A curious identity in connection with saddle-point method and Stirlings formula Hsien-Kuei Hwang ABSTRACT . We prove the curious identity in the sense of formal power series
2025-04-30
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A curious identity in connection with saddle-point method and Stirling’s formula
Hsien-Kuei Hwang
ABSTRACT. We prove the curious identity in the sense of formal power series:
Z∞
−∞
[ym] exp
−t2
2+X
j>3
(it)j
j!yj−2
dt=Z∞
−∞
[ym] exp
−t2
2+X
j>3
(it)j
jyj−2
dt,
for m= 0,1, . . . , where [ym]f(y)denotes the coefficient of ymin the Taylor expansion of f.
The generality of this identity from the perspective of saddle-point method is also examined.
1. INTRODUCTION
The following unusual identity was discovered through different manipulations of the saddle-
point method in order to derive Stirling’s formula, which has a huge literature since de Moivre’s
and Stirling’s pioneering analysis in the early eighteenth century; see for example the survey [1]
(and the references therein) and the book [9] for five different analytic proofs. While the identity
can be deduced from known expansions for n!(e.g., [3,23]), the formulation, as well as the
proof given here, is of independent interest per se. Denote by [ym]f(y)the coefficient of ymin
the Taylor expansion of f.
Theorem 1.1. Let
cm:= 1
√2πZ∞
−∞
[ym] exp−t2
2+X
j>3
(it)j
j!yj−2dt, (1)
and
dm:= 1
√2πZ∞
−∞
[ym] exp−t2
2+X
j>3
(it)j
jyj−2dt. (2)
Then
cm=dm(m= 0,1, . . . ).(3)
When mis odd, cm=dm= 0 because the coefficient of ymcontains only odd powers of t.
When m= 2lis even, the identity (3) can be written explicitly as follows:
c2l=X
16h62l
(−1)l+h(2l+ 2h)!
(l+h)!2l+hX
j1+2j2+···+2lj2l=2l
j1+···+j2l=h
1
j1!···j2l!·3!j14!j2···(2l+ 2)!j2l
=X
16h62l
(−1)l+h(2l+ 2h)!
(l+h)!2l+hX
j1+2j2+···+2lj2l=2l
j1+···+j2l=h
1
j1!···j2l!·3j14j2···(2l+ 2)j2l
=d2l.
Date: October 21, 2022.
The research of the author was partially supported by Taiwan Ministry of Science and Technology under the
Grant MOST 108-2118-M-001-005-MY3.
1
arXiv:2210.10989v1 [math.CO] 20 Oct 2022
In particular,
{c2l}l>0=1,−1
12,1
288,139
51840,−571
2488320,−163879
209018880,···,
which are modulo sign the coefficients appearing in the asymptotic expansion of Stirling’s
formula; see [8, §1.18] or [22,A001163,A001164]:
1
n!∼enn−n−1
2
√2πX
m>0
c2mn−m,or n!∼√2πe−nnn+1
2X
m>0
(−1)mc2mn−m.(4)
These Stirling coefficients have been extensively studied in the literature; see, e.g., [6, §8.2],
and [2,15,18,23], and the references cited there.
On the other hand, the integral in (1) without the coefficient-extraction operator [ym]is di-
vergent for y∈Rdue to periodicity:
Z∞
−∞
exp−t2
2+X
j>3
(it)j
j!yj−2dt=Z∞
−∞
expeity −1−ity
y2dt,
while that in (2) is absolutely convergent for real |y|<1:
Z∞
−∞
exp−t2
2+X
j>3
(it)j
jyj−2dt=Z∞
−∞1−ity−y−2
e−it/y dt.
Proof of Theorem 1.1.For convenience, we write
fn≈gnwhen fn=gn+Oe−εn,
for some generic ε > 0whose value is immaterial.
