EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
Vol. 16, No. 2, 2023, 791-805
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

Asymptotic Approximations for generalized
Apostol-Bernoulli, Apostol-Euler and Apostol-Genocchi

Polynomials in terms of Hyperbolic Functions

Cristina B. Corcino1,2,∗, Roberto 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 approximation formulas for polynomials of the type Apostol-Bernoulli,
Apostol-Euler and Apostol-Genocchi with integer order and real parameters are obtained via hy-
perbolic functions. The derivation of the formulas is done using the principle of saddle point and
expansion of appropriate hyperbolic function about a saddle point.

2020 Mathematics Subject Classifications: 11B68, 42A16, 11M35

Key Words and Phrases: Approximations, Bernoulli polynomials, Euler polynomials, Genocchi
polynomials

1. Introduction

Let α ∈ Z+, λ ∈ C\{0}, a, b, c ∈ R+, b ̸= 1, c ̸= 1, a ̸= b and x ∈ R. The generalized
Apostol-Bernoulli, Euler and Genocchi polynomials with parameters α, λ, a, b, c, are given
by means of the following generating functions (see [1]).(

t

λbt − at

)α

cxt =

∞∑
n=0

B(α)
n (x;λ; a, b, c)

tn

n!
,

∣∣∣∣t ln b

a

∣∣∣∣ < 2π, (1.1)

(
2

λbt + at

)α

cxt =
∞∑
n=0

E(α)
n (x;λ; a, b, c)

tn

n!
,

∣∣∣∣t ln b

a

∣∣∣∣ < π, (1.2)

and (
2t

λbt + at

)α

cxt =
∞∑
n=0

G(α)
n (x;λ; a, b, c)

tn

n!
,

∣∣∣∣t ln b

a

∣∣∣∣ < π. (1.3)

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v16i2.4703

Email addresses: corcinoc@cnu.edu.ph (C. Corcino), corcinor@cnu.edu.ph (R. Corcino)

https://www.ejpam.com 791 © 2023 EJPAM All rights reserved.



C. Corcino, R. Corcino / Eur. J. Pure Appl. Math, 16 (2) (2023), 791-805 792

The polynomials defined above will also be referred to as Apostol-Bernoulli type, Apostol-
Euler type and Apostol-Genocchi type polynomials in the discussion below. When α =
1, λ = 1, b = c = e and a = 1, these polynomials will reduce to the classical Bernoulli,
Euler and Genocchi polynomials.

Asymptotic approximations for higher order Genocchi polynomials using residues were
done in [2] and [3]. Approximations for the Bernoulli and Euler polynomials using hyper-
bolic functions were obtained in [4] and approximations for Genocchi polynomials in terms
of hyperbolic functions were obtained in [5]. At the time of the search, there were no ap-
proximations for the generalized Apostol-Bernoulli, Apostol-Euler and Apostol-Genocchi
polynomials found in the literature.

In this paper asymptotic approximations for these polynomials will be derived using
the method of [4]. In particular, the following results in [4] will be utilized.

Lemma 1.1. For z ∈ C\{0}, the function Φk(n, z) defined by

Φk(n, z) =
n!

(nz)n
1

2πi

∫
C
(w − z−1)kenzw

dw

wn+1
, (1.4)

where C is a circle with center at the origin and radius ϵ1, can be represented in the form

Φk(n, z) =
pk(n)

(nz)k
(1.5)

with

p0(n) = 1, p1(n) = 0, p2(n) = −n, p3(n) = 2n, (1.6)

and the remaining polynomials are given by the recurrence

pk(n) = (1− k)pk−1(n) + npk−2(n). (1.7)

Lemma 1.2. For fixed z ̸= 0, the sequence Φk(n, z) is an asymptotic sequence for n →
+∞ that satisfies Φk(n, z) = O(n[ k

2
]−k).

Theorem 1.3. Let f(w) be a meromorphic function with simple poles w1, w2, ... and an-
alytic at the origin. Let the contour C be a circle whose center is at the origin and which
contains no poles of f(w) inside. The polynomials Pn(nz) defined by

Pn(nz) =
n!

2πi

∫
C
f(w)enwz dw

wn+1
(1.8)

may be expanded as the infinite sum

Pn(nz) = (nz)n
∞∑
k=0

f (k)(z−1)

k!

pk(n)

(nz)k
, (1.9)

valid for z ∈ C\{0} such that |z−1| < |z−1 − wk| for all k = 1, 2, ... where pk(n) are the
polynomials given in Lemma 1.1.



C. Corcino, R. Corcino / Eur. J. Pure Appl. Math, 16 (2) (2023), 791-805 793

2. Proof of Theorem 1.3

The following proof of Theorem 1.3 is an expository of the proof presented in [4]. This
is being provided to aid the derivation of the asymptotic formulas in Section 3.

Proof. Write

Pn(nz) =
n!

2πi

∫
C
f(w)enwz−n logw dw

w
. (2.1)

The key observation used for obtaining approximations of Pn(nz) for large n and fixed z is
that the main contribution of the integrand to the integral originates at the saddle point
of the argument of the exponential (for a discussion of saddle point method see [6]), that
is, at w = z−1. If z−1 is not a pole of f(w), then f(w) can be expanded around z−1 as
follows:

f(w) =

∞∑
k=0

f (k)(z−1)

k!
(w − z−1)k,

∣∣w − z−1
∣∣ < r, (2.2)

where r is the distance from the z−1 to the nearest singularity of f(w). The radius ϵ1 of
the contour C in the definition of Pn(z) can be chosen as close to 0 as necessary. Then for
w ∈ C(C : |w| = ϵ1), the above series is absolutely convergent if

∣∣z−1
∣∣ < ∣∣z−1 − wk

∣∣ for all
k = 1, 2, · · · .

Substituting the expansion to f(w) yields

Pn(nz) =
n!

2πi

∫
C

∞∑
k=0

f (k)(z−1)

k!
(w − z−1)kenwz dw

wn+1
, (2.3)

where

f (k)(z−1) =
k!

2π

∫
C′

f(t)dt

(t− z−1)k+1
, (2.4)

and C ′ is a circle around z−1 whose radius R ≡
∣∣t− z−1

∣∣ < ∣∣z−1 − wk

∣∣ for all k. That is,
R ≡ min

∣∣z−1 − wk

∣∣− ϵ2 for some ϵ2 > 0.

Since f(t) is bounded on C ′ there exists M1 such that |f(t)| < M1 for t ∈ C ′. Therefore,∣∣∣f (k)(z−1)
∣∣∣ ≤ k!

2π

∫
C′

|f(t)|
|t− z−1|k+1

|dt|

≤ k!

2π

M1

Rk+1
2πR = M1

k!

Rk
. (2.5)

Let a = max
w∈C

|w−z−1|
R . Note that a depends only on z, ϵ, and R and we can make a < 1.

Then

|Pn(nz)| ≤ M1

∞∑
k=0

n!

2π

∫
C

(∣∣w − z−1
∣∣

R

)k

|enwz| |dw|
|wn+1|

(2.6)



C. Corcino, R. Corcino / Eur. J. Pure Appl. Math, 16 (2) (2023), 791-805 794

The function enwz is bounded on C for finite n and fixed z. Thus, there exists M2 such
that |enwz| ≤ M2, for w ∈ C. Hence,

|Pn(nz)| ≤ M1M2

∞∑
k=0

n!

2π

∫
C
ak

|dw|
|wn+1|

= M1M2

∞∑
k=0

ak
n!

2π

1

ϵn+1
1

2π ϵ1

= M1M2
n!

ϵn1

∞∑
k=0

ak < ∞ . (2.7)

Evaluating the integral in (2.3),

Pn(nz) =
∞∑
k=0

f (k)(z−1)

k!

n!

2πi

∫
C
(w − z−1)kenwz dw

wn+1

= (nz)n
∞∑
k=0

f (k)(z−1)

k!

n!

(nz)n
1

2πi

∫
C
(w − z−1)kenwz dw

wn+1

= (nz)n
∞∑
k=0

f (k)(z−1)

k!
Φk(n, z), (2.8)

where

Φk(n, z) =
n!

(nz)n
1

2πi

∫
C
(w − z−1)kenwz dw

wn+1
. (2.9)

From Lemma 1.1 and Lemma 1.2 , the functions Φk(n, z) are polynomials in n divided by
powers of nz and constitute an asymptotic sequence for n → +∞. The desired asymptotic
sequence is

Pn(nz) = (nz)n
∞∑
k=0

f (k)(z−1)

k!

pk(n)

(nz)k
, (2.10)

where pk(n) are defined in Lemma 1.1.

Remark 2.1. As can be seen in the proof of Theorem 1.3, the results of the theorem still
hold for f(t) having poles w1, w2, . . . of order greater than 1.

3. The Asymptotic Approximations

The following are the main results of the study. In the discussion below, δ = log λ, λ ∈
C\{0} where the logarithm is taken to be the principal branch and ρ = (δ+µ ln(ba−1))/2.

Theorem 3.1. (Apostol-Bernoulli type polynomials of order 1)
Let a, b, c ∈ R+\{1}, a ̸= b and µ = (x ln c)−1. For x ∈ C\{0}, such that |µ| < |µ± δ

ln(ba−1)
|,



C. Corcino, R. Corcino / Eur. J. Pure Appl. Math, 16 (2) (2023), 791-805 795

the following formula holds,

Bn(nx;λ; a, b, c) =
(nx ln c)n

2
√
λ

µ(ab)
−µ
2

sinh ρ

{
1− A

2n(x ln c)2
+O(n−2)

}
, (3.1)

where,

A =

(
ln(ab)

2
− 1

µ
+

ln(ba−1)

2
coth ρ

)(
ln(ab)

2
+

ln(ba−1)

2
coth ρ

)
− ln(ab)

2µ

+
ln(ba−1)

2

(
csch2ρ

ln(ba−1)

2
− coth ρ

µ

)
. (3.2)

Proof. Taking α = 1, (1.1) reduces to(
t

λbt − at

)
cxt =

∞∑
n=0

Bn(x;λ; a, b, c)
tn

n!
.

Applying the Cauchy Integral Formula (for a discussion about Cauchy Integral For-
mula, see [7], [8]),

Bn(x;λ; a, b, c)

n!
=

1

2πi

∫
C

tcxt

λbt − at
dt

tn+1
, (3.3)

where C is a circle with center at the origin and radius <
∣∣∣ δ
ln(ba−1)

∣∣∣. Note that −δ/ ln(ba−1)

is the simple pole of the integrand of (3.3) different from zero and nearest to the origin as
can be seen in the computation of the singularities below.

Rewriting

λbt − at = eδet ln b − et ln a

=
(
eδ+t ln b − et ln a

) e−t ln a

e−t ln a

=
(
eδ+t(ln ba−1) − 1

)
et ln a

=

[
2e

δ+t ln(ba−1)
2 sinh

(
δ + t ln(ba−1)

2

)]
et ln a.

Then (3.3) becomes

Bn(x;λ; a, b, c)

n!
=

1
2λ

−1/2

2πi

∫
C

t(ab)
−t
2

sinh
(
δ+t ln(ba−1)

2

)cxt dt

tn+1
,



C. Corcino, R. Corcino / Eur. J. Pure Appl. Math, 16 (2) (2023), 791-805 796

from which,

Bn(nx;λ; a, b, c)

n!
=

1
2λ

− 1
2

2πi

∫
C
g(t)etnx ln c−n log tdt

t
, (3.4)

where

g(t) =
t(ab)

−t
2

sinh
(
δ+t ln(ba−1)

2

) . (3.5)

The saddle-point at which the major contribution to the integral in (3.4) occurs is µ =
(x ln c)−1. The singularities of g(t) are computed as follows:

sinh

(
δ + t ln(ba−1)

2

)
⇔ δ + t ln(ba−1)

2
= kπi, k ∈ Z

δ + t ln(ba−1) = 2kπi

t ln(ba−1) = 2kπi− δ

tk := t =
2kπi− δ

ln(ba−1)
, k ∈ Z.

Assume that µ = (x ln c)−1 is not a singularity of g. Then g(t) can be expanded about µ.
That is,

g(t) =
∞∑
k=0

g(k)(µ)

k!
(t− µ)k, |t− µ| < r

where r is the distance from µ to the nearest singularity of g(t). The derivatives of g(t)
for k = 1, 2 evaluated at t = µ are given below:

g′(µ) =

(
1 +

−µ ln(ab)

2
− µ

ln(ba−1)

2
coth ρ

)
e

−µ
2

ln(ab)

sinh ρ
, (3.6)

g′′(µ) =

{(
ln(ab)

2
− 1

µ
+

ln(ba−1)

2
coth ρ

)(
ln(ab)

2
+

ln(ba−1)

2
coth ρ

)
− ln(ab)

2µ
+

[ln(ba−1)]2

4
csch2ρ− ln(ba−1)

2µ
coth ρ

}
× µ e

−µ
2

ln(ab)

sinh ρ
. (3.7)

Using (3.4) and applying Theorem 1.3,

Bn(nx;λ; a, b, c) =
(nx ln c)n

2
√
λ

{
g(µ)− g′′(µ)

2n(x ln c)2
+O(n−2)

}
=

(nx ln c)n

2
√
λ

{
µ(ab)

−µ
2

sinh ρ
− µ(ab)

−µ
2

sinh ρ

A

2n(x ln c)2
+O(n−2)

}



C. Corcino, R. Corcino / Eur. J. Pure Appl. Math, 16 (2) (2023), 791-805 797

=
(nx ln c)n

2
√
λ

µ(ab)
−µ
2

sinh ρ

{
1− A

2n(x ln c)2
+O(n−2)

}
,

where A is as given in (3.2).

Asymptotic formula for the Apostol-Bernoulli type polynomials of order α > 1 is given
in the next theorem.

Theorem 3.2. (Apostol-Bernoulli type polynomials of order α ≥ 2)
Let a, b, c ∈ R+\{1}, a ̸= b and µ = (x ln c)−1. For x ∈ C\{0} such that |µ| <

∣∣µ± δ
ln(ba−1)

∣∣,
δ = log λ, n ≥ α, the following holds,

B(α)
n (nx;λ; a, b, c) =

(nx ln c)n

2αλ
α
2

(
µ(ab)−

µ
2

sinh ρ

)α{
1− α(A+ (α− 1)J2)

2n(x ln c)2
+O(n−2)

}
, (3.8)

where A is given in (3.2) and J is given by

J = − ln(ab)

2
+

1

µ
− coth ρ

ln(ba−1)

2
. (3.9)

Proof. Applying the Cauchy Integral Formula to (1.1) yields

B
(α)
n (x;λ; a, b, c)

n!
=

1

2πi

∫
C

(
t

λbt − at

)α

cxt
dt

tn+1
,

where C is a circle around the origin with radius <
∣∣ δ
ln(ba−1)

∣∣.
Writing (

t

λbt − at

)α

=
tαa−αt

(eδ(ba−1)t − 1)α
=

tαa−αt

[exp (t ln(ba−1) + δ)− 1]α

=
tαa−αt[

2 exp
(
t ln(ba−1)+δ

2

)
sinh

(
t ln(ba−1)+δ

2

)]α
=

tα(ab)
−α
2

t

2αλ
α
2 sinhα

(
t ln(ba−1)+δ

2

)
Thus,

B
(α)
n (nx;λ; a, b, c)

n!
=

(
2−αλ

−α
2

) 1

2πi

∫
C

tα(ab)−
α
2
tcnxt

sinhα
(
t ln(ba−1)+δ

2

) dt

tn+1

= 2−αλ
−α
2

1

2πi

∫
C
gα(t)c

nxt dt

tn+1
,



C. Corcino, R. Corcino / Eur. J. Pure Appl. Math, 16 (2) (2023), 791-805 798

where,

gα(t) = [g(t)]α =

 t(ab)
−t
2

sinh
(
t ln(ba−1)+δ

2

)
α

. (3.10)

The saddle-point is still µ = (x ln c)−1. The function gα(t) has poles of order α at tk =
2kπi−δ
ln(ba−1)

, k ∈ Z. Assuming that µ = (x ln c)−1 is not a singularity of gα(t). Then gα(t) can

be expanded about µ. That is,

gα(t) =
∞∑
k=0

g
(k)
α (µ)

k!
(t− µ)k, |t− µ| < r

where r is the distance from µ to the nearest singularity of gα(t).

The derivatives for k = 1, 2 are

g′α(t) = α[g(t)]α−1g′(t), (3.11)

g′′α(t) = α
{
g(t)α−1g′′(t) + (α− 1)g(t)α−2[g′(t)]2

}
, (3.12)

where g(t) is defined in (3.5). Evaluating at t = µ,

g′α(µ) = α

(
µ(ab)−

µ
2

sinh ρ

)α (
− ln(ab)

2
+

1

µ
− coth ρ

ln(ba−1)

2

)
, (3.13)

g′′α(µ) = α


(
µ(ab)−

µ
2

sinh ρ

)α−1
µe−

µ ln(ab)
2

sinh ρ
A+ (α− 1)

(
µ(ab)−

µ
2

sinh ρ

)α−2 (
µe−

µ ln(ab)
2

sinh ρ
J

)2


(3.14)

= α

(
µ(ab)−

µ
2

sinh ρ

)α{
A+ (α− 1)

(
− ln(ab)

2
+

1

µ
− coth ρ

ln(ba−1)

2

)2
}
. (3.15)

It follows from Theorem 1.3 that

B
(α)
n (nx;λ; a, b, c)

2−αλ
−α
2

= (nx ln c)n
∞∑
k=0

g
(k)
α (n)

k!

pk(n)

(nx ln c)k

= (nx ln c)n
{
gα(µ) +

g′′α(µ)

2(nx ln c)2
p2(n) +O(n−2)

}
.

Then,

B(α)
n (nx;λ; a, b, c) = 2−αλ−α

2 (nx ln c)n

{(
µ(ab)

−µ
2

sinh ρ

)α

+
g′′α(µ)

2n(x ln c)2
+O(n−2)

}



C. Corcino, R. Corcino / Eur. J. Pure Appl. Math, 16 (2) (2023), 791-805 799

= 2−αλ−α
2 (nx ln c)n

(
µ(ab)

−µ
2

sinh ρ

)α{
1− αA+ α(α− 1)J2

2n(x ln c)2
+O(n−2)

}
,

where A is given in(3.2) and

J = − ln(ab)

2
+

1

µ
− coth ρ

ln(ba−1)

2
.

Theorem 3.3. (Apostol-Euler type polynomials of order 1)
Let a, b, c ∈ R+\{1}, a ̸= b and µ = (x ln c)−1. For x ∈ C\{0} such that |µ| < |µ± πi−δ

ln(ba−1)
|,

the following holds,

En(nx;λ; a, b, c) =
(nx ln c)n(ab)

−µ
2 λ

−1
2

cosh ρ

{
1− F

2n(x ln c)2
+O(n−2)

}
(3.16)

where

F =

(
ln(ab)

2
+

ln(ba−1)

2
tanh ρ

)2

− ln2(ba−1)

4
sech2 ρ. (3.17)

Proof. Taking α = 1, (1.2) reduces to(
2

λbt + at

)
cxt =

∞∑
n=0

En(x;λ; a, b, c)
tn

n!
,

∣∣∣∣t ln b

a

∣∣∣∣ < π.

Applying the Cauchy Integral Formula,

En(x;λ; a, b, c)

n!
=

1

2πi

∫
C

2cxt

[λbt + at]

dt

tn+1
,

where C is a circle around the origin with radius <
∣∣∣ πi−δ
ln(ba−1)

∣∣∣.
Writing

1

λbt + at
=

a−t

eδ(ba−1)t + 1
,

then

En(x;λ; a, b, c)

n!
=

1

2πi

∫
C

2(a−1cx)t

[eδ(ba−1)t + 1]

dt

tn+1
.

With



C. Corcino, R. Corcino / Eur. