EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
Vol. 14, No. 3, 2021, 666-684
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

Asymptotic Approximations of Apostol-Genocchi
Numbers and Polynomials

Cristina B. Corcino1,2

1 Research Institute for Computational Mathematics and Physics, Cebu Normal University,
6000 Cebu City, Philippines
2 Mathematics Department, Cebu Normal University, 6000 Cebu City, Philippines

Abstract. Asymptotic approximations of the Apostol-Genocchi numbers and polynomials are
derived using Fourier series and ordering of poles of the generating function. Asymptotic formulas
for the Apostol-Euler numbers and polynomials are obtained as consequence. Asymptotic formulas
for special cases which include the Genocchi numbers and polynomials are also explicitly stated.

2020 Mathematics Subject Classifications: 11B68, 41A60

Key Words and Phrases: Asymptotic approximations, Genocchi polynomials, Bernoulli poly-
nomials, Euler polynomials, Apostol-Bernoulli polynomials, Apostol-Euler polynomials, Apostol-
Genocchi polynomials

1. Introduction

The Apostol-Genocchi polynomials Gn(x;λ) are defined by the generating function

2text

λet + 1
=
∞∑
n=0

Gn(x;λ)
tn

n!
, (1.1)

where |t| < π when λ = 1 and |t+log λ| < π when λ 6= 1. When λ = 1, the above equation
gives the generating function of the Genocchi polynomials [3].

When x = 0, (1.1) reduces to the generating function of the Apostol-Genocchi numbers
Gn(0;λ) given by

2t

λet + 1
=
∞∑
n=0

Gn(0;λ)
tn

n!
. (1.2)

For λ not zero, the set of poles of the generating function (1.1) is

Tλ := {(2k + 1)πi− log λ : k ∈ Z}, (1.3)

DOI: https://doi.org/10.29020/nybg.ejpam.v14i3.3976

Email address: corcinoc@cnu.edu.ph (C. Corcino)

http://www.ejpam.com 666 © 2021 EJPAM All rights reserved.



C. Corcino / Eur. J. Pure Appl. Math, 14 (3) (2021), 666-684 667

which is also the set of poles of (1.2), where the logarithm is taken to be the principal
branch.

Bayad [2] and Luo [13] derived Fourier series of Apostol-Genocchi polynomials ex-
pressed in terms of these poles. The Fourier series they obtained is given in the next
section. Fourier expansion of higher-order Apostol-Genocchi polynomials was derived in
[4] and was shown to be reducible to those obtained in [2] and [13] when the order is 1.

New identities involving the Apostol-Genocchi polynomials were established in [9].
Some generalizations and properties of these polynomials were presented in [14]. Multipli-
cation and explicit recursive formulas of higher-order Apostol-Genocchi polynomials were
obtained in [12]. A new generalization of Apostol type Hermite-Genocchi polynomials is
studied in [1] while products of the Apostol-Genocchi polynomials were studied in [10].
Moreover, the higher-order convolutions of these polynomials using generating-function
methods and summation-transform techniques were established in [11].

Inspired by the work of Kim and Kim [7], a new class of the Frobenius-Genocchi
polynomials was considered in [6] by means of the polyexponential function and new
relations and properties were obtained. New relations on q-Genocchi polynomials where
the relations were stated by symmetric group of degree n were done in [5].

Navas, Ruiz and Varona [15] obtained asymptotic estimates of the Apostol-Bernoulli
and Apostol-Euler numbers and polynomials and further analyzed the asymptotic behavior
of the Apostol-Bernoulli polynomials in detail. The starting point of their analysis is the
Fourier series of the polynomials on the closed interval [0, 1] followed by ordering the poles
of the generating function.

In this paper, asymptotic approximations of the Apostol-Genocchi numbers and poly-
nomials for λ ∈ C\{0} are obtained. The method used in [15] is applied to the Apostol-
Genocchi numbers and polynomials to obtain asymptotic formulas of these numbers and
polynomials. A more detailed proof of the results is provided so as to reach a bigger group
of readers. Asymptotic formulas of Genocchi numbers and Euler numbers are obtained as
special cases. Asymptotic formulas of the Apostol-Euler numbers and Apostol-Euler poly-
nomials are also derived. The results in this paper will complete the results of [15] as the
latter considered only the Apostol-Bernoulli and Apostol-Euler polynomials. Moreover,
the results can be used as check formulas of those in [15].

2. Asymptotic Approximations

Fourier series of the Apostol-Genocchi polynomials in terms of the poles in Tλ is given
in the following theorem.

Theorem 2.1. ([2], [13]) Let λ ∈ C\{0}. For n ≥ 1, 0 ≤ x ≤ 1,

Gn(x;λ)

n!
=

2

λx

∑
k∈Z

e(2k+1)πix

[(2k + 1)πi− log λ]n
, (2.1)

where the logarithm is taken to be the principal branch.



C. Corcino / Eur. J. Pure Appl. Math, 14 (3) (2021), 666-684 668

Taking x = 0 in (2.1) gives the Fourier series of the Apostol-Genocchi numbers given
by

Gn(0;λ)

n!
= 2

∑
k∈Z

1

[(2k + 1)πi− log λ]n
, (2.2)

where the logarithm is taken to be the principal branch.

Proceeding as in [15], ordering of the poles of the generating function (1.1) is done in
the following lemma.

Lemma 2.2. Let uk = (2k + 1)πi − log λ with k ∈ Z, λ ∈ C\{0} and γ = (log λ)/2πi,
where the logarithm is taken to be the principal branch.

a) If Im λ > 0 then 0 < Re γ < 1
2 and for k ≥ 1,

|u0| < |u−1| < |u1| < |u−2| < |u2| < · · · < |u−k| < |uk| < · · · (2.3)

b) If Im λ < 0 then −1
2 < Re γ < 0 and for k ≥ 1,

|u−1| < |u0| < |u−2| < |u1| < |u−3| < · · ·
< |u−k| < |uk−1| < |u−(k+1)| < |uk| < · · · . (2.4)

c) If λ > 0 (positive real number), then Re γ = 0, and for k ≥ 1,

|u0| = |u−1| < |u1| = |u−2| < |u2| < · · ·
< |u−k| < |uk| = |u−(k+1)| < |uk+1| < · · · . (2.5)

d) If λ < 0 (negative real number), then Re γ = 1
2 , and for k ≥ 1,

|u0| < |u1| = |u−1| < |u2| = |u−2| < · · · < |uk| = |u−k| < |uk+1| < · · · . (2.6)

Moreover, |uk| ≥ 2π(|k| − 1) if |k| ≥ 1.

Proof. With the logarithm taken to be the principal branch, γ (as a function of λ)
maps λ ∈ C\{0} to the strip −1

2 < Re γ ≤ 1
2 (see [15]). To see this write

γ =
θ

2π
− i ln |λ|

2π
,

from which we have

Re γ =
θ

2π
and Im γ = − ln |λ|

2π
.

With −π < θ ≤ π,

−π
2π
≤ Re γ =

θ

2π
≤ π

2π
⇒ −1

2
< Re γ ≤ 1

2
,



C. Corcino / Eur. J. Pure Appl. Math, 14 (3) (2021), 666-684 669

where Re γ = 0 when λ > 0 and Re γ = 1
2 when λ < 0.

If Im λ > 0, then 0 < θ < π, hence 0 < Re γ < 1
2 . If Im λ < 0, then −π < θ < 0,

hence −1
2 < Re γ < 0.

To verify the chains in (2.3), (2.4), (2.5), (2.6), let x = Re γ and y = Im γ. Then for
k ∈ Z,

uk = 2π

√(
k +

1

2
− x
)2

+ y2.

a) If Im λ > 0, then 0 < x < 1
2 and

|u0| = 2π

√(
1

2
− x
)2

+ y2

|u1| = 2π

√(
3

2
− x
)2

+ y2

|u2| = 2π

√(
5

2
− x
)2

+ y2

|u−1| = 2π

√(
−1

2
− x
)2

+ y2 = 2π

√(
1

2
+ x

)2

+ y2

|u−2| = 2π

√(
−3

2
− x
)2

+ y2 = 2π

√(
3

2
+ x

)2

+ y2

|u−3| = 2π

√(
−5

2
− x
)2

+ y2 = 2π

√(
5

2
+ x

)2

+ y2

|u3| = 2π

√(
7

2
− x
)2

+ y2

· · ·

From which one can see that the order of magnitude of uk, k ∈ Z given in (2.3)
holds.

b) The second case can be derived similarly.

The last two cases are belonging to the case Im λ = 0. This means that λ is a real
number which is either positive or negative but not zero. Hence the cases c and d.

c) If λ > 0, then Re γ = 0. For k ≥ 0,

|uk| = 2π

√(
k +

1

2

)2

+ y2.



C. Corcino / Eur. J. Pure Appl. Math, 14 (3) (2021), 666-684 670

In particular,

|u0| = 2π

√(
1

2

)2

+ y2

|u1| = 2π

√(
1 +

1

2

)2

+ y2

|u−1| = 2π

√(
−1 +

1

2

)2

+ y2

|u2| = 2π

√(
2 +

1

2

)2

+ y2

|u−2| = 2π

√(
−2 +

1

2

)2

+ y2

|u3| = 2π

√(
3 +

1

2

)2

+ y2

From which we have the chain

|u0| = |u−1| < |u1| = |u−2| < |u2| < · · ·
< |uk| = |u−(k+1)| < |uk+1| < · · · ,

which is exactly (2.5).

d) If λ < 0, θ = π, hence x = 1
2 . For k ≥ 0,

|uk| = 2π
√
k2 + y2 = |u−k|,

from which it can be observed easily that

|u0| < |u1| = |u−1| < |u2| = |u−2| < |u3| = |u−3|
< · · · < |uk| = |u−k| < · · · ,

which is exactly the chain in (2.6).

Moreover,

|uk| = 2π

∣∣∣∣k +
1

2
− γ
∣∣∣∣

= 2π

√(
k +

1

2
− x

)2

+ y2

≥ 2π

√(
k +

1

2
− x

)2



C. Corcino / Eur. J. Pure Appl. Math, 14 (3) (2021), 666-684 671

= 2π

∣∣∣∣k +
1

2
− x

∣∣∣∣ , with − 1

2
≤ x ≤ 1

2

= 2π

∣∣∣∣k − (x − 1

2

)∣∣∣∣
≥ 2π

(
|k| −

∣∣∣∣x− 1

2

∣∣∣∣)
≥ 2π

(
|k| −

∣∣∣∣12 − x

∣∣∣∣)
≥ 2π (|k| − 1) .

An asymptotic expansion of the Apostol-Genocchi numbers Gn(0;λ) is given in the
next theorem.

Theorem 2.3. Given λ ∈ C\{0}, let H be a finite subset of Tλ satisfying

max {|u| : u ∈ H} < min {|u| : u ∈ Tλ\H} := ν.

For all integers n ≥ 2,
Gn(0;λ)

n!
= 2

∑
u∈H

1

un
+O(ν−n).

Proof. Write the series in (2.2) as
∑

k
1

(uk)n
. By Lemma 2.2 we can relabel the set of

poles in increasing order of magnitude as

|µ0| ≤ |µ1| ≤ · · · ≤ |µM | ≤ · · · .

Since |µk| ≥ 2π(|k| − 1), for k ≥ 2, the series
∑

k
1

(µk)n
is absolutely convergent for n ≥ 2.

For any M > 2, the tail of the series is

∞∑
k=M+1

1

|µk|n
=

1

|µM+1|n
∞∑

k=M+1

∣∣∣∣µM+1

µk

∣∣∣∣n .
Since for k > M + 1,

∣∣∣µM+1

µk

∣∣∣ ≤ 1, we have
∣∣∣µM+1

µk

∣∣∣n ≤ ∣∣∣µM+1

µk

∣∣∣2 for n ≥ 2.

Hence,
∞∑

k=M+1

1

|µk|n
≤ 1

|µM+1|n
∞∑

k=M+1

∣∣∣∣µM+1

µk

∣∣∣∣2 .
Let

CM,λ =
∞∑

k=M+1

∣∣∣∣µM+1

µk

∣∣∣∣2 .
Then

∞∑
k=M+1

1

|µk|n
≤

CM,λ

|µM+1|n
.



C. Corcino / Eur. J. Pure Appl. Math, 14 (3) (2021), 666-684 672

Consider CM,λ:

CM,λ =

∞∑
k=M+1

|µM+1|2

|µk|2

= |µM+1|2
∞∑

k=M+1

1

|µk|2

= (2π)2
∣∣∣∣M + 1 +

1

2
− γ
∣∣∣∣2 ∞∑
k=M+1

1

(2π)2
∣∣k + 1

2 − γ
∣∣2

≤
∣∣∣∣M +

3

2
− γ
∣∣∣∣2 ∞∑
k=M+1

1

(|k| − 1)2

≤ 2

∣∣∣∣M +
3

2
− γ
∣∣∣∣2 ∞∑

l=0

1

(M + l)2

≤ 2

∣∣∣∣M +
3

2
− γ
∣∣∣∣2
(

1

M2
+

∞∑
l=1

1

(M + l)2

)
.

With
∞∑
l=1

1

(M + l)2
≤
∫ ∞
1

1

(M + x)2
dx =

1

M + 1
,

CM,λ ≤ 2

∣∣∣∣M +
3

2
− γ
∣∣∣∣2( 1

M2
+

1

M + 1

)
=

2
∣∣M + 3

2 − γ
∣∣2

M2
+

2
∣∣M + 3

2 − γ
∣∣2

M + 1
.

Let

ε1 =

∣∣M + 3
2 − γ

∣∣2
M2

≤
∣∣∣∣52 − γ

∣∣∣∣2 ,
and

ε2 =

∣∣M + 3
2 − γ

∣∣
M + 1

≤ 1 +
|1/2− γ|
|M + 1|

≤ 1 +

∣∣∣∣12 − γ
∣∣∣∣ .

Consequently,

Cm,λ
|µM+1|n

≤ 2
ε1

|µM+1|n
+ 2

ε2
|µM+1|n

·
∣∣∣∣M +

3

2
− γ
∣∣∣∣

≤ 2ε1
|µM+1|n

+
2ε2 · |M + 3/2− γ|

|µM+1|n
,



C. Corcino / Eur. J. Pure Appl. Math, 14 (3) (2021), 666-684 673

where

|µM+1| =
∣∣∣∣M +

3

2
− γ
∣∣∣∣ =

√(
M +

3

2
−Re γ

)2

+ (Im γ)2 ≥ |M | − 2.

CM,λ ≤
ε1

2n−1πn |M + 3/2− γ|n
+

ε2
2n−1πn |M + 3/2− γ|n−1

≤ ε1
2n−1πn (|M | − 2)n

+
ε2

2n−1πn (|M | − 2)n

≤ |5/2− γ|2

2n−1πn (|M | − 2)n
+

1 + |1/2− γ|
2n−1πn (|M | − 2)n

≤ |5/2− γ|
2

2n−1πn
+

1 + |1/2− γ|
2n−1πn

.

We can see that CM,λ → 0 as n→∞ for |M | > 2. Thus, the tail of the series,

∞∑
k=M+1

1

|µk|n
→ 0 as n→∞.

Moreover, for fixed M > 2 and n � 0, CM,λ is bounded and independent of M . Hence,
we can replace CM,λ by Cλ. This completes the proof of the theorem.

When λ = 1, log λ = 0 and uk = (2k+ 1)πi, k ∈ Z. Take H = {πi,−πi}. Then ν = 3π
and the ordinary Genocchi numbers Gn = Gn(0; 1) satisfy

Gn
2(n!)

=
Gn(0; 1)

2(n!)
=

1

(πi)n
+

1

(−πi)n
+O((3π)−n). (2.7)

An approximation of Gn(0; 1) is given by

Gn
2(n!)

≈ 1

(πi)n
+

1

(−πi)n
. (2.8)

For odd n, n ≥ 3, it is known that Gn = 0 which is also true when we use (2.8). For even
indices,

G2n ≈
(−1)n4((2n)!)

π2n
, n ≥ 2 (2.9)

Taking n = 4,

G8 ≈
4(8!)

π8
≈ 16.99.

This value is very close to the exact value of G8 which is 17.

It is proved in the next theorem that an asymptotic approximation of the Apostol-
Genocchi polynomials can be obtained from its Fourier series (2.1) by choosing an appro-
priate subset of Tλ.



C. Corcino / Eur. J. Pure Appl. Math, 14 (3) (2021), 666-684 674

Theorem 2.4. Given λ ∈ C\{0}, let H be a finite subset of Tλ satisfying

max{|u| : u ∈ H} < min{|u| : u ∈ Tλ \H} := ν.

For all integers n ≥ 2, we have, uniformly for x in a compact subset K of C,

Gn(x;λ)

n!
= 2

∑
u ∈ H

eux

un
+O

(
eν|x|

νn

)
,

where the constant implicit in the order term depends on λ, H and K. Moreover, for
n� 0, this constant can be made independent of K, equal to the constant for the Apostol-
Genocchi numbers, corresponding to the case x = 0.

Proof. From the generating function (1.1) we have

2ze(x+y)z

λez + 1
=
∞∑
n=0

Gn(x+ y;λ)
zn

n!
.

The LHS can be written

2zexz

λez + 1
· eyz =

( ∞∑
n=0

Gn(x;λ)
zn

n!

)( ∞∑
n=0

(yz)n

n!

)

=
∞∑
n=0

n∑
k=0

Gn−k(x;λ)
zn−k

(n− k)!

(yz)k

k!

=
∞∑
n=0

(
n∑
k=0

(
n

k

)
Gn−k(x;λ)yk

)
zn

n!
,

from which

Gn(x+ y;λ) =
n∑
k=0

(
n

k

)
Gn−k(x;λ)yk.

For z ∈ C, writing z = 0 + z (here y = z, x = 0),

Gn(z;λ) =
n∑
k=0

(
n

k

)
Gn−k(0, λ)zk,

Gn(z;λ)

n!
=

n∑
k=0

Gn−k(0;λ)

(n− k)!

zk

k!

= 2
n∑
k=0

( ∑
u ∈ H

1

un−k
+O(ν−(n−k))

)
zk

k!
(by Theorem 2.3)

= 2
n∑
k=0

(∑
u∈ H

1

un−k
zk

k!

)
+

n∑
k=0

O(ν−(n−k))
zk

k!
,



C. Corcino / Eur. J. Pure Appl. Math, 14 (3) (2021), 666-684 675

where the implicit constant c in the order term is that corresponding to z = 0 and only
depends on H and λ. Note also that∣∣∣∣∣

n∑
k=0

O(ν−n+k)
zk

k!

∣∣∣∣∣ ≤
n∑
k=0

cν−n+k
|zk|
k!

= cν−n
n∑
k=0

νk
|zk|
k!

≤ cv−nen(ν|z|),

where en =
∑n

k=0
wk

k! .

To prove the theorem, it remains to show that

e∗n(uz)

un
=
euz − en(uz)

un

is bounded.

Using MVT for Banach spaces (see also [15])

e∗n(w) =
wn+1

(n+ 1)!
+

wn+2

(n+ 2)!
+ · · ·

=
wn+1

(n+ 1)!

{
1 +

w

n+ 2
+

w2

(n+ 3)(n+ 2)
+ · · ·

}
,

from which

|e∗n(w)| ≤
∣∣∣∣ wn+1

(n+ 1)!

∣∣∣∣ ∣∣∣∣1 +
w

n+ 2
+

w2

(n+ 3)(n+ 2)
+ · · ·

∣∣∣∣
≤ |w|

n+1

(n+ 1)!
eRe+(w),

where Re+(w) = max{Re(w), 0}.
Since |u| ≤ ν, for all u ∈ H, we have

|e∗n(uz)|
|un|

≤ e|uz||uz|n+1

|un|(n+ 1)!

= |u|e|uz| |z
n+1|

(n+ 1)!

< νeν|z|
|z|n+1

(n+ 1)!
,

so that ∣∣∣∣∣∑
u∈H

e∗n(uz)

un

∣∣∣∣∣ ≤ ∑
u ∈ H

|e∗n(uz)|
|un|



C. Corcino / Eur. J. Pure Appl. Math, 14 (3) (2021), 666-684 676

< #Hνeν|z|
|z|n+1

(n+ 1)!
,

where #H = no. of elements in H.

We give the argument that

#Hνeν|z|
|z|n+1

(n+ 1)!
< ceν|z|ν−n

if

#H
(ν|z|)n+1

(n+ 1)!
< c,

which certainly holds for n� 0, uniformly for z in a compact subset K ⊂ C.

Corollary 2.5. Let K be an arbitrary compact subset of C. The Genocchi polynomials
satisfy uniformly on K the estimates

G2n(x)

(2n)!
=

(−1)n 4 cosπx

π2n
+O

(
e3π|x|

(3π)n

)
, n ≥ 2,

G2n+1(x)

(2n+ 1)!
=

(−1)n 4 sinπx

π2n+1
+O

(
e3π|x|

(3π)n

)
, n ≥ 3,

where the implicit constant in the order term depends on the set K. Moreover, for n �
0, this constant can be made independent of K, equal to the constant for the Genocchi
numbers, corresponding to the case x = 0.

Proof. The Genocchi polynomials correspond to the case λ = 1 so that uk = (2k+1)πi,
for k ∈ Z. Thus, T1 = {(2k + 1)πi : k ∈ Z}. Taking H = {(2k + 1)πi | k = −1, 0} =
{−πi, πi}, then ν = |3πi| = 3π. From Theorem 2.4,

Gn(x; 1)

n!
= 2

∑
u∈H

eux

un
+O

(
eν|x|

νn

)

= 2

(
e−πix

(−πi)n
+

eπix

(πi)n

)
+O

(
e3π|x|

(3π)n

)
.

For even indices,

G2n(x)

(2n)!
=
G2n(x; 1)

(2n)!

= 2

(
e−πix

(πi)2n
+

eπix

(πi)2n

)
+O

(
e3π|x|

(3π)2n

)

=
4 cosπx

(πi)2n
+O

(
e3π|x|

(3π)2n

)



C. Corcino / Eur. J. Pure Appl. Math, 14 (3) (2021), 666-684 677

=
(−1)n 4 cosπx

π2n
+O

(
e3π|x|

(3π)n

)
.

For odd indices,

G2n+1(x)

(2n+ 1)!
=
G2n+1(x; 1)

(2n+ 1)!

= 2

(
e−πix

(−πi)2n+1
+

eπix

(πi)2n+1

)
+O

(
e3π|x|

(3π)2n+1

)

= 2

(
(−1)n 2 sinπx

(π)2n+1

)
+O

(
e3π|x|

(3π)2n+1

)

=
(−1)n(4 sinπx)

π2n+1
+O

(
e3π|x|

(3π)n

)
.

Notice the resemblance of the results in Corollary 2.5 and of (33) in [3].
Since, for k = 2n,

cos (πx− kπ

2
) = ± cosπx = (−1)n cosπx,

(33) in [3] can be written as

G2n(x) =
4((2n)!)

π2n
[
(−1)n cosπx+O(3−n)

]
G2n(x)

(2n)!
=

(−1)n4 cosπx

π2n
+O

(
3−n

π2n

)
=

(−1)n4 cosπx

π2n
+O

(
1

(3π)n

)
=

(−1)n4 cosπx

π2n
+O

(
e3π|x|

(3π)n

)
, for x ∈ K.

For odd k (k = 2n+ 1),

cosπx− kπ

2
= (−1)n sinπx.

Then (33) in [3] can be written as

G2n+1(x) =
4((2n+ 1)!)

π2n+1

[
(−1)n sinπx+O

(
3−(2n+1)

)]
G2n+1(x)

(2n+ 1)!
=

(−1)n 4 sinπx

π2n+1
+O

(
3−(2n+1)

π2n+1

)

=
(−1)n 4 sinπx

π2n+1
+O

(
1

(3π)2n+1

)



C. Corcino / Eur. J. Pure Appl. Math, 14 (3) (2021), 666-684 678

=
(−1)n 4 sinπx

π2n+1
+O

(
e3π|x|

(3π)2n+1

)

=
(−1)n 4 sinπx

π2n+1
+O

(
e3π|x|

(3π)n

)
.

Thus, the asymptotic formulas in Corollary 2.5 are equivalent to (33) in [3].

3. λ is a negative real number

When λ is a negative real number, writing λ = −|λ|, the generating function is given
by

2text

−|λ|et + 1
=

∞∑
n=0

Gn(x;λ)
tn

n!
. (3.1)

The poles of the generating function (3.1) is

T−|λ| = {2kπi− log |λ| : k ∈ Z}.

The next theorem follows from Theorem 2.4.

Theorem 3.1. Given that λ is a negative real number, let F be a finite subset of T−|λ|
satisfying

max {|a| : a ∈ F} < min {|a| : a ∈ T−|λ|\F} := µ.

For all integers n ≥ 2, we have, uniformly for x in a compact subset K of C,

Gn(x;λ)

n!
= 2

∑
a∈F

eax

an
+O

(
eµ|x|

µn

)
, (3.2)

where the constant implicit in the order term depends on λ, F and K.

The Apostol-Genocchi numbers Gn(0;−1) corresponding to the case λ = −1 has
generating function

2t

−et + 1
=
∞∑
n=0

Gn(0;−1)
tn

n!
, (3.3)

The set of poles is T−1 = {2kπi : k ∈ Z\{0}}. An asymptotic formula for Gn(0;−1) is
given in the following theorem.

Theorem 3.2. For n ≥ 3, the Apostol-Genocchi numbers Gn(0;−1) satisfy

Gn(0;−1)

n!
= 2

(
1

(−2πi)n
+

1

(2πi)n

)
+O

(
(4π)−n

)
. (3.4)

In particular,
G2n(0;−1)

(2n)!
=

(−1)n4

(2π)2n
+O

(
(4π)−2n

)
, n ≥ 2. (3.5)



C. Corcino / Eur. J. Pure Appl. Math, 14 (3) (2021), 666-684 679

Proof. Taking x = 0, F = {−2πi, 2πi} in Theorem 3.1, then µ = 4π. Hence,

−1
2Gn(0;−1)

n!
= −

(
1

(−2πi)n
+

1

(2πi)n

)
+O

(
(4π)−n

)
, (3.6)

from which (3.4) follows.

For (n ≥ 3), (3.6) gives G2n+1(0;−1) ≈ 0. Indeed G2n+1(0;−1) = 0, ∀n ≥ 1.

For n ≥ 2,
G2n(0;−1)

(2n)!
= 4

(
(−1)n

(2π)2n

)
+O

(
(4π)−2n

)
. (3.7)

From (3.7) we have the approximation

G2n(0;−1) ≈ (−1)n 4(2n)!

(2π)2n
. (3.8)

Taking n = 4,

G8(0;−1) =
4(8!)

(2π)8
≈ .06638.

The actual value of G8(0;−1) = −2B8 = 1
15 ≈ .06667.

The Apostol-Genocchi polynomials, Gn(x;−1) correspond to the case λ = −1. These
polynomials have generating function

2text

−et + 1
=
∞∑
n=0

Gn(x;−1)
tn

n!
. (3.9)

We will prove the following theorem.

Theorem 3.3. Let K be a compact subset of C. The Apostol-Genocchi polynomials
Gn(x;−1) satisfy uniformly on K the estimates

G2n(x;−1)

(2n)!
=

(−1)n4 cos 2πx

(2π)2n
+O

(
e4π|x|

(4π)n

)
, (3.10)

G2n+1(x;−1)

(2n+ 1)!
=

(−1)n4 sin 2πx

(2π)2n+1
+O

(
e4π|x|

(4π)n

)
, (3.11)

where the implicit constant in the order term depends on the set K. Moreover, for n� 0,
this constant can be made independent of K, equal to the constant for the Apostol-Genocchi
numbers Gn(0;−1) corresponding to the case x = 0.



C. Corcino / Eur. J. Pure Appl. Math, 14 (3) (2021), 666-684 680

Proof. Taking F = {−2πi, 2πi}, then µ = 4π. Hence, it follows from Theorem 3.1 that

−1
2 Gn(x;−1)

n!
= − e2πix

(2πi)n
− e−2πix

(−2πi)n
+O

(
e4π|x|

(4π)n

)
. (3.12)

For odd indices,

−1
2 G2n+1(x;−1)

(2n+ 1)!
= −

(
e2πix

(2πi)2n+1
+

e−2πix

(−2πi)2n+1

)
+O

(
e4π|x|

(4π)2n+1

)
(3.13)

G2n+1(x;−1)

(2n+ 1)!
=

(−1)n4 sin 2πx

(2π)2n+1
+O

(
e4π|x|

(4π)n

)
. (3.14)

For even indices,

G2n(x;−1)

(2n)!
= 2

(
e2πix

(2πi)2n
+

e−2πix

(−2πi)2n

)
+O

(
e4π|x|

(4π)2n

)
(3.15)

=
(−1)n4 cos 2πx

(2π)2n
+O

(
e4π|x|

(4π)n

)
. (3.16)

4. Apostol-Euler Numbers and Polynomials

The Apostol-Euler numbers are defined by the generating function

2

λet + 1
=

∞∑
n=0

En(0;λ)
tn

n!
. (4.1)

Multiplying both sides of (4.1) by t gives

∞∑
n=0

Gn(0;λ)
tn

n!
=

∞∑
n=0

(n+ 1)En(0;λ)
tn+1

(n+ 1)!
,

from which we have, for n ≥ 1

En−1(0;λ) =
Gn(0;λ)

n
= (n− 1)!

Gn(0;λ)

n!
. (4.2)

Thus, from Theorem 2.3,

En−1(0;λ) = 2(n− 1)!

[ ∞∑
n=0

1

un
+O

(
ν−n

)]
, (4.3)

where F ⊆ Tλ = {(2k + 1)πi− log λ | k ∈ Z} and F satisfies



C. Corcino / Eur. J. Pure Appl. Math, 14 (3) (2021), 666-684 681

max{|u| : u ∈ F} < min{|u| : u ∈ Tλ\F} = ν.

For odd n, say n = 2k + 1, from (4.2), we have

E2k(0;λ) =
G2k+1(0;λ)

2k + 1
, (4.4)

while for even n, say n = 2k,

E2k−1(0;λ) =
G2k(0;λ)

2k
. (4.5)

The case λ = 1, corresponds to the Euler numbers En. From (4.2),

En−1 =
Gn
n
. (4.6)

Since Gn = 0 for all odd n ≥ 3, E2k = 0 for k ≥ 1.

For odd indices, using (2.9) we have

E2n−1 = (2n− 1)!
G2n

(2n)!
= (2n− 1)!

(
(−1)n(4)

π2n
+O

(
(3π)−n

))
, n ≥ 2. (4.7)

Taking n = 2,

E3 ≈ 3!

(
4

π4

)
=

24

π4
= 0.24638.

The Actual value of E3 = 0.25.

The Apostol-Euler Polynomials En(x;λ) are defined by the generating function

2ext

λet + 1
=
∞∑
n=0

En(x;λ)
tn

n!
, (4.8)

which can be written
∞∑
n=0

Gn(x;λ)tn

n!
=
∞∑
n=0

(n+ 1)En(x;λ)
tn+1

(n+ 1)!
. (4.9)

Thus,

En−1(x;λ) =
Gn(x;λ)

n
. (4.10)

From Theorem 2.4,

En−1(x;λ) =
Gn(x;λ)

n
· (n− 1)!

(n− 1)!

= (n− 1)!
Gn(x;λ)

n!

= (n− 1)!

(
2
∑
u∈F

euz

un
+O

(
eν|x|

νn

))
.

Hence, we have the following corollary.



C. Corcino / Eur. J. Pure Appl. Math, 14 (3) (2021), 666-684 682

Corollary 4.1. Given λ ∈ C\{0}, let F be a finite subset of Tλ satisfying

max{|u| : u ∈ F} < min{|u| : u ∈ Tλ\F} = ν.

Let K be an arbitrary compact subset of C. The Apostol-Euler polynomials satisfy uni-
formly on K the estimates,

En−1(x;λ)

(n− 1)!
= 2

∑
u∈F

eux

un
+O

(
eν|x|

νn

)
,

where the constant implicit in the order term depends on λ, F and K. Moreover, for n� 0,
this constant can be made independent of K, equal to the constant for the Apostol-Euler
numbers, corresponding to the case x = 0.

It follows from Corollary 2.5 that the Euler polynomials which correspond to λ = 1,
satisfy, uniformly on a compact subset K of C the estimates

E2n−1(x)

(2n− 1)!
=
G2n(x)

(2n)!
=

(−1)n4 cosπx

π2n
+O

(
e3π|x|

(3π)n

)
, (4.11)

E2n(x)

(2n)!
=
G2n+1(x)

(2n+ 1)!
=

(−1)n4 sinπx

π2n+1
+O

(
e3π|x|

(3π)n

)
, (4.12)

as n→∞, for n ≥ 1.

The Apostol-Euler polynomials En−1(x;−1) correspond to the special case λ = −1.
From (4.10),

En−1(x;−1) =
Gn(x;−1)

n
. (4.13)

It follows from (3.10) and (3.11), respectively that

E2n(x;−1)

(2n)!
=

(−1)n4 sin 2πx

(2π)2n+1
+O

(
e4π|x|

(4π)n

)
, (4.14)

E2n−1(x;−1)

(2n− 1)!
=

(−1)n4 cos 2πx

(2π)2n
+O

(
e4π|x|

(4π)n

)
, (4.15)

on a compact subset K of C.

5. Conclusion

Asymptotic approximations of the Apostol-Genocchi numbers and polynomials were
obtained for values of the parameter λ in C\{0}. Unlike in [15] we have considered
explicitly the case when λ is negative and obtained corresponding asymptotic formulas.



REFERENCES 683

Moreover, the asymptotic formulas for λ = 1 are explicitly obtained for each of the Apostol-
Genocchi and Apostol-Euler numbers and polynomials. The tangent polynomials [8] have
generating function very similar to that of the Apostol-Genocchi polynomials. The author
recommends finding Fourier expansion and asymptotic approximations of these polynomi-
als.

Acknowledgements

This research project is funded by Cebu Normal University through its Center for
Research and Development.

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