_{k}$-contour.

Throughout the paper, we will frequently use the connection formula

(3.8)
$\displaystyle\begin{split}F^{(m)}\\!\left(z\mathrm{e}^{-2\pi\mathrm{i}};\\!\begin{array}[]{c}N_{1},\,\ldots,\\!\\!\\!\\!\\\
\,\sigma_{1},\,\,\ldots,\\!\\!\\!\\!\\\
\end{array}\begin{array}[]{c}N_{m}\\!\\!\\\ \sigma_{m}\\!\\!\\\
\end{array}\right)=\;&F^{(m)}\\!\left(z;\\!\begin{array}[]{c}N_{1},\,\ldots,\\!\\!\\!\\!\\\
\,\sigma_{1},\,\,\ldots,\\!\\!\\!\\!\\\
\end{array}\begin{array}[]{c}N_{m}\\!\\!\\\ \sigma_{m}\\!\\!\\\
\end{array}\right)\\\
&-2\pi\mathrm{i}\mathrm{e}^{\sigma_{1}z}z^{N_{1}-1}F^{(m-1)}\\!\left(z;\\!\begin{array}[]{c}N_{2},\,\ldots,\\!\\!\\!\\!\\\
\,\sigma_{2},\,\,\ldots,\\!\\!\\!\\!\\\
\end{array}\begin{array}[]{c}N_{m}\\!\\!\\\ \sigma_{m}\\!\\!\\\
\end{array}\right),\end{split}$

valid when $m\geq 1$ [15, Eq. (2.6)].

The numerical evaluation of the hyperterminants is discussed in the paper by
T. B. Bennett et al. [2, Theorem A.1] (see [15] for basic properties).

###  3.1. Level 1 hyperasymptotics

We now derive the Level 1 hyperasymptotic expansions. In the integral
representations (3.2) and (3.3) for the remainders $R_{N}(z)$ and
$\widetilde{R}_{N}(z)$ we substitute (3.1) (with $M$ in place of $N$) into
$\Gamma^{\ast}(t)$ and $\Gamma^{\ast}(t\mathrm{e}^{\mp\pi\mathrm{i}})$. We
obtain, after term-by-term integration, the re-expansions

(3.9)
$\displaystyle\begin{split}R_{N}(z)=\;&\frac{1}{2\pi\mathrm{i}}\frac{1}{z^{N-1}}\sum\limits_{m=0}^{M-1}(-1)^{m}\gamma_{m}F^{(1)}\\!\left(z;\\!\begin{array}[]{c}N-m\\!\\\
2\pi\mathrm{e}^{\frac{\pi}{2}\mathrm{i}}\\!\\\ \end{array}\right)\\\
&-\frac{1}{2\pi\mathrm{i}}\frac{1}{z^{N-1}}\sum\limits_{m=0}^{M-1}(-1)^{m}\gamma_{m}F^{(1)}\\!\left(z;\\!\begin{array}[]{c}N-m\\!\\\
2\pi\mathrm{e}^{-\frac{\pi}{2}\mathrm{i}}\\!\\\
\end{array}\right)+R_{N,M}(z)\end{split}$

and

(3.10)
$\displaystyle\begin{split}\widetilde{R}_{N}(z)=&-\frac{1}{2\pi\mathrm{i}}\frac{1}{z^{N-1}}\sum\limits_{m=0}^{M-1}\gamma_{m}F^{(1)}\\!\left(z;\\!\begin{array}[]{c}N-m\\!\\\
2\pi\mathrm{e}^{\frac{\pi}{2}\mathrm{i}}\\!\\\ \end{array}\right)\\\
&+\frac{1}{2\pi\mathrm{i}}\frac{1}{z^{N-1}}\sum\limits_{m=0}^{M-1}\gamma_{m}F^{(1)}\\!\left(z;\\!\begin{array}[]{c}N-m\\!\\\
2\pi\mathrm{e}^{-\frac{\pi}{2}\mathrm{i}}\\!\\\
\end{array}\right)+\widetilde{R}_{N,M}(z),\end{split}$

with

(3.11)
$R_{N,M}(z)=\frac{1}{2\pi\mathrm{i}}\frac{1}{z^{N}}\int_{0}^{[\pi/2^{-}]}\frac{\mathrm{e}^{2\pi\mathrm{i}t}t^{N-1}}{1-t/z}R_{M}(t)\mathrm{d}t-\frac{1}{2\pi\mathrm{i}}\frac{1}{z^{N}}\int_{0}^{[-\pi/2^{+}]}\frac{\mathrm{e}^{-2\pi\mathrm{i}t}t^{N-1}}{1-t/z}R_{M}(t)\mathrm{d}t$

and

$\widetilde{R}_{N,M}(z)=-\frac{1}{2\pi\mathrm{i}}\frac{1}{z^{N}}\int_{0}^{[\pi/2^{-}]}\frac{\mathrm{e}^{2\pi\mathrm{i}t}t^{N-1}}{1-t/z}R_{M}(t\mathrm{e}^{-\pi\mathrm{i}})\mathrm{d}t+\frac{1}{2\pi\mathrm{i}}\frac{1}{z^{N}}\int_{0}^{[-\pi/2^{+}]}\frac{\mathrm{e}^{-2\pi\mathrm{i}t}t^{N-1}}{1-t/z}R_{M}(t\mathrm{e}^{\pi\mathrm{i}})\mathrm{d}t,$

provided $M<N$. The remainders $R_{N,M}(z)$ and $\widetilde{R}_{N,M}(z)$ are
defined initially for $|\arg z|<\frac{\pi}{2}$, but can be extended by a
standard analytic continuation argument to $|\arg z|<\pi$.

For the sake of completeness we shall now reproduce the results of [8, 14]. We
will show that the first few terms of the series in (2.7) and (2.8) with the
exponential pre-factor $\mathrm{e}^{2\pi\mathrm{i}z}$ are switched on smoothly
across the Stokes line $\arg z=\frac{\pi}{2}$ according to Berry’s universal
error function law. For this we examine the behaviour of the remainder terms
$R_{N}(z)$ and $\widetilde{R}_{N}(z)$ in the vicinity of the ray $\arg
z=\frac{\pi}{2}$. Assume that the asymptotic expansions (2.5) and (2.6) are
truncated so that the optimal value for $N$, $N=\left\lfloor
2\pi|z|\right\rfloor$ is chosen in (3.1) (cf. [8, p. 621]). Under this
assumption it can be shown that for each fixed $M$,

(3.12)
$R_{N,M}(z),\;\widetilde{R}_{N,M}(z)=\begin{cases}\mathcal{O}(\left|z\right|^{-M}\mathrm{e}^{-2\pi\left|z\right|})&\text{if
}\;|\arg z|\leq\frac{\pi}{2},\\\\[5.0pt]
\mathcal{O}(\left|z\right|^{-M}\mathrm{e}^{-2\pi\operatorname{Im}(z)})&\text{if
}\;\frac{\pi}{2}<\arg z\leq\pi-\delta<\pi,\end{cases}$

as $z\to\infty$, uniformly with respect to $\arg z$ [14, Theorem 1.4]. The
behaviour of $R_{N}(z)$ and $\widetilde{R}_{N}(z)$ as $z\to\infty$ in the
neighbourhood of the line $\arg z=\frac{\pi}{2}$ can now be deduced from
(3.9), (3.10), and the asymptotic properties of the function $F^{(1)}$. With
$\varphi=\arg(\sigma z)$, $N\sim|\sigma z|$ and fixed $n$, F. W. J. Olver [20]
showed that $F^{(1)}$ possesses the following asymptotic behaviour for large
$|\sigma z|$:

(3.13) $\frac{\mathrm{e}^{-\sigma
z}}{2\pi\mathrm{i}z^{N-1-n}}F^{(1)}\\!\left(z;\\!\begin{array}[]{c}N-n\\!\\\
\sigma\\!\\\
\end{array}\right)=-\frac{\mathrm{i}\mathrm{e}^{(\pi-\varphi)\mathrm{i}N}}{1+\mathrm{e}^{-\varphi\mathrm{i}}}\frac{\mathrm{e}^{-\sigma
z-\left|\sigma z\right|}}{\sqrt{2\pi\left|\sigma
z\right|}}(1+\mathcal{O}(|\sigma z|^{-1}))$

when $|\varphi|\leq\pi-\delta<\pi$, and

(3.14) $\frac{\mathrm{e}^{-\sigma
z}}{2\pi\mathrm{i}z^{N-1-n}}F^{(1)}\\!\left(z;\\!\begin{array}[]{c}N-n\\!\\\
\sigma\\!\\\
\end{array}\right)=\frac{1}{2}\operatorname{erfc}\Big{(}c(\varphi)\sqrt{\tfrac{1}{2}|\sigma
z|}\Big{)}+\mathcal{O}\bigg{(}\bigg{|}\frac{\mathrm{e}^{-\frac{1}{2}|\sigma
z|c^{2}(\varphi)}}{\sqrt{\left|\sigma z\right|}}\bigg{|}\bigg{)},$

when $-\pi<-\pi+\delta\leq\varphi\leq 3\pi-\delta<3\pi$, with a conjugate
behaviour in the sector $-3\pi<-3\pi+\delta\leq\varphi\leq\pi-\delta<\pi$. The
quantity $c(\varphi)$ is defined by

(3.15)
$\frac{1}{2}c^{2}(\varphi)=1+\mathrm{i}(\varphi-\pi)-\mathrm{e}^{\mathrm{i}(\varphi-\pi)},$

and corresponds to the branch of $c(\varphi)$ whose behaviour is

(3.16)
$c(\varphi)=-(\varphi-\pi)-\frac{\mathrm{i}}{6}(\varphi-\pi)^{2}+\frac{1}{36}(\varphi-\pi)^{3}+\ldots$

near $\varphi=\pi$. Graphs of $\frac{1}{2}c^{2}(\varphi)$ and $c(\varphi)$ are
given in the paper by Olver [20, Figs. 3 and 4].

We now observe that the main contributions to $R_{N}(z)$ and
$\widetilde{R}_{N}(z)$ arise from the first sums in (3.9) and (3.10),
respectively. Neglecting error terms and retaining only the leading term in
the expansion (3.16), we find that as $z\to\infty$ within a region centred on
the positive imaginary axis of angular width $\mathcal{O}(|z|^{-1/2})$,

(3.17)
$R_{N}(z)\sim\mathrm{e}^{2\pi\mathrm{i}z}\sum\limits_{m=0}^{M-1}(-1)^{m}\frac{\gamma_{m}}{z^{m}}\times\frac{1}{2}\operatorname{erfc}\left(\left(\tfrac{\pi}{2}-\theta\right)\sqrt{\pi|z|}\right)$

and

(3.18)
$\widetilde{R}_{N}(z)\sim-\mathrm{e}^{2\pi\mathrm{i}z}\sum\limits_{m=0}^{M-1}\frac{\gamma_{m}}{z^{m}}\times\frac{1}{2}\operatorname{erfc}\left(\left(\tfrac{\pi}{2}-\theta\right)\sqrt{\pi|z|}\right)$

with $\theta=\arg z$, provided $M\geq 1$ is fixed. The effect of the
complementary error function in (3.17) and (3.18) is to switch on the leading
terms of the series pre-factored by the exponential
$\mathrm{e}^{2\pi\mathrm{i}z}$ in (2.7) and (2.8) in the manner described by
Berry as $\theta$ increases through $\frac{\pi}{2}$. Note that in Berry’s
original formulation the argument of the complementary error function is
replaced by the approximate quantity

$\left(\tfrac{\pi}{2}-\theta\right)\sqrt{\pi|z|}\approx\frac{\operatorname{Im}(2\pi\mathrm{i}z)}{\sqrt{2\operatorname{Re}(-2\pi\mathrm{i}z)}}.$

A similar result may be obtained for the Stokes line $\arg z=-\frac{\pi}{2}$,
where the dominant contributions to $R_{N}(z)$ and $\widetilde{R}_{N}(z)$ come
from the second sums in (3.9) and (3.10).

It is clear from the estimates (3.12) that the sub-subdominant contributions
(corresponding to $k\geq 2$ in (2.7) and (2.8)) cannot be detected by
truncating the asymptotic expansions (2.5) and (2.6) near their least terms
and keeping the number of terms in the re-expansions fixed. We expect that the
contributions scaled by the factors $\mathrm{e}^{\pm 4\pi\mathrm{i}z}$ are
born from the remainders of the optimally truncated Level 1 hyperasymptotic
expansions (3.9) and (3.10). That this is indeed the case may be verified
numerically and graphically as follows. Optimal truncation is achieved when
$N=\lfloor 4\pi|z|\rfloor$ and $M=\lfloor 2\pi|z|\rfloor$.111Generally, the
optimal truncation indices at Level $k$ form a sequence in which the
difference between consecutive terms is about $2\pi|z|$. This is because the
difference between the exponents of two consecutive subdominant exponentials
in (2.7) and (2.8) is precisely $\pm 2\pi\mathrm{i}z$. See [2, §4.1] for
truncation schemes in general. The modified Stokes multipliers
$\mathcal{S}_{N,M}^{(2)}$ and $\widetilde{\mathcal{S}}_{N,M}^{(2)}$ associated
with the small exponential $\mathrm{e}^{4\pi\mathrm{i}z}$ for the Stokes line
$\arg z=\frac{\pi}{2}$ are then defined exactly by the identities

$R_{N,M}(z)=\mathcal{S}_{N,M}^{(2)}\mathrm{e}^{4\pi\mathrm{i}z}\quad\text{ and
}\quad\widetilde{R}_{N,M}(z)=-\widetilde{\mathcal{S}}_{N,M}^{(2)}\mathrm{e}^{4\pi\mathrm{i}z}.$

Both $\mathcal{S}_{N,M}^{(2)}=\mathcal{S}_{N,M}^{(2)}(\theta)$ and
$\widetilde{\mathcal{S}}_{N,M}^{(2)}=\widetilde{\mathcal{S}}_{N,M}^{(2)}(\theta)$
are piecewise continuous functions of $|z|$, and continuous functions of
$\theta=\arg z$. The real parts of these modified Stokes multipliers with
$|z|=5$ are plotted against $\arg z$ on Figure 1. The corresponding imaginary
parts are negligible in magnitude. It is seen that $\mathcal{S}_{N,M}^{(2)}$
increases smoothly from approximately $0$ to approximately $1$ as $\theta$
passes continuously from values somewhat below $\frac{\pi}{2}$ to values
somewhat above $\frac{\pi}{2}$. On the Stokes line $\theta=\frac{\pi}{2}$,
$\mathcal{S}_{N,M}^{(2)}\approx\frac{3}{8}$, in agreement with the
discontinuous treatment (2.9). Similarly, if $\theta$ changes from values
somewhat below $\frac{\pi}{2}$ to values somewhat above $\frac{\pi}{2}$, then
$\widetilde{\mathcal{S}}_{N,M}^{(2)}$ increases from approximately $0$ to
approximately $\frac{1}{8}$ (when $\theta=\frac{\pi}{2}$) and then soon
decreases to approximately $0$ again. This is consistent with the result
(2.10) above. A similar analysis may be carried out for the other Stokes line
$\arg z=-\frac{\pi}{2}$.

(a)

(b)

Figure 1. (a) The behaviour of $\operatorname{Re}(\mathcal{S}_{N,M}^{(2)})$
with $|z|=5$, $N=\lfloor 4\pi|z|\rfloor=62$ and $M=\lfloor 2\pi|z|\rfloor=31$
in the phase range $0<\arg z<\frac{3\pi}{4}$. (b) The behaviour of
$\operatorname{Re}(\widetilde{\mathcal{S}}_{N,M}^{(2)})$ with $|z|=5$,
$N=\lfloor 4\pi|z|\rfloor=62$ and $M=\lfloor 2\pi|z|\rfloor=31$ in the phase
range $0<\arg z<\pi$.

To obtain a rigorous analytical description of these smooth transitions, it is
necessary to go to the second stage of hyperasymptotics.

###  3.2. Level 2 hyperasymptotics

The Level 2 hyperasymptotic expansions are derived by re-expanding the Level 1
expansions. Thus, in the integral representation (3.11) for the remainder
$R_{N,M}(z)$ we substitute (3.9) (with $M$ in place of $N$, and $K$ in place
of $M$) into $R_{M}(t)$. We obtain, using the definitions (3.5), (3.7) and
term-by-term integration, the re-expansion

(3.19)
$\displaystyle\begin{split}R_{N,M}(z)=\;&\frac{1}{(2\pi\mathrm{i})^{2}}\frac{1}{z^{N-1}}\sum\limits_{k=0}^{K-1}(-1)^{k}\gamma_{k}F^{(2)}\\!\left(z;\\!\begin{array}[]{c}N-M+1,\\!\\!\\!\\!\\\
2\pi\mathrm{e}^{\frac{\pi}{2}\mathrm{i}},\\!\\!\\!\\!\\\
\end{array}\begin{array}[]{c}M-k\\\
2\pi\mathrm{e}^{\frac{\pi}{2}\mathrm{i}}\\\ \end{array}\right)\\\
&+\frac{1}{(2\pi\mathrm{i})^{2}}\frac{1}{z^{N-1}}\sum\limits_{k=0}^{K-1}(-1)^{k}\gamma_{k}F^{(2)}\\!\left(z;\\!\begin{array}[]{c}N-M+1,\\!\\!\\!\\!\\\
2\pi\mathrm{e}^{-\frac{\pi}{2}\mathrm{i}},\\!\\!\\!\\!\\\
\end{array}\begin{array}[]{c}M-k\\\
2\pi\mathrm{e}^{-\frac{\pi}{2}\mathrm{i}}\\\ \end{array}\right)\\\
&+\frac{1}{(2\pi\mathrm{i})^{2}}\frac{1}{z^{N-1}}\sum\limits_{k=0}^{K-1}(-1)^{k}\gamma_{k}F^{(2)}\\!\left(z;\\!\begin{array}[]{c}N-M+1,\\!\\!\\!\\!\\\
2\pi\mathrm{e}^{\frac{\pi}{2}\mathrm{i}},\\!\\!\\!\\!\\\
\end{array}\begin{array}[]{c}M-k\\\
2\pi\mathrm{e}^{-\frac{\pi}{2}\mathrm{i}}\\\ \end{array}\right)\\\
&+\frac{1}{(2\pi\mathrm{i})^{2}}\frac{1}{z^{N-1}}\sum\limits_{k=0}^{K-1}(-1)^{k}\gamma_{k}F^{(2)}\\!\left(z;\\!\begin{array}[]{c}N-M+1,\\!\\!\\!\\!\\\
2\pi\mathrm{e}^{-\frac{\pi}{2}\mathrm{i}},\\!\\!\\!\\!\\\
\end{array}\begin{array}[]{c}M-k\\\
2\pi\mathrm{e}^{\frac{\pi}{2}\mathrm{i}}\\\
\end{array}\right)+R_{N,M,K}(z),\end{split}$

with

$R_{N,M,K}(z)=\frac{1}{2\pi\mathrm{i}}\frac{1}{z^{N}}\int_{0}^{[\pi/2^{-}]}\frac{\mathrm{e}^{2\pi\mathrm{i}t}t^{N-1}}{1-t/z}R_{M,K}(t)\mathrm{d}t-\frac{1}{2\pi\mathrm{i}}\frac{1}{z^{N}}\int_{0}^{[-\pi/2^{+}]}\frac{\mathrm{e}^{-2\pi\mathrm{i}t}t^{N-1}}{1-t/z}R_{M,K}(t)\mathrm{d}t,$

provided $K<M<N$. The corresponding expansion for $\widetilde{R}_{N,M}(z)$ can
be derived in a similar manner, with the exception that special attention
should be paid when identifying the various $F^{(2)}$ hyperterminants. In the
instance that two singulants are identical, the argument of the function
$F^{(1)}$ arising from the application of (3.9) will lie outside the principal
domain of definition specified in (3.4). In these cases, the application of
the connection formula (3.8), with $m=1$, becomes necessary. Then we find,
after some algebraic computation, that the required re-expansion is

(3.20)
$\displaystyle\begin{split}\widetilde{R}_{N,M}(z)=&-\frac{1}{2\pi\mathrm{i}}\frac{1}{z^{N-1}}\sum\limits_{k=0}^{K-1}\gamma_{k}F^{(1)}\\!\left(z;\\!\begin{array}[]{c}N-k\\!\\\
4\pi\mathrm{e}^{\frac{\pi}{2}\mathrm{i}}\\!\\\ \end{array}\right)\\\
&+\frac{1}{(2\pi\mathrm{i})^{2}}\frac{1}{z^{N-1}}\sum\limits_{k=0}^{K-1}\gamma_{k}F^{(2)}\\!\left(z;\\!\begin{array}[]{c}N-M+1,\\!\\!\\!\\!\\\
2\pi\mathrm{e}^{\frac{\pi}{2}\mathrm{i}},\\!\\!\\!\\!\\\
\end{array}\begin{array}[]{c}M-k\\\
2\pi\mathrm{e}^{\frac{\pi}{2}\mathrm{i}}\\\ \end{array}\right)\\\
&+\frac{1}{2\pi\mathrm{i}}\frac{1}{z^{N-1}}\sum\limits_{k=0}^{K-1}\gamma_{k}F^{(1)}\\!\left(z;\\!\begin{array}[]{c}N-k\\!\\\
4\pi\mathrm{e}^{-\frac{\pi}{2}\mathrm{i}}\\!\\\ \end{array}\right)\\\
&+\frac{1}{(2\pi\mathrm{i})^{2}}\frac{1}{z^{N-1}}\sum\limits_{k=0}^{K-1}\gamma_{k}F^{(2)}\\!\left(z;\\!\begin{array}[]{c}N-M+1,\\!\\!\\!\\!\\\
2\pi\mathrm{e}^{-\frac{\pi}{2}\mathrm{i}},\\!\\!\\!\\!\\\
\end{array}\begin{array}[]{c}M-k\\\
2\pi\mathrm{e}^{-\frac{\pi}{2}\mathrm{i}}\\\ \end{array}\right)\\\
&+\frac{1}{(2\pi\mathrm{i})^{2}}\frac{1}{z^{N-1}}\sum\limits_{k=0}^{K-1}\gamma_{k}F^{(2)}\\!\left(z;\\!\begin{array}[]{c}N-M+1,\\!\\!\\!\\!\\\
2\pi\mathrm{e}^{\frac{\pi}{2}\mathrm{i}},\\!\\!\\!\\!\\\
\end{array}\begin{array}[]{c}M-k\\\
2\pi\mathrm{e}^{-\frac{\pi}{2}\mathrm{i}}\\\ \end{array}\right)\\\
&+\frac{1}{(2\pi\mathrm{i})^{2}}\frac{1}{z^{N-1}}\sum\limits_{k=0}^{K-1}\gamma_{k}F^{(2)}\\!\left(z;\\!\begin{array}[]{c}N-M+1,\\!\\!\\!\\!\\\
2\pi\mathrm{e}^{-\frac{\pi}{2}\mathrm{i}},\\!\\!\\!\\!\\\
\end{array}\begin{array}[]{c}M-k\\\
2\pi\mathrm{e}^{\frac{\pi}{2}\mathrm{i}}\\\
\end{array}\right)+\widetilde{R}_{N,M,K}(z),\end{split}$

where

$\displaystyle\widetilde{R}_{N,M,K}(z)=$
$\displaystyle-\frac{1}{2\pi\mathrm{i}}\frac{1}{z^{N}}\int_{0}^{[\pi/2^{-}]}\frac{\mathrm{e}^{2\pi\mathrm{i}t}t^{N-1}}{1-t/z}R_{M,K}(t\mathrm{e}^{-\pi\mathrm{i}})\mathrm{d}t$
$\displaystyle+\frac{1}{2\pi\mathrm{i}}\frac{1}{z^{N}}\int_{0}^{[-\pi/2^{+}]}\frac{\mathrm{e}^{-2\pi\mathrm{i}t}t^{N-1}}{1-t/z}R_{M,K}(t\mathrm{e}^{\pi\mathrm{i}})\mathrm{d}t,$

with the constraint that $K<M<N$. The remainder terms $R_{N,M,K}(z)$ and
$\widetilde{R}_{N,M,K}(z)$ are defined initially for $|\arg z|<\frac{\pi}{2}$,
but can be extended via analytic continuation to $|\arg z|<\pi$.

The presence of the $F^{(2)}$ hyperterminants with identical singulants in
(3.19) and (3.20) indicates that higher-order Stokes phenomena occur on the
rays $\arg z=\pm\frac{\pi}{2}$ (cf. [13, §3]). We shall now provide a smooth
interpretation of these phenomena by describing the birth of the subdominant
contributions with the exponentially small pre-factor
$\mathrm{e}^{4\pi\mathrm{i}z}$ in the neighbourhood of the positive imaginary
axis. The procedure is similar to that presented above in connection with the
leading subdominant exponential $\mathrm{e}^{2\pi\mathrm{i}z}$. Suppose that
the hyperasymptotic expansions (3.9) and (3.10) are truncated near their least
terms, e.g., the truncation indices are $N=\lfloor 4\pi|z|\rfloor$ and
$M=\lfloor 2\pi|z|\rfloor$, respectively. With this assumption, it can be
shown that for any fixed $K$,

$R_{N,M,K}(z),\;\widetilde{R}_{N,M,K}(z)=\begin{cases}\mathcal{O}(\left|z\right|^{-K}\mathrm{e}^{-4\pi\left|z\right|})&\text{if
}\;|\arg z|\leq\frac{\pi}{2},\\\\[5.0pt]
\mathcal{O}(\left|z\right|^{-K}\mathrm{e}^{-4\pi\operatorname{Im}(z)})&\text{if
}\;\frac{\pi}{2}<\arg z\leq\pi-\delta<\pi,\end{cases}$

as $z\to\infty$, uniformly with respect to $\arg z$ (cf. [2, §4.3]). The
behaviour of $R_{N,M}(z)$ and $\widetilde{R}_{N,M}(z)$ for large $z$ in the
vicinity of the line $\arg z=\frac{\pi}{2}$ can now be inferred from (3.19),
(3.20), and the asymptotic properties of the hyperterminants $F^{(1)}$ and
$F^{(2)}$. With $\varphi=\arg(\sigma z)$, $N\sim|\sigma z|$ and fixed non-
negative integer $n$, we show in Theorem 5.1 that $F^{(2)}$ admits the
following asymptotic behaviour for large $|\sigma z|$:

$\frac{\mathrm{e}^{-2\sigma
z}}{(2\pi\mathrm{i})^{2}z^{2N-2-n}}F^{(2)}\\!\left(z;\\!\begin{array}[]{c}N,\\!\\!\\!\\!\\\
\sigma,\\!\\!\\!\\!\\\ \end{array}\begin{array}[]{c}N-n\\!\\\ \sigma\\!\\\
\end{array}\right)=\frac{1}{4}\operatorname{erfc}\Big{(}c(\varphi)\sqrt{|\sigma
z|}\Big{)}+\frac{1}{8}\operatorname{erfc}^{2}\Big{(}c(\varphi)\sqrt{\tfrac{1}{2}|\sigma
z|}\Big{)}\\\
+\begin{dcases}\mathcal{O}\bigg{(}\bigg{|}\cfrac{\mathrm{e}^{-\left|\sigma
z\right|c^{2}(\varphi)}}{\sqrt{|\sigma z|}}\bigg{|}\bigg{)},&\text{if
}\;-\pi<-\pi+\delta\leq\varphi\leq\pi,\\\
\mathcal{O}\bigg{(}\bigg{|}\cfrac{\mathrm{e}^{-\frac{1}{2}\left|\sigma
z\right|c^{2}(\varphi)}}{\sqrt{|\sigma z|}}\bigg{|}\bigg{)},&\text{if
}\;\pi<\varphi\leq 3\pi-\delta<3\pi,\end{dcases}$

with a conjugate behaviour in the sector
$-3\pi<-3\pi+\delta\leq\varphi\leq\pi-\delta<\pi$. The quantity $c(\varphi)$
is defined by (3.15) and (3.16). To estimate the $F^{(2)}$ hyperterminants
with “mixed” singulants, we use

$\frac{\mathrm{e}^{-2\sigma
z}}{(2\pi\mathrm{i})^{2}z^{2N-2-n}}F^{(2)}\\!\left(z;\\!\begin{array}[]{c}N,\\!\\!\\!\\!\\\
\sigma,\\!\\!\\!\\!\\\ \end{array}\begin{array}[]{c}N-n\\!\\\
\sigma\mathrm{e}^{\pm\pi\mathrm{i}}\\!\\\
\end{array}\right)=\begin{dcases}\mathcal{O}\bigg{(}\bigg{|}\cfrac{\mathrm{e}^{-\left|\sigma
z\right|c^{2}(\varphi)}}{\sqrt{|\sigma z|}}\bigg{|}\bigg{)},&\text{if
}\;|\varphi|\leq\pi,\\\
\mathcal{O}\bigg{(}\bigg{|}\cfrac{\mathrm{e}^{-\frac{1}{2}\left|\sigma
z\right|c^{2}(\varphi)}}{\sqrt{|\sigma z|}}\bigg{|}\bigg{)},&\text{if
}\;\pi<|\varphi|\leq 2\pi-\delta<2\pi\end{dcases}$

(see Proposition 5.2). Accordingly, the main contribution to $R_{N,M}(z)$
comes from the first sum in (3.19). Neglecting error terms and keeping only
the leading term in the expansion (3.16), we find that

$R_{N,M}(z)\sim\mathrm{e}^{4\pi\mathrm{i}z}\sum\limits_{k=0}^{K-1}(-1)^{k}\frac{\gamma_{k}}{z^{k}}\times\left(\frac{1}{4}\operatorname{erfc}\left(\left(\tfrac{\pi}{2}-\theta\right)\sqrt{2\pi|z|}\right)+\frac{1}{8}\operatorname{erfc}^{2}\left(\left(\tfrac{\pi}{2}-\theta\right)\sqrt{\pi|z|}\right)\right)$

for large $z$ such that $\theta=\arg z=\frac{\pi}{2}+\mathcal{O}(|z|^{-1/2})$,
and with $K\geq 1$ held fixed. The combination of the $\operatorname{erfc}$
functions increases rapidly but smoothly from being exponentially small for
$\theta<\frac{\pi}{2}$, to $\frac{3}{8}$ when $\theta=\frac{\pi}{2}$, and
finally to almost $1$ when $\theta>\frac{\pi}{2}$. This is in complete
accordance with the numerical results presented in the preceding subsection.
The treatment in the neighbourhood of $\arg z=-\frac{\pi}{2}$ is similar, with
the third sum in (3.19) then controlling the dominant behaviour of
$R_{N,M}(z)$. Similarly, the transitional behaviour of
$\widetilde{R}_{N,M}(z)$ from $\theta\leq\frac{\pi}{2}$ to
$\theta\geq\frac{\pi}{2}$ is essentially described by the first two sums in
(3.20). Therefore,

$\widetilde{R}_{N,M}(z)\sim-\mathrm{e}^{4\pi\mathrm{i}z}\sum\limits_{k=0}^{K-1}\frac{\gamma_{k}}{z^{k}}\times\left(\frac{1}{4}\operatorname{erfc}\left(\left(\tfrac{\pi}{2}-\theta\right)\sqrt{2\pi|z|}\right)-\frac{1}{8}\operatorname{erfc}^{2}\left(\left(\tfrac{\pi}{2}-\theta\right)\sqrt{\pi|z|}\right)\right)$

for large $z$ such that $\theta=\frac{\pi}{2}+\mathcal{O}(|z|^{-1/2})$, and
with fixed $K\geq 1$. The expression in the large parentheses increases
rapidly but smoothly from approximately $0$ for $\theta<\frac{\pi}{2}$, to
$\frac{1}{8}$ when $\theta=\frac{\pi}{2}$, and then drops down to $0$ again
when $\theta>\frac{\pi}{2}$. This agrees with the numerical observation
discussed in the previous subsection. It is thus seen that the role of the
$F^{(1)}$ functions in the first line of (3.20) is to cancel the contributions
switched on by the corresponding $F^{(2)}$ functions in the second line of
(3.20) across the line $\arg z=\frac{\pi}{2}$. A similar result is obtained
for the Stokes line $\arg z=-\frac{\pi}{2}$, where the third and fourth sums
in (3.20) are dominant.

This process can be continued to produce a sequence of re-expanded remainder
terms, each of which is exponentially smaller than its predecessor. The
contributions scaled by the pre-factors $\mathrm{e}^{\pm 2\pi\mathrm{i}kz}$
are then born from the remainders of the optimally truncated Level $(k-1)$
hyperasymptotic expansions. Note that, in contrast to the case of
$\log\Gamma(z)$, the smooth transition of successively smaller exponential
contributions can only be demonstrated one at a time since each time the level
is increased, the number of terms at each previous level in the
hyperasymptotic expansions increases by approximately $2\pi|z|$.

##  4\. Howls’ conjecture

Howls (C. J. Howls, personal communication, December, 2017) conjectured the
following formal power series identity involving hyperterminants:

(4.1)
$\exp\left(\sum\limits_{k=1}^{\infty}\frac{(2\pi\mathrm{i})^{k-1}}{k}F^{(1)}\\!\left(z;\\!\begin{array}[]{c}kN-k+1\\!\\\
k\sigma\\!\\\
\end{array}\right)t^{k}\right)=\sum\limits_{m=0}^{\infty}F^{(m)}\\!\left(z;\\!\begin{array}[]{c}N,\,\ldots,\\!\\!\\!\\!\\\
\,\sigma,\,\,\ldots,\\!\\!\\!\\!\\\ \end{array}\begin{array}[]{c}N\\!\\!\\\
\sigma\\!\\!\\\ \end{array}\right)t^{m}.$

This rather unexpected identity is suggested by the exponentially improved
asymptotic expansion of $\log\Gamma(z)$ [24, §6.4.2] and the hyperasymptotic
expansion (to all levels) of $\Gamma(z)$. Our aim in this section is to
rigorously interpret and verify this conjecture. To this end, it proves
convenient to introduce the complete Bell polynomials ${\bf
Y}_{m}(y_{1},\ldots,y_{m})$ of $m$ complex variables $y_{1},\ldots,y_{m}$.
They may be defined by

(4.2) ${\bf
Y}_{m}(y_{1},\ldots,y_{m})=\sum\limits_{\pi(m)}\prod\limits_{j=1}^{m}\frac{y_{j}^{k_{j}}}{k_{j}!},$

where $\pi(m)$ denotes a partition of $m$, usually denoted by
$1^{k_{1}}2^{k_{2}}\cdots m^{k_{m}}$, with $k_{1}+2k_{2}+\ldots+mk_{m}=m$;
$k_{j}\geq 0$ being the number of parts of size $j$. By convention, ${\bf
Y}_{0}=1$. The complete Bell polynomials satisfy the generating function
identity

(4.3)
$\exp\left(\sum\limits_{k=0}^{\infty}y_{k}t^{k}\right)=\sum\limits_{m=0}^{\infty}{\bf
Y}_{m}(y_{1},\ldots,y_{m})t^{m}.$

It should be noted that our notation differs from that used commonly in the
literature. For example, the polynomials $Y_{m}$ discussed in [1] or [26,
§5.2] are related to ours via $Y_{m}(1!y_{1},\ldots,m!y_{m})=m!{\bf
Y}_{m}(y_{1},\ldots,y_{m})$. In view of (4.1) and (4.3), we may reformulate
Howls’ conjecture as

(4.4) $F^{(m)}\\!\left(z;\\!\begin{array}[]{c}N,\,\ldots,\\!\\!\\!\\!\\\
\,\sigma,\,\,\ldots,\\!\\!\\!\\!\\\ \end{array}\begin{array}[]{c}N\\!\\!\\\
\sigma\\!\\!\\\ \end{array}\right)={\bf
Y}_{m}\left(F^{(1)}\\!\left(z;\\!\begin{array}[]{c}N\\!\\\ \sigma\\!\\\
\end{array}\right),\ldots,\frac{(2\pi\mathrm{i})^{m-1}}{m}F^{(1)}\\!\left(z;\\!\begin{array}[]{c}mN-m+1\\!\\\
m\sigma\\!\\\ \end{array}\right)\right).$

Notice that this identity, unlike (4.1), does not suffer from possible
divergence issues. We can now formulate the following theorem, which is the
rigorous form of Howls’ conjecture.

######  Theorem 4.1.

Let $N$ be a complex number such that $\operatorname{Re}(N)>1$, and let
$\sigma$ be an arbitrary element of $\widehat{\mathbb{C}}$. Then the identity
(4.4) holds for any $m\geq 0$ and for all values of $\arg z$.

###### Proof.

For the sake of brevity, let us denote the right-hand side of (4.4) by
$f^{(m)}(z;N,\sigma)$. Thus, we are required to prove that

(4.5)
$f^{(m)}(z;N,\sigma)=F^{(m)}\\!\left(z;\\!\begin{array}[]{c}N,\,\ldots,\\!\\!\\!\\!\\\
\,\sigma,\,\,\ldots,\\!\\!\\!\\!\\\ \end{array}\begin{array}[]{c}N\\!\\!\\\
\sigma\\!\\!\\\ \end{array}\right)$

for all $m\geq 0$. We proceed by induction on $m$. It is readily seen that the
equality (4.5) holds when $m=0$ or $1$. Assume that it holds for
$0,1,\ldots,m-1$ ($m\geq 1$). Employing the recurrence relation

${\bf Y}_{m}(y_{1},\ldots,y_{m})=\frac{1}{m}\sum\limits_{k=1}^{m}ky_{k}{\bf
Y}_{m-k}(y_{1},\ldots,y_{m-k})$

(cf. [26, Eq. (3), §5.2]) and the induction hypothesis, we obtain

$f^{(m)}(z;N,\sigma)=\frac{1}{m}\sum\limits_{k=1}^{m}(2\pi\mathrm{i})^{k-1}F^{(1)}\\!\left(z;\\!\begin{array}[]{c}kN-k+1\\!\\\
k\sigma\\!\\\
\end{array}\right)F^{(m-k)}\\!\left(z;\\!\begin{array}[]{c}N,\,\ldots,\\!\\!\\!\\!\\\
\,\sigma,\,\,\ldots,\\!\\!\\!\\!\\\ \end{array}\begin{array}[]{c}N\\!\\!\\\
\sigma\\!\\!\\\ \end{array}\right).$

If we replace $z$ by $z\mathrm{e}^{-2\pi\mathrm{i}}$ in this equality, apply
the connection formula (3.8) for each hyperterminant on the right-hand side,
and use (4.5) with $m-1$ in place of $m$, we obtain after some simplification
that

(4.6)
$f^{(m)}(z\mathrm{e}^{-2\pi\mathrm{i}};N,\sigma)=f^{(m)}(z;N,\sigma)-2\pi\mathrm{i}\mathrm{e}^{\sigma
z}z^{N-1}F^{(m-1)}\\!\left(z;\\!\begin{array}[]{c}N,\,\ldots,\\!\\!\\!\\!\\\
\,\sigma,\,\,\ldots,\\!\\!\\!\\!\\\ \end{array}\begin{array}[]{c}N\\!\\!\\\
\sigma\\!\\!\\\ \end{array}\right).$

Now let us assume, for simplicity, that $|\arg z|<\pi$ and $\arg\sigma=0$. Let
$\gamma$ be the positively oriented key-hole contour depicted in Figure 2. By
Cauchy’s integral theorem, we can write

Figure 2. The contour of integration $\gamma$.

$f^{(m)}(z;N,\sigma)=\frac{1}{2\pi\mathrm{i}}\oint_{\gamma}\frac{f^{(m)}(t;N,\sigma)}{t-z}\mathrm{d}t$

for all $z$ lying inside the contour $\gamma$. By virtue of the equality (3.6)
and the standard asymptotic expansion of the incomplete gamma function (see,
e.g., [21, §8.11(i)]), each hyperterminant on the right-hand side of (4.4) is
$\mathcal{O}(|z|^{-1})$ as $z\to\infty$ in $|\arg z|\leq\pi$, provided all the
other parameters held fixed. Consequently, for each $m\geq 2$ and fixed $N$,
$\sigma$, $f^{(m)}(z;N,\sigma)=\mathcal{O}(|z|^{-1})$ as $z\to\infty$ in the
sector $|\arg z|\leq\pi$. The hyperterminants remain bounded as $z\to 0$ in
$|\arg z|\leq\pi$, and thus so does $f^{(m)}(z;N,\sigma)$. Therefore, when the
radius of the large circular portion of the contour $\gamma$ approaches
$+\infty$, the integral along it tends to zero. Similarly, when the radius of
the small circular arc tends to $0$, the integral along it tends to zero. If
now we write $t=s\mathrm{e}^{\pm\pi\mathrm{i}}$ on the two rays, we find that

$f^{(m)}(z;N,\sigma)=\frac{1}{2\pi\mathrm{i}}\int_{0}^{+\infty}\frac{f^{(m)}(s\mathrm{e}^{-\pi\mathrm{i}};N,\sigma)-f^{(m)}(s\mathrm{e}^{\pi\mathrm{i}};N,\sigma)}{s+z}\mathrm{d}s$

provided $|\arg z|<\pi$. Finally, the integrand may be expressed in terms of
the hyperterminant $F^{(m-1)}$ by using (4.6), and hence

$\displaystyle f^{(m)}(z;N,\sigma)$
$\displaystyle=\mathrm{e}^{\pi\mathrm{i}N}\int_{0}^{+\infty}\frac{\mathrm{e}^{-\sigma
s}s^{N-1}}{s+z}F^{(m-1)}\\!\left(s\mathrm{e}^{\pi\mathrm{i}};\\!\begin{array}[]{c}N,\,\ldots,\\!\\!\\!\\!\\\
\,\sigma,\,\,\ldots,\\!\\!\\!\\!\\\ \end{array}\begin{array}[]{c}N\\!\\!\\\
\sigma\\!\\!\\\ \end{array}\right)\mathrm{d}s$
$\displaystyle=\int_{0}^{[\pi^{-}]}\frac{\mathrm{e}^{\sigma
t}t^{N-1}}{z-t}F^{(m-1)}\\!\left(t;\\!\begin{array}[]{c}N,\,\ldots,\\!\\!\\!\\!\\\
\,\sigma,\,\,\ldots,\\!\\!\\!\\!\\\ \end{array}\begin{array}[]{c}N\\!\\!\\\
\sigma\\!\\!\\\ \end{array}\right)\mathrm{d}t.$

It is seen from (3.5) and (3.7) that this integral is precisely the
hyperterminant $F^{(m)}$ appearing on the right-hand side of the equality
(4.5). The restrictions on $\arg z$ and $\arg\sigma$ can now be dropped by
appealing to analytic continuation. ∎

######  Remark.

Note that there is an alternative way to finish the proof of the theorem.
Denote by $g^{(m)}(z;N,\sigma)$ the difference of the two sides of (4.5).
Assume for a moment that $N>2$. Then the functional equations (3.8) and (4.6),
Lemma 5.4 and Riemann’s theorem on removable singularities together imply that
$g^{(m)}(z;N,\sigma)$ is an entire function. From Lemma 5.3 we can assert that
$g^{(m)}(z;N,\sigma)=\mathcal{O}(|z|^{-1})$ as $z\to\infty$. Hence by
Liouville’s theorem, $g^{(m)}(z;N,\sigma)$ is identically zero. The
restriction $N>2$ can be removed by analytic continuation.

##  5\. Uniform asymptotic approximations for the hyperterminants

For the gamma function and its reciprocal, the smoothing of the (higher-order)
Stokes phenomenon is described in terms of $F^{(m)}$ hyperterminants of the
form

$F^{(m)}\\!\left(z;\\!\begin{array}[]{c}N,\,\ldots,\\!\\!\\!\\!\\\
\,\sigma,\,\,\ldots,\\!\\!\\!\\!\\\
\end{array}\begin{array}[]{c}N,\\!\\!\\!\\\ \sigma,\\!\\!\\!\\\
\end{array}\begin{array}[]{c}N-n\\!\\!\\\ \sigma\\!\\!\\\ \end{array}\right).$

Of most interest for our purpose is their behaviour when $N\sim|\sigma z|\gg
1$, $n$ is a fixed non-negative integer, and $\varphi=\arg(\sigma z)$
increases (resp. decreases) through $\pi$ (resp. $-\pi$). In these
circumstances, it is seen from (3.14) and (3.16) that, when suitably
normalised, $F^{(1)}$ possesses the property of changing rapidly, but
smoothly, from being exponentially small to being approximately $1$ as
$\varphi$ increases continuously through $\pi$. A similar behaviour is present
near the ray $\varphi=-\pi$. In Theorem 5.1 below we show that each of the
higher-level terminants $F^{(m)}$, $m\geq 2$, exhibits this behaviour, which
is, however, no longer described by a single complementary error function but
rather by a multivariate polynomial in such functions with properly scaled
arguments. That the leading-order asymptotics should take this form is, of
course, no surprise, in view of the identity (4.4) which plays an essential
role in our proof.

######  Theorem 5.1.

Let $m$ be a positive integer and $\sigma$ be any element of
$\widehat{\mathbb{C}}$. Let $N=|\sigma z|+\rho$, $\varphi=\arg(\sigma z)$, and
define $c(\varphi)$ by (3.15) and (3.16). Then for any fixed non-negative
integer $n$ and large $|\sigma z|$,

(5.1) $\begin{multlined}\frac{\mathrm{e}^{-m\sigma
z}}{(2\pi\mathrm{i})^{m}z^{mN-
m-n}}F^{(m)}\\!\left(z;\\!\begin{array}[]{c}N,\,\ldots,\\!\\!\\!\\!\\\
\,\sigma,\,\,\ldots,\\!\\!\\!\\!\\\
\end{array}\begin{array}[]{c}N,\\!\\!\\!\\\ \sigma,\\!\\!\\!\\\
\end{array}\begin{array}[]{c}N-n\\!\\!\\\ \sigma\\!\\!\\\
\end{array}\right)=\sum\limits_{\pi(m)}\prod\limits_{j=1}^{m}\frac{1}{(2j)^{k_{j}}k_{j}!}\operatorname{erfc}^{k_{j}}\Big{(}c(\varphi)\sqrt{\tfrac{j}{2}\left|\sigma
z\right|}\Big{)}\\\
+\begin{dcases}\mathcal{O}\bigg{(}\bigg{|}\cfrac{\mathrm{e}^{-\frac{m}{2}\left|\sigma
z\right|c^{2}(\varphi)}}{\sqrt{|\sigma z|}}\bigg{|}\bigg{)},&\text{if
}\;-\pi<-\pi+\delta\leq\varphi\leq\pi,\\\
\mathcal{O}\bigg{(}\bigg{|}\cfrac{\mathrm{e}^{-\frac{1}{2}\left|\sigma
z\right|c^{2}(\varphi)}}{\sqrt{|\sigma z|}}\bigg{|}\bigg{)},&\text{if
}\;\pi<\varphi\leq
3\pi-\delta<3\pi,\end{dcases}\end{multlined}\frac{\mathrm{e}^{-m\sigma
z}}{(2\pi\mathrm{i})^{m}z^{mN-
m-n}}F^{(m)}\\!\left(z;\\!\begin{array}[]{c}N,\,\ldots,\\!\\!\\!\\!\\\
\,\sigma,\,\,\ldots,\\!\\!\\!\\!\\\
\end{array}\begin{array}[]{c}N,\\!\\!\\!\\\ \sigma,\\!\\!\\!\\\
\end{array}\begin{array}[]{c}N-n\\!\\!\\\ \sigma\\!\\!\\\
\end{array}\right)=\sum\limits_{\pi(m)}\prod\limits_{j=1}^{m}\frac{1}{(2j)^{k_{j}}k_{j}!}\operatorname{erfc}^{k_{j}}\Big{(}c(\varphi)\sqrt{\tfrac{j}{2}\left|\sigma
z\right|}\Big{)}\\\
+\begin{dcases}\mathcal{O}\bigg{(}\bigg{|}\cfrac{\mathrm{e}^{-\frac{m}{2}\left|\sigma
z\right|c^{2}(\varphi)}}{\sqrt{|\sigma z|}}\bigg{|}\bigg{)},&\text{if
}\;-\pi<-\pi+\delta\leq\varphi\leq\pi,\\\
\mathcal{O}\bigg{(}\bigg{|}\cfrac{\mathrm{e}^{-\frac{1}{2}\left|\sigma
z\right|c^{2}(\varphi)}}{\sqrt{|\sigma z|}}\bigg{|}\bigg{)},&\text{if
}\;\pi<\varphi\leq 3\pi-\delta<3\pi,\end{dcases}$

and

(5.2)
$\begin{multlined}\frac{\mathrm{e}^{-2\pi\mathrm{i}mN}\mathrm{e}^{-m\sigma
z}}{(2\pi\mathrm{i})^{m}z^{mN-
m-n}}F^{(m)}\\!\left(z;\\!\begin{array}[]{c}N,\,\ldots,\\!\\!\\!\\!\\\
\,\sigma,\,\,\ldots,\\!\\!\\!\\!\\\
\end{array}\begin{array}[]{c}N,\\!\\!\\!\\\ \sigma,\\!\\!\\!\\\
\end{array}\begin{array}[]{c}N-n\\!\\!\\\ \sigma\\!\\!\\\
\end{array}\right)=\sum\limits_{\pi(m)}\prod\limits_{j=1}^{m}\frac{(-1)^{k_{j}}}{(2j)^{k_{j}}k_{j}!}\operatorname{erfc}^{k_{j}}\Big{(}\overline{c(-\varphi)}\sqrt{\tfrac{j}{2}\left|\sigma
z\right|}\Big{)}\\\
+\begin{dcases}\mathcal{O}\bigg{(}\bigg{|}\cfrac{\mathrm{e}^{-\frac{m}{2}\left|\sigma
z\right|c^{2}(-\varphi)}}{\sqrt{|\sigma z|}}\bigg{|}\bigg{)},&\text{if
}\;-\pi\leq\varphi\leq\pi-\delta<\pi,\\\
\mathcal{O}\bigg{(}\bigg{|}\cfrac{\mathrm{e}^{-\frac{1}{2}\left|\sigma
z\right|c^{2}(-\varphi)}}{\sqrt{|\sigma z|}}\bigg{|}\bigg{)},&\text{if
}\;-3\pi<-3\pi+\delta\leq\varphi<-\pi,\end{dcases}\end{multlined}\frac{\mathrm{e}^{-2\pi\mathrm{i}mN}\mathrm{e}^{-m\sigma
z}}{(2\pi\mathrm{i})^{m}z^{mN-
m-n}}F^{(m)}\\!\left(z;\\!\begin{array}[]{c}N,\,\ldots,\\!\\!\\!\\!\\\
\,\sigma,\,\,\ldots,\\!\\!\\!\\!\\\
\end{array}\begin{array}[]{c}N,\\!\\!\\!\\\ \sigma,\\!\\!\\!\\\
\end{array}\begin{array}[]{c}N-n\\!\\!\\\ \sigma\\!\\!\\\
\end{array}\right)=\sum\limits_{\pi(m)}\prod\limits_{j=1}^{m}\frac{(-1)^{k_{j}}}{(2j)^{k_{j}}k_{j}!}\operatorname{erfc}^{k_{j}}\Big{(}\overline{c(-\varphi)}\sqrt{\tfrac{j}{2}\left|\sigma
z\right|}\Big{)}\\\
+\begin{dcases}\mathcal{O}\bigg{(}\bigg{|}\cfrac{\mathrm{e}^{-\frac{m}{2}\left|\sigma
z\right|c^{2}(-\varphi)}}{\sqrt{|\sigma z|}}\bigg{|}\bigg{)},&\text{if
}\;-\pi\leq\varphi\leq\pi-\delta<\pi,\\\
\mathcal{O}\bigg{(}\bigg{|}\cfrac{\mathrm{e}^{-\frac{1}{2}\left|\sigma
z\right|c^{2}(-\varphi)}}{\sqrt{|\sigma z|}}\bigg{|}\bigg{)},&\text{if
}\;-3\pi<-3\pi+\delta\leq\varphi<-\pi,\end{dcases}$

uniformly with respect to $\varphi$ and bounded real values of $\rho$. The
index $\pi(m)$ runs through all partitions of $m$ into non-negative parts,
i.e., over all non-negative integer solutions of the equation
$k_{1}+2k_{2}+\ldots+mk_{m}=m$.

######  Remark.

If $m=1$, then the $\varphi$-intervals of validity for (5.1) and (5.2) are
maximal (see [20, Theorem 1]). We expect this property to hold generally for
all $m\geq 1$, but we shall not pursue a proof of this claim here.

Successive hyperasymptotic re-expansions of $\Gamma^{\ast}(z)$ and its
reciprocal also involve hyperterminants $F^{(m)}$ with “mixed” singulants. We
expect that the contribution from such hyperterminants is of the same order of
magnitude as the error terms in (5.1) and (5.2). We therefore anticipate that
the smooth transition of the higher-order Stokes discontinuities is
essentially described by (5.1) and (5.2). This is supported by the fact that
(5.1) yields, to leading order, $\frac{1}{k!}\left(\frac{1}{2}\right)_{k}$ on
the Stokes line $\varphi=\pi$, in agreement with (2.9). In the following
proposition, we verify this assertion in the particular case of $m=2$. We
believe that the general case can be established using an appropriate
inductive argument, but we leave it as an open question for further research.

######  Proposition 5.2.

Let $\sigma$ be any element of $\widehat{\mathbb{C}}$. Let $N=|\sigma
z|+\rho$, $\varphi=\arg(\sigma z)$, and define $c(\varphi)$ by (3.15) and
(3.16). Then for any fixed non-negative integer $n$ and large $|\sigma z|$,

(5.3) $\frac{\mathrm{e}^{-2\sigma
z}}{(2\pi\mathrm{i})^{2}z^{2N-2-n}}F^{(2)}\\!\left(z;\\!\begin{array}[]{c}N,\\!\\!\\!\\!\\\
\sigma,\\!\\!\\!\\!\\\ \end{array}\begin{array}[]{c}N-n\\!\\\
\sigma\mathrm{e}^{\pm\pi\mathrm{i}}\\!\\\
\end{array}\right)=\begin{dcases}\mathcal{O}\bigg{(}\bigg{|}\cfrac{\mathrm{e}^{-\left|\sigma
z\right|c^{2}(\varphi)}}{\sqrt{|\sigma z|}}\bigg{|}\bigg{)},&\text{if
}\;|\varphi|\leq\pi,\\\
\mathcal{O}\bigg{(}\bigg{|}\cfrac{\mathrm{e}^{-\frac{1}{2}\left|\sigma
z\right|c^{2}(\varphi)}}{\sqrt{|\sigma z|}}\bigg{|}\bigg{)},&\text{if
}\;\pi<|\varphi|\leq 2\pi-\delta<2\pi,\end{dcases}$

uniformly with respect to $\varphi$ and bounded real values of $\rho$.

To prove Theorem 5.1 and Proposition 5.2 we first need to establish some
lemmata.

######  Lemma 5.3.

Let $m$ be a positive integer and $N_{1},\ldots,N_{m}$ be an arbitrary set of
real numbers such that $N_{k}>1$ for $k=1,\ldots,m$. Let $\sigma$ be any
element of $\widehat{\mathbb{C}}$. Then there exists a positive constant
$c_{m}$, depending only on $m$, such that

(5.4)
$\displaystyle\begin{split}\left|F^{(m)}\\!\left(z;\\!\begin{array}[]{c}N_{1},\\!\\!\\!\\\
\sigma,\\!\\!\\!\\\ \end{array}\begin{array}[]{c}\ldots,\\!\\!\\!\\\
\ldots,\\!\\!\\!\\\ \end{array}\begin{array}[]{c}N_{m}\\!\\!\\\
\sigma\\!\\!\\\
\end{array}\right)\right|\leq\;&c_{m}\frac{1}{|z|}\frac{\sqrt{N_{m}}\Gamma(N_{m})}{|\sigma|^{N_{m}}}\prod\limits_{k=1}^{m-1}\frac{\sqrt{\Sigma_{k}}\Gamma(N_{k}-1)}{|\sigma|^{N_{k}-1}}\\\
&\times\begin{cases}1&\text{if }\;|\varphi|\leq\pi,\\\\[2.5pt]
\left|\cos\varphi\right|^{-\Sigma_{1}}&\text{if
}\;\pi<\varphi<\frac{3\pi}{2},\end{cases}\end{split}$

where $\varphi=\arg(\sigma z)$ and $\Sigma_{k}=N_{k}+N_{k+1}+\ldots+N_{m}$.

###### Proof.

Throughout the proof, we shall use the following inequality:

(5.5) $\left|1+\frac{t}{w}\right|\geq\begin{cases}1&\text{if }\;|\arg
w|\leq\frac{\pi}{2},\\\\[2.5pt] |\sin(\arg w)|&\text{if }\;\frac{\pi}{2}<|\arg
w|<\pi,\end{cases}$

where $t>0$. The proof of (5.5) is elementary and is therefore left to the
reader.

We proceed by induction on $m$. The base case $m=1$ was proved in [2,
Proposition B.1]. Assume that the statement holds for $1,2,\ldots,m-1$ ($m\geq
2$). If $|\varphi|<\pi$, we can write, using (3.5) and (3.7), that

(5.6) $F^{(m)}\\!\left(z;\\!\begin{array}[]{c}N_{1},\\!\\!\\!\\\
\sigma,\\!\\!\\!\\\ \end{array}\begin{array}[]{c}\ldots,\\!\\!\\!\\\
\ldots,\\!\\!\\!\\\ \end{array}\begin{array}[]{c}N_{m}\\!\\!\\\
\sigma\\!\\!\\\
\end{array}\right)=\mathrm{e}^{\pi\mathrm{i}N_{1}}\frac{1}{z}\frac{1}{\sigma^{N_{1}}}\int_{0}^{+\infty}\frac{\mathrm{e}^{-t}t^{N_{1}-1}}{1+t/(\sigma
z)}F^{(m-1)}\\!\left(\frac{t}{\sigma}\mathrm{e}^{\pi\mathrm{i}};\\!\begin{array}[]{c}N_{2},\\!\\!\\!\\\
\sigma,\\!\\!\\!\\\ \end{array}\begin{array}[]{c}\ldots,\\!\\!\\!\\\
\ldots,\\!\\!\\!\\\ \end{array}\begin{array}[]{c}N_{m}\\!\\!\\\
\sigma\\!\\!\\\ \end{array}\right)\mathrm{d}t.$

Then by the induction hypothesis and the inequality (5.5), we readily find
that

$\displaystyle\left|F^{(m)}\\!\left(z;\\!\begin{array}[]{c}N_{1},\\!\\!\\!\\\
\sigma,\\!\\!\\!\\\ \end{array}\begin{array}[]{c}\ldots,\\!\\!\\!\\\
\ldots,\\!\\!\\!\\\ \end{array}\begin{array}[]{c}N_{m}\\!\\!\\\
\sigma\\!\\!\\\ \end{array}\right)\right|$ $\displaystyle\leq
c_{m-1}\frac{1}{\left|z\right|}\frac{\sqrt{N_{m}}\Gamma(N_{m})}{\left|\sigma\right|^{N_{m}}}\frac{\Gamma(N_{1}-1)}{\left|\sigma\right|^{N_{1}-1}}\prod\limits_{k=2}^{m-1}\frac{\sqrt{\Sigma_{k}}\Gamma(N_{k}-1)}{\left|\sigma\right|^{N_{k}-1}}$
$\displaystyle<c_{m-1}\frac{1}{\left|z\right|}\frac{\sqrt{N_{m}}\Gamma(N_{m})}{\left|\sigma\right|^{N_{m}}}\prod\limits_{k=1}^{m-1}\frac{\sqrt{\Sigma_{k}}\Gamma(N_{k}-1)}{\left|\sigma\right|^{N_{k}-1}},$

provided $|\varphi|\leq\frac{\pi}{2}$.

Consider now the phase range $\frac{\pi}{2}<\varphi\leq\pi$. We deform the
contour of integration in (5.6) by rotating it through an acute angle
$\alpha$. Thus, by appealing to Cauchy’s theorem and analytic continuation, we
have, for an arbitrary $0<\alpha<\frac{\pi}{2}$, that

$\displaystyle F^{(m)}\\!\left(z;\\!\begin{array}[]{c}N_{1},\\!\\!\\!\\\
\sigma,\\!\\!\\!\\\ \end{array}\begin{array}[]{c}\ldots,\\!\\!\\!\\\
\ldots,\\!\\!\\!\\\ \end{array}\begin{array}[]{c}N_{m}\\!\\!\\\
\sigma\\!\\!\\\ \end{array}\right)=\;$
$\displaystyle\mathrm{e}^{\pi\mathrm{i}N_{1}}\frac{1}{z}\frac{1}{\sigma^{N_{1}}}\left(\frac{\mathrm{e}^{\mathrm{i}\alpha}}{\cos\alpha}\right)^{N_{1}}$
$\displaystyle\times\int_{0}^{+\infty}\frac{\mathrm{e}^{-\frac{s\mathrm{e}^{\mathrm{i}\alpha}}{\cos\alpha}}s^{N_{1}-1}}{1+s\mathrm{e}^{\mathrm{i}\alpha}/(\sigma
z\cos\alpha)}F^{(m-1)}\\!\left(\frac{s\mathrm{e}^{\mathrm{i}(\alpha+\pi)}}{\sigma\cos\alpha};\\!\begin{array}[]{c}N_{2},\\!\\!\\!\\\
\sigma,\\!\\!\\!\\\ \end{array}\begin{array}[]{c}\ldots,\\!\\!\\!\\\
\ldots,\\!\\!\\!\\\ \end{array}\begin{array}[]{c}N_{m}\\!\\!\\\
\sigma\\!\\!\\\ \end{array}\right)\mathrm{d}s$

when $\frac{\pi}{2}<\varphi\leq\pi$. Employing the inequality (5.5), we then
obtain the bound

$\displaystyle\left|F^{(m)}\\!\left(z;\\!\begin{array}[]{c}N_{1},\\!\\!\\!\\\
\sigma,\\!\\!\\!\\\ \end{array}\begin{array}[]{c}\ldots,\\!\\!\\!\\\
\ldots,\\!\\!\\!\\\ \end{array}\begin{array}[]{c}N_{m}\\!\\!\\\
\sigma\\!\\!\\\ \end{array}\right)\right|\leq\;$
$\displaystyle\frac{1}{|z|}\frac{1}{|\sigma|^{N_{1}}}\frac{1}{(\cos\alpha)^{N_{1}}}$
$\displaystyle\times\int_{0}^{+\infty}\mathrm{e}^{-s}s^{N_{1}-1}\left|F^{(m-1)}\\!\left(\frac{s\mathrm{e}^{\mathrm{i}(\alpha+\pi)}}{\sigma\cos\alpha};\\!\begin{array}[]{c}N_{2},\\!\\!\\!\\\
\sigma,\\!\\!\\!\\\ \end{array}\begin{array}[]{c}\ldots,\\!\\!\\!\\\
\ldots,\\!\\!\\!\\\ \end{array}\begin{array}[]{c}N_{m}\\!\\!\\\
\sigma\\!\\!\\\ \end{array}\right)\right|\mathrm{d}s$
$\displaystyle\times\begin{cases}1&\text{if
}\;\frac{\pi}{2}<\varphi\leq\frac{\pi}{2}+\alpha,\\\\[2.5pt]
\csc(\varphi-\alpha)&\text{if
}\;\frac{\pi}{2}+\alpha<\varphi\leq\pi.\end{cases}$

We can simplify this result further by using the induction hypothesis to
deduce

(5.7)
$\displaystyle\begin{split}\left|F^{(m)}\\!\left(z;\\!\begin{array}[]{c}N_{1},\\!\\!\\!\\\
\sigma,\\!\\!\\!\\\ \end{array}\begin{array}[]{c}\ldots,\\!\\!\\!\\\
\ldots,\\!\\!\\!\\\ \end{array}\begin{array}[]{c}N_{m}\\!\\!\\\
\sigma\\!\\!\\\
\end{array}\right)\right|\leq\;&c_{m-1}\frac{1}{\left|z\right|}\frac{\sqrt{N_{m}}\Gamma(N_{m})}{\left|\sigma\right|^{N_{m}}}\frac{\Gamma(N_{1}-1)}{\left|\sigma\right|^{N_{1}-1}}\prod\limits_{k=2}^{m-1}\frac{\sqrt{\Sigma_{k}}\Gamma(N_{k}-1)}{\left|\sigma\right|^{N_{k}-1}}\\\
&\times\frac{1}{(\cos\alpha)^{\Sigma_{1}-1}}\times\begin{cases}1&\text{if
}\;\frac{\pi}{2}<\varphi\leq\frac{\pi}{2}+\alpha,\\\\[2.5pt]
\csc(\varphi-\alpha)&\text{if
}\;\frac{\pi}{2}+\alpha<\varphi\leq\pi.\end{cases}\end{split}$

We now choose the value of $\alpha$ which minimises the right-hand side of
this inequality when $\varphi=\pi$, namely
$\alpha=\operatorname{arccot}(\sqrt{\Sigma_{1}-1})$. With this choice of
$\alpha$, the factor in the second line of (5.7) may be bounded by
$\sqrt{\mathrm{e}\,\Sigma_{1}}$ for all $\frac{\pi}{2}<\varphi\leq\pi$ (see
the proof of Proposition B.1 in [2]). A similar proof holds for the conjugate
sector $-\pi\leq\varphi<-\frac{\pi}{2}$.

Finally, it remains to consider the sector $\pi<\varphi<\frac{3\pi}{2}$. The
proof is based on the functional relation (cf. (3.8))

$\displaystyle F^{(m)}\\!\left(z;\\!\begin{array}[]{c}N_{1},\\!\\!\\!\\\
\sigma,\\!\\!\\!\\\ \end{array}\begin{array}[]{c}\ldots,\\!\\!\\!\\\
\ldots,\\!\\!\\!\\\ \end{array}\begin{array}[]{c}N_{m}\\!\\!\\\
\sigma\\!\\!\\\ \end{array}\right)=\;$ $\displaystyle
F^{(m)}\\!\left(z\mathrm{e}^{-2\pi\mathrm{i}};\\!\begin{array}[]{c}N_{1},\\!\\!\\!\\\
\sigma,\\!\\!\\!\\\ \end{array}\begin{array}[]{c}\ldots,\\!\\!\\!\\\
\ldots,\\!\\!\\!\\\ \end{array}\begin{array}[]{c}N_{m}\\!\\!\\\
\sigma\\!\\!\\\ \end{array}\right)$
$\displaystyle+2\pi\mathrm{i}\mathrm{e}^{\sigma
z}z^{N_{1}-1}F^{(m-1)}\\!\left(z;\\!\begin{array}[]{c}N_{2},\\!\\!\\!\\\
\sigma,\\!\\!\\!\\\ \end{array}\begin{array}[]{c}\ldots,\\!\\!\\!\\\
\ldots,\\!\\!\\!\\\ \end{array}\begin{array}[]{c}N_{m}\\!\\!\\\
\sigma\\!\\!\\\ \end{array}\right).$

The first term on the right-hand side can be estimated by applying the result
of the previous paragraph. To estimate the second term, we use the induction
hypothesis. Accordingly,

$\left|F^{(m)}\\!\left(z;\\!\begin{array}[]{c}N_{1},\\!\\!\\!\\\
\sigma,\\!\\!\\!\\\ \end{array}\begin{array}[]{c}\ldots,\\!\\!\\!\\\
\ldots,\\!\\!\\!\\\ \end{array}\begin{array}[]{c}N_{m}\\!\\!\\\
\sigma\\!\\!\\\
\end{array}\right)\right|\leq\sqrt{\mathrm{e}}c_{m-1}\frac{1}{|z|}\frac{\sqrt{N_{m}}\Gamma(N_{m})}{|\sigma|^{N_{m}}}\prod\limits_{k=1}^{m-1}\frac{\sqrt{\Sigma_{k}}\Gamma(N_{k}-1)}{|\sigma|^{N_{k}-1}}\\\
+\frac{2\pi\left|\sigma z\right|^{N_{1}-1}\mathrm{e}^{-\left|\sigma
z\right|\left|\cos\varphi\right|}}{\sqrt{\Sigma_{1}}\Gamma(N_{1}-1)}\frac{1}{\left|\cos\varphi\right|^{\Sigma_{2}}}c_{m-1}\frac{1}{\left|z\right|}\frac{\sqrt{N_{m}}\Gamma(N_{m})}{\left|\sigma\right|^{N_{m}}}\prod\limits_{k=1}^{m-1}\frac{\sqrt{\Sigma_{k}}\Gamma(N_{k}-1)}{\left|\sigma\right|^{N_{k}-1}}.$

Notice that the quantity $r^{M}\mathrm{e}^{-ar}$, as a function of $r>0$,
takes its maximum value at $r=M/a$ when $a>0$ and $M>0$. We therefore find
that

$\displaystyle\frac{2\pi\left|\sigma
z\right|^{N_{1}-1}\mathrm{e}^{-\left|\sigma
z\right|\left|\cos\varphi\right|}}{\sqrt{\Sigma_{1}}\Gamma(N_{1}-1)}\frac{1}{\left|\cos\varphi\right|^{\Sigma_{2}}}\leq\frac{\sqrt{2\pi(N_{1}-1)}}{\sqrt{\Sigma_{1}}\Gamma^{\ast}(N_{1}-1)}\frac{1}{\left|\cos\varphi\right|^{\Sigma_{1}-1}}<\sqrt{2\pi}\frac{1}{\left|\cos\varphi\right|^{\Sigma_{1}}}.$

The second inequality can be obtained from the fact that $\Gamma^{\ast}(M)\geq
1$ for any $M>0$ (see, for instance, [21, Eq. 5.6.1]). This completes the
proof of the lemma. ∎

We now define the value of the hyperterminant $F^{(m)}$ at the origin to be

$F^{(m)}\\!\left(0;\\!\begin{array}[]{c}N_{1},\,\ldots,\\!\\!\\!\\!\\\
\,\sigma_{1},\,\,\ldots,\\!\\!\\!\\!\\\
\end{array}\begin{array}[]{c}N_{m}\\!\\!\\\ \sigma_{m}\\!\\!\\\
\end{array}\right)=\mathop{\lim}\limits_{\begin{subarray}{c}\left|z\right|\to
0\\\
\left|\arg(\sigma_{1}z)\right|<\pi\end{subarray}}F^{(m)}\\!\left(z;\\!\begin{array}[]{c}N_{1},\,\ldots,\\!\\!\\!\\!\\\
\,\sigma_{1},\,\,\ldots,\\!\\!\\!\\!\\\
\end{array}\begin{array}[]{c}N_{m}\\!\\!\\\ \sigma_{m}\\!\\!\\\
\end{array}\right),$

provided that this limit exists.

######  Lemma 5.4.

Let $m\geq 2$ be a positive integer and $N_{1},\ldots,N_{m}$ be an arbitrary
set of real numbers such that $N_{1}>2$ and $N_{k}>1$ for $k=2,\ldots,m$. Let
$\sigma$ be any element of $\widehat{\mathbb{C}}$. Then

$\left|F^{(m)}\\!\left(0;\\!\begin{array}[]{c}N_{1},\\!\\!\\!\\\
\sigma,\\!\\!\\!\\\ \end{array}\begin{array}[]{c}\ldots,\\!\\!\\!\\\
\ldots,\\!\\!\\!\\\ \end{array}\begin{array}[]{c}N_{m}\\!\\!\\\
\sigma\\!\\!\\\ \end{array}\right)\right|\leq
c_{m}\frac{\sqrt{N_{m}}\Gamma(N_{m})}{\left|\sigma\right|^{N_{m}}}\frac{\Gamma(N_{1}-2)}{\left|\sigma\right|^{N_{1}-2}}\prod\limits_{k=2}^{m-1}\frac{\sqrt{\Sigma_{k}}\Gamma(N_{k}-1)}{\left|\sigma\right|^{N_{k}-1}}$

where $c_{m}$ is the constant appearing in Lemma 5.3 and, as before,
$\Sigma_{k}=N_{k}+N_{k+1}+\ldots+N_{m}$.

###### Proof.

From (5.6), we can infer that

$F^{(m)}\\!\left(0;\\!\begin{array}[]{c}N_{1},\\!\\!\\!\\\ \sigma,\\!\\!\\!\\\
\end{array}\begin{array}[]{c}\ldots,\\!\\!\\!\\\ \ldots,\\!\\!\\!\\\
\end{array}\begin{array}[]{c}N_{m}\\!\\!\\\ \sigma\\!\\!\\\
\end{array}\right)=\mathrm{e}^{\pi\mathrm{i}N_{1}}\frac{1}{\sigma^{N_{1}-1}}\int_{0}^{+\infty}\mathrm{e}^{-t}t^{N_{1}-2}F^{(m-1)}\\!\left(\frac{t}{\sigma}\mathrm{e}^{\pi\mathrm{i}};\\!\begin{array}[]{c}N_{2},\\!\\!\\!\\\
\sigma,\\!\\!\\!\\\ \end{array}\begin{array}[]{c}\ldots,\\!\\!\\!\\\
\ldots,\\!\\!\\!\\\ \end{array}\begin{array}[]{c}N_{m}\\!\\!\\\
\sigma\\!\\!\\\ \end{array}\right)\mathrm{d}t.$

The desired result now follows by estimating the right-hand side using the
inequality (5.4). ∎

###### Proof of Theorem 5.1.

Throughout the proof, we will frequently make use of the following inequality:

(5.8)
$\left|\operatorname{erfc}(w)\right|\leq\begin{cases}K|\mathrm{e}^{-w^{2}}|,&\text{if
}\;\operatorname{Re}(w)\geq 0,\\\ K,&\text{if
}\;\operatorname{Re}(w)<0,\end{cases}$

with an absolute constant $K>0$. This inequality can be verified by reference
to the large-$w$ behaviour of $\operatorname{erfc}(w)$ (see, e.g., [21,
§7.12(i)]) and the fact that $\operatorname{erfc}$ is an entire function. We
leave the details to the interested reader.

We begin with the proof of (5.1). The proof is by induction on $n$. For the
base case $n=0$, we combine (4.4), (4.2) and (3.14) to obtain

$\frac{\mathrm{e}^{-m\sigma
z}}{(2\pi\mathrm{i})^{m}z^{mN-m}}F^{(m)}\\!\left(z;\\!\begin{array}[]{c}N,\,\ldots,\\!\\!\\!\\!\\\
\,\sigma,\,\,\ldots,\\!\\!\\!\\!\\\ \end{array}\begin{array}[]{c}N\\!\\!\\\
\sigma\\!\\!\\\ \end{array}\right)\\\
=\sum\limits_{\pi(m)}\prod\limits_{j=1}^{m}\frac{1}{j^{k_{j}}k_{j}!}\left(\frac{1}{2}\operatorname{erfc}\Big{(}c(\varphi)\sqrt{\tfrac{j}{2}\left|\sigma
z\right|}\Big{)}+\mathcal{O}\bigg{(}\bigg{|}\frac{\mathrm{e}^{-\frac{j}{2}\left|\sigma
z\right|c^{2}(\varphi)}}{\sqrt{j\left|\sigma
z\right|}}\bigg{|}\bigg{)}\right)^{k_{j}}.$

In arriving at this expression, we made use of the fact that
$k_{1}+2k_{2}+\ldots+mk_{m}=m$. Expanding the right-hand side using the
binomial theorem gives (5.1) but with the error term

(5.9)
$\mathcal{O}(1)\sum\limits_{\pi(m)}\prod\limits_{j=1}^{m}\frac{1}{j^{k_{j}}k_{j}!}\frac{\mathrm{e}^{-\frac{jk_{j}}{2}\left|\sigma
z\right|c^{2}(\varphi)}}{\sqrt{j\left|\sigma
z\right|}}\sum\limits_{r=0}^{k_{j}-1}\binom{k_{j}}{r}\frac{1}{2^{r}}\operatorname{erfc}^{r}\Big{(}c(\varphi)\sqrt{\tfrac{j}{2}\left|\sigma
z\right|}\Big{)}\frac{\mathrm{e}^{\frac{jr}{2}\left|\sigma
z\right|c^{2}(\varphi)}}{(j\left|\sigma z\right|)^{(k_{j}-1-r)/2}}.$

Consider first the case that $-\pi<-\pi+\delta\leq\varphi\leq\pi$. In this
case $\operatorname{Re}(c(\varphi))\geq 0$ (see [20, Fig. 4]), and therefore
(5.8) implies that the inner sum in (5.9) is $\mathcal{O}(1)$ for large
$|\sigma z|$. Since $k_{1}+2k_{2}+\ldots+mk_{m}=m$, (5.9) can then be
simplified to the form given in (5.1). If $\pi<\varphi\leq 3\pi-\delta<3\pi$,
then $\operatorname{Re}(c(\varphi))<0$ (cf. [20, Fig. 4]). Hence, in this case
the inner sum in (5.9) is

$\mathcal{O}(1)\mathrm{e}^{\frac{j(k_{j}-1)}{2}\left|\sigma
z\right|c^{2}(\varphi)},$

giving the desired error term in (5.1).

Suppose now that (5.1) holds up to $n-1$, where $n$ is a positive integer. The
induction step relies upon the identity (see [15, Eq. (2.8)])

(5.10)
$\begin{multlined}zF^{(m)}\\!\left(z;\\!\begin{array}[]{c}N,\,\ldots,\\!\\!\\!\\!\\\
\,\sigma,\,\,\ldots,\\!\\!\\!\\!\\\
\end{array}\begin{array}[]{c}N,\\!\\!\\!\\\ \sigma,\\!\\!\\!\\\
\end{array}\begin{array}[]{c}N-n\\!\\!\\\ \sigma\\!\\!\\\
\end{array}\right)=F^{(m)}\\!\left(z;\\!\begin{array}[]{c}N,\,\ldots,\\!\\!\\!\\!\\\
\,\sigma,\,\,\ldots,\\!\\!\\!\\!\\\
\end{array}\begin{array}[]{c}N,\\!\\!\\!\\\ \sigma,\\!\\!\\!\\\
\end{array}\begin{array}[]{c}N-n+1\\!\\!\\\ \sigma\\!\\!\\\
\end{array}\right)\\\
-\sum_{k=0}^{m-1}F^{(k)}\\!\left(z;\\!\begin{array}[]{c}N,\,\ldots,\\!\\!\\!\\!\\\
\,\sigma,\,\,\ldots,\\!\\!\\!\\!\\\ \end{array}\begin{array}[]{c}N\\!\\!\\\
\sigma\\!\\!\\\
\end{array}\right)F^{(m-k)}\\!\left(0;\\!\begin{array}[]{c}N+1,\\!\\!\\!\\\
\sigma,\\!\\!\\!\\\ \end{array}\begin{array}[]{c}N,\,\ldots,\\!\\!\\!\\!\\\
\,\sigma,\,\,\ldots,\\!\\!\\!\\!\\\ \end{array}\begin{array}[]{c}N-n\\!\\!\\\
\sigma\\!\\!\\\
\end{array}\right).\end{multlined}zF^{(m)}\\!\left(z;\\!\begin{array}[]{c}N,\,\ldots,\\!\\!\\!\\!\\\
\,\sigma,\,\,\ldots,\\!\\!\\!\\!\\\
\end{array}\begin{array}[]{c}N,\\!\\!\\!\\\ \sigma,\\!\\!\\!\\\
\end{array}\begin{array}[]{c}N-n\\!\\!\\\ \sigma\\!\\!\\\
\end{array}\right)=F^{(m)}\\!\left(z;\\!\begin{array}[]{c}N,\,\ldots,\\!\\!\\!\\!\\\
\,\sigma,\,\,\ldots,\\!\\!\\!\\!\\\
\end{array}\begin{array}[]{c}N,\\!\\!\\!\\\ \sigma,\\!\\!\\!\\\
\end{array}\begin{array}[]{c}N-n+1\\!\\!\\\ \sigma\\!\\!\\\
\end{array}\right)\\\
-\sum_{k=0}^{m-1}F^{(k)}\\!\left(z;\\!\begin{array}[]{c}N,\,\ldots,\\!\\!\\!\\!\\\
\,\sigma,\,\,\ldots,\\!\\!\\!\\!\\\ \end{array}\begin{array}[]{c}N\\!\\!\\\
\sigma\\!\\!\\\
\end{array}\right)F^{(m-k)}\\!\left(0;\\!\begin{array}[]{c}N+1,\\!\\!\\!\\\
\sigma,\\!\\!\\!\\\ \end{array}\begin{array}[]{c}N,\,\ldots,\\!\\!\\!\\!\\\
\,\sigma,\,\,\ldots,\\!\\!\\!\\!\\\ \end{array}\begin{array}[]{c}N-n\\!\\!\\\
\sigma\\!\\!\\\ \end{array}\right).$

We will estimate the sum in (5.10) under the assumptions that $|\sigma z|$ is
large, $N\sim|\sigma z|$, and $m$ and $n$ are both fixed. From Lemma 5.4, we
can assert that

$\left|F^{(m-k)}\\!\left(0;\\!\begin{array}[]{c}N+1,\\!\\!\\!\\\
\sigma,\\!\\!\\!\\\ \end{array}\begin{array}[]{c}N,\,\ldots,\\!\\!\\!\\!\\\
\,\sigma,\,\,\ldots,\\!\\!\\!\\!\\\ \end{array}\begin{array}[]{c}N-n\\!\\!\\\
\sigma\\!\\!\\\ \end{array}\right)\right|\\\ \leq
c_{m-k}\frac{\sqrt{N-n}\Gamma(N-n)}{\left|\sigma\right|^{N-n}}\left(\frac{\Gamma(N-1)}{\left|\sigma\right|^{N-1}}\right)^{m-k-1}\prod\limits_{j=1}^{m-k-2}\sqrt{jN-n}$

for any $0\leq k\leq m-2$. On replacing the gamma functions by Stirling’s
approximation, we obtain

(5.11) $F^{(m-k)}\\!\left(0;\\!\begin{array}[]{c}N+1,\\!\\!\\!\\\
\sigma,\\!\\!\\!\\\ \end{array}\begin{array}[]{c}N,\,\ldots,\\!\\!\\!\\!\\\
\,\sigma,\,\,\ldots,\\!\\!\\!\\!\\\ \end{array}\begin{array}[]{c}N-n\\!\\!\\\
\sigma\\!\\!\\\
\end{array}\right)=\mathcal{O}(1)\left|z\right|^{(m-k)(N-1)-n+1}\mathrm{e}^{-(m-k)\left|\sigma
z\right|}\frac{1}{\sqrt{\left|\sigma z\right|}}.$

By [18, Eq. (2.2)] this estimate is also valid for $k=m-1$. Assume that
$\varphi$ is confined to the sector $-\pi<-\pi+\delta\leq\varphi\leq\pi$ (and
so, in particular, $\operatorname{Re}(c(\varphi))\geq 0$). The combination of
the base case (with $k$ in place of $m$) and the inequality (5.8) yields the
crude estimate

(5.12) $\frac{\mathrm{e}^{-k\sigma
z}}{(2\pi\mathrm{i})^{k}z^{kN-k}}F^{(k)}\\!\left(z;\\!\begin{array}[]{c}N,\,\ldots,\\!\\!\\!\\!\\\
\,\sigma,\,\,\ldots,\\!\\!\\!\\!\\\ \end{array}\begin{array}[]{c}N\\!\\!\\\
\sigma\\!\\!\\\
\end{array}\right)=\mathcal{O}(1)\mathrm{e}^{-\frac{k}{2}\left|\sigma
z\right|c^{2}(\varphi)}.$

Note that (5.12) remains true when $k=0$. Since
$\operatorname{Re}\left(\frac{1}{2}\left|\sigma
z\right|c^{2}(\varphi)\right)=\left|\sigma
z\right|(1+\cos\varphi)=\left|\sigma z\right|+\operatorname{Re}(\sigma z)$, we
can infer from (5.11) and (5.12) that the summation in (5.10) is

$\sum\limits_{k=0}^{m-1}\mathcal{O}(1)\mathrm{e}^{k\sigma
z}z^{kN-k}\mathrm{e}^{-\frac{k}{2}\left|\sigma
z\right|c^{2}(\varphi)}\left|z\right|^{(m-k)(N-1)-n+1}\mathrm{e}^{-(m-k)\left|\sigma
z\right|}\frac{1}{\sqrt{\left|\sigma z\right|}}\\\ =z^{mN-
m-n+1}\mathrm{e}^{m\sigma
z}\mathcal{O}\bigg{(}\bigg{|}\cfrac{\mathrm{e}^{-\frac{m}{2}\left|\sigma
z\right|c^{2}(\varphi)}}{\sqrt{|\sigma z|}}\bigg{|}\bigg{)}.$

Thus, after re-normalisation, (5.10) becomes

$\frac{\mathrm{e}^{-m\sigma z}}{(2\pi\mathrm{i})^{m}z^{mN-
m-n}}F^{(m)}\\!\left(z;\\!\begin{array}[]{c}N,\,\ldots,\\!\\!\\!\\!\\\
\,\sigma,\,\,\ldots,\\!\\!\\!\\!\\\
\end{array}\begin{array}[]{c}N,\\!\\!\\!\\\ \sigma,\\!\\!\\!\\\
\end{array}\begin{array}[]{c}N-n\\!\\!\\\ \sigma\\!\\!\\\
\end{array}\right)\\\ =\frac{\mathrm{e}^{-m\sigma
z}}{(2\pi\mathrm{i})^{m}z^{mN-
m-n+1}}F^{(m)}\\!\left(z;\\!\begin{array}[]{c}N,\,\ldots,\\!\\!\\!\\!\\\
\,\sigma,\,\,\ldots,\\!\\!\\!\\!\\\
\end{array}\begin{array}[]{c}N,\\!\\!\\!\\\ \sigma,\\!\\!\\!\\\
\end{array}\begin{array}[]{c}N-n+1\\!\\!\\\ \sigma\\!\\!\\\
\end{array}\right)+\mathcal{O}\bigg{(}\bigg{|}\frac{\mathrm{e}^{-\frac{m}{2}\left|\sigma
z\right|c^{2}(\varphi)}}{\sqrt{|\sigma z|}}\bigg{|}\bigg{)}.$

The desired result now follows by applying the induction hypothesis on the
right-hand side of this equality. If $\pi<\varphi\leq 3\pi-\delta<3\pi$, then
$\operatorname{Re}(c(\varphi))<0$ and the left-hand side of (5.12) is
$\mathcal{O}(1)$. Thus, the sum in (5.10) may be estimated as

$\sum\limits_{k=0}^{m-1}\mathcal{O}(1)\mathrm{e}^{k\sigma
z}z^{kN-k}\left|z\right|^{(m-k)(N-1)-n+1}\mathrm{e}^{-(m-k)\left|\sigma
z\right|}\frac{1}{\sqrt{\left|\sigma z\right|}}\\\ =z^{mN-
m-n+1}\mathrm{e}^{m\sigma
z}\sum\limits_{k=0}^{m-1}\mathcal{O}\bigg{(}\bigg{|}\frac{\mathrm{e}^{-\frac{m-k}{2}\left|\sigma
z\right|c^{2}(\varphi)}}{\sqrt{|\sigma z|}}\bigg{|}\bigg{)}=z^{mN-
m-n+1}\mathrm{e}^{m\sigma
z}\mathcal{O}\bigg{(}\bigg{|}\frac{\mathrm{e}^{-\frac{1}{2}\left|\sigma
z\right|c^{2}(\varphi)}}{\sqrt{|\sigma z|}}\bigg{|}\bigg{)}.$

We can now proceed analogously to the preceding case and finish the proof of
(5.1).

The proof of (5.2) is completely analogous, except that one uses the
asymptotic formula (cf. [20, Eq. (5.11)])

$\frac{\mathrm{e}^{-2\pi\mathrm{i}N}\mathrm{e}^{-\sigma
z}}{2\pi\mathrm{i}z^{N-1-n}}F^{(1)}\\!\left(z;\\!\begin{array}[]{c}N-n\\!\\\
\sigma\\!\\\
\end{array}\right)=-\frac{1}{2}\operatorname{erfc}\Big{(}\overline{c(-\varphi)}\sqrt{\tfrac{1}{2}|\sigma
z|}\Big{)}+\mathcal{O}\bigg{(}\bigg{|}\frac{\mathrm{e}^{-\frac{1}{2}|\sigma
z|c^{2}(-\varphi)}}{\sqrt{\left|\sigma z\right|}}\bigg{|}\bigg{)},$

valid when $-3\pi<-3\pi+\delta\leq\varphi\leq\pi-\delta<\pi$, in place of
(3.14). We omit the details. ∎

###### Proof of Proposition 5.2.

Consider first the case when the singulant pair is
$(\sigma,\sigma\mathrm{e}^{-\pi\mathrm{i}})$. If we make a change of
integration variable from $t_{2}$ to $u$ by
$u=t_{2}/(t_{1}\mathrm{e}^{\pi\mathrm{i}})$ and then from $t_{1}$ to $s$ by
$s=t_{1}(1+u)$, (3.5) is recast as

$F^{(2)}\\!\left(z;\\!\begin{array}[]{c}N,\\!\\!\\!\\!\\\
\sigma,\\!\\!\\!\\!\\\ \end{array}\begin{array}[]{c}N-n\\!\\\
\sigma\mathrm{e}^{-\pi\mathrm{i}}\\!\\\
\end{array}\right)=(-1)^{n}\mathrm{e}^{\pi\mathrm{i}N}\int_{0}^{+\infty}\frac{u^{N-n-1}}{(1+u)^{2N-n-1}}F^{(1)}\\!\left((1+u)z;\\!\begin{array}[]{c}2N-n-1\\!\\\
\sigma\\!\\\ \end{array}\right)\mathrm{d}u.$

Using the inequality (5.4) and the known integral representation of the beta
function [21, Eq. 5.12.3], we deduce

(5.13) $\left|F^{(2)}\\!\left(z;\\!\begin{array}[]{c}N,\\!\\!\\!\\!\\\
\sigma,\\!\\!\\!\\!\\\ \end{array}\begin{array}[]{c}N-n\\!\\\
\sigma\mathrm{e}^{-\pi\mathrm{i}}\\!\\\ \end{array}\right)\right|\leq
c_{1}\frac{1}{\left|z\right|}\frac{1}{\left|\sigma\right|^{2N-n-1}}\frac{\Gamma(N-n)\Gamma(N)}{\sqrt{2N-n-1}}$

provided $|\varphi|\leq\pi$. If $|\sigma z|$ is large, $N\sim|\sigma z|$, and
$n$ is fixed, then, with the aid of Stirling’s formula, we find from (5.13)
that

(5.14) $F^{(2)}\\!\left(z;\\!\begin{array}[]{c}N,\\!\\!\\!\\!\\\
\sigma,\\!\\!\\!\\!\\\ \end{array}\begin{array}[]{c}N-n\\!\\\
\sigma\mathrm{e}^{-\pi\mathrm{i}}\\!\\\
\end{array}\right)=\mathcal{O}(1)\left|z\right|^{2N-2-n}\mathrm{e}^{-2\left|\sigma
z\right|}\frac{1}{\sqrt{\left|\sigma z\right|}}=z^{2N-2-n}\mathrm{e}^{2\sigma
z}\mathcal{O}\bigg{(}\bigg{|}\frac{\mathrm{e}^{-\left|\sigma
z\right|c^{2}(\varphi)}}{\sqrt{|\sigma z|}}\bigg{|}\bigg{)}$

uniformly in the sector $|\varphi|\leq\pi$.

Suppose now that $\pi<\varphi\leq 2\pi-\delta<2\pi$. From (3.8), one infers

$F^{(2)}\\!\left(z;\\!\begin{array}[]{c}N,\\!\\!\\!\\!\\\
\sigma,\\!\\!\\!\\!\\\ \end{array}\begin{array}[]{c}N-n\\!\\\
\sigma\mathrm{e}^{-\pi\mathrm{i}}\\!\\\
\end{array}\right)=F^{(2)}\\!\left(z\mathrm{e}^{-2\pi\mathrm{i}};\\!\begin{array}[]{c}N,\\!\\!\\!\\!\\\
\sigma,\\!\\!\\!\\!\\\ \end{array}\begin{array}[]{c}N-n\\!\\\
\sigma\mathrm{e}^{-\pi\mathrm{i}}\\!\\\
\end{array}\right)+2\pi\mathrm{i}\mathrm{e}^{\sigma
z}z^{N-1}F^{(1)}\\!\left(z;\\!\begin{array}[]{c}N-n\\!\\\
\sigma\mathrm{e}^{-\pi\mathrm{i}}\\!\\\ \end{array}\right).$

If we again take $N\sim|\sigma z|\gg 1$, and keep $n$ fixed, then we can
estimate the first term on the right-hand side by the quantity on the right-
hand side of (5.14). Regarding the second term, since
$0<\arg(\sigma\mathrm{e}^{-\pi\mathrm{i}}z)\leq\pi-\delta<\pi$, Olver’s
formula (3.13) implies that

$2\pi\mathrm{i}\mathrm{e}^{\sigma
z}z^{N-1}F^{(1)}\\!\left(z;\\!\begin{array}[]{c}N-n\\!\\\
\sigma\mathrm{e}^{-\pi\mathrm{i}}\\!\\\
\end{array}\right)=z^{2N-2-n}\mathrm{e}^{2\sigma
z}\mathcal{O}\bigg{(}\bigg{|}\frac{\mathrm{e}^{-\frac{1}{2}\left|\sigma
z\right|c^{2}(\varphi)}}{\sqrt{|\sigma z|}}\bigg{|}\bigg{)}.$

The conjugate sector $-2\pi<-2\pi+\delta\leq\varphi<-\pi$ may be treated in a
similar manner.

If the singulant pair is $(\sigma,\sigma\mathrm{e}^{\pi\mathrm{i}})$, the
estimate (5.3) can simply be deduced from the previous case and the equality

$F^{(2)}\\!\left(z;\\!\begin{array}[]{c}N,\\!\\!\\!\\!\\\
\sigma,\\!\\!\\!\\!\\\ \end{array}\begin{array}[]{c}N-n\\!\\\
\sigma\mathrm{e}^{\pi\mathrm{i}}\\!\\\
\end{array}\right)=\mathrm{e}^{-2\pi\mathrm{i}N}F^{(2)}\\!\left(z;\\!\begin{array}[]{c}N,\\!\\!\\!\\!\\\
\sigma,\\!\\!\\!\\!\\\ \end{array}\begin{array}[]{c}N-n\\!\\\
\sigma\mathrm{e}^{-\pi\mathrm{i}}\\!\\\ \end{array}\right),$

which follows readily from the definition (3.5). ∎

##  6\. Conclusions

The Stokes phenomenon concerns the sudden change across certain rays in the
complex plane, known as Stokes lines, exhibited by the coefficients
multiplying exponentially small terms in compound asymptotic expansions.
Dingle introduced a set of rules for locating Stokes lines and continuing
asymptotic expansions across them. Included among these rules is the “final
main rule” stating that half the discontinuity in form occurs on reaching the
Stokes line, and half on leaving it the other side. Berry showed that, if an
asymptotic expansion is terminated at or near its numerically least term, the
transition between two different asymptotic forms across a Stokes line is
effected smoothly and not discontinuously. Furthermore, for a wide class of
functions, the coefficient multiplying a subdominant exponential contribution
(a Stokes multiplier) possesses a universal structure expressed approximately
in terms of an error function whose argument is an appropriate variable
describing the transition across a Stokes line.

In this paper, we revisited the well-known asymptotic expansions of the gamma
function and its reciprocal. These expansions do not share the simple
properties above. In the neighbourhood of a Stokes line, not one but
infinitely many exponentially small contributions appear, each associated with
its own Stokes multiplier. Moreover, these multipliers may no longer obey
Dingle’s rule: their values can differ from $\frac{1}{2}$ on a Stokes line and
can be non-zero only on the line itself. This unconventional behaviour of the
multipliers is a manifestation of an infinite number of higher-order Stokes
phenomena. We demonstrated that these phenomena are rapid but smooth
transitions in the remainder terms of successive hyperasymptotic re-
expansions. The approximate functional form of the Stokes multipliers is,
however, no longer described by a single error function but rather by a
multivariate polynomial in such functions. The basis of our proof was an
identity between hyperterminants, which was originally conjectured by Howls.

The treatment presented here is limited to the situation where all the
singulants are equal to each other. In the language of Borel summation, this
means that the singularities of the Borel transform are collinear and are
equally spaced. This is a common phenomenon, e.g., for transseries solutions
to non-linear difference and differential equations (cf. [7, 9, 17]). The work
in this paper can certainly be extended to cover such problems. A more
interesting extension is to allow the singulants to have different magnitude
but remain collinear. We believe that the smooth interpretation of the higher-
order Stokes phenomenon remains possible in such circumstances, although the
approximate functional form may involve transcendental functions that are more
complicated than an error function.

## Acknowledgement

The author thanks the referees for helpful comments and suggestions for
improving the presentation. The author wishes to thank A. B. Olde Daalhuis for
useful discussions. The author’s research was supported by a Premium
Postdoctoral Fellowship of the Hungarian Academy of Sciences and by the JSPS
Postdoctoral Research Fellowship No. P21020.

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