The standard asymptotic expansion. We begin with the Cauchy integral representation for
n!−1:1
n!=1
2πi I|z|=n
z−n−1ezdz,
where the integration contour is the circle with radius |z|=n. The standard application of the
saddle-point method (see [9, p. 555]) proceeds by first making the change of variables z7→ neu,
giving
e−nnn
n!=1
2πi Zπi
−πi
en(eu−1−u)du≈1
2πi Zεi
−εi
en(eu−1−u)du. (5)
Now by the change of variables u=it
√n, we have
1
2πi Zεi
−εi
en(eu−1−u)du=1
2π√nZε√n
−ε√n
exp−t2
2+X
j>3
(it)j
j!n−1
2j+1dt.
If we choose ε=εn=n−2
5, say, then nε2
n→ ∞ and nεj
n→0for j>3, so that the series on
the right-hand side is small on the integration path; we can then expand the exponential of this
series in decreasing powers of n, and then extending the integration limits to infinity, yielding
the expansion (4) with c2mexpressed in the formal power series form (1). See [9, Ex. VIII.3;
p. 555 et seq.] for technical details.
On the other hand, a more effective means of computing c2mis to make first the change of
variables eu−1−u=1
2v2in the rightmost integral in (5), where u=u(v)is positive when v
is and is analytic in |v|6ε; see [26, § 3.6.3]. Then
e−nnn
n!≈1
2πi Zεi
−εi
e1
2nv2g(v) dv=1
2πZε
−ε
e−1
2nt2g(it) dt,
2
where g(v) := du
dvis analytic in |v|6ε. By Lagrange inversion formula (see [9, p. 732]),
gm:= [vm]g(v)=[tm]1
2t2
et−1−t1
2(m+1)
(m= 0,1, . . . ).(6)
Then a direct application of Watson’s Lemma (see [28, §1.5]) gives the asymptotic expansion
1
n!≈enn−n
2πX
m>0
g2mZ∞
−∞
e−1
2nt2(it)2mdt≈enn−n
√2πn X
m>0
¯g2mn−m,(7)
where
{¯g2m}m>0:= ng2m
(−1)m(2m)!
m!2mom>0=n1,−1
12,1
288,139
51840,−571
2488320,···o.
We then obtain the relation
c2m= ¯g2m=g2m
(−1)m(2m)!
m!2m.(8)
Second asymptotic expansion. It is well known that n!−1has the alternative Laplace integral
representation (see [27, p. 246]):
1
n!=1
2πi ZR+i∞
R−i∞
z−n−1ezdz(R > 0),
so that, by the change of variables z=R(1 + x), where R=n+ 1,
e−n−1(n+ 1)n
n!=1
2πi Zi∞
−i∞
e−(n+1)(log(1+x)−x)dx
≈1
2πi Zεi
−εi
e−(n+1)(log(1+x)−x)dx.
Now by the change of variables x=it
√n+1 , we have
1
2πi Zεi
−εi
e−(n+1)(log(1+x)−x)dx
=1
2π√n+ 1 Zε√n+1
−ε√n+1
exp−t2
2+X
j>3
(it)j
j(n+ 1)−1
2j+1dt.
By a similar procedure described above, we then deduce the asymptotic expansion
1
n!∼en+1(n+ 1)−n−1
2
√2πX
m>0
d2m(n+ 1)−m,(9)
where dmis given in (2); compare (7).
On the other hand, by the change of variables log(1 + x)−x=−1
2y2(y > 0when x > 0),
we have
e−n−1(n+ 1)n
n!≈1
2πi Zεi
−εi
e1
2(n+1)y2h(y) dy=1
2πZε
−ε
e−1
2(n+1)t2h(it) dt,
where h(y) = dx
dyis analytic in |y|6ε. Again, by Lagrange inversion formula,
hm:= [ym]h(y)=[ym]1
2y2
y−log(1 + y)1
2(m+1)
(m= 0,1, . . . ).(10)
Although the definition of hmlooks very different from that of gm(see (6)), their numerical
values coincide except for m= 1:
3
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时间：2025-04-30
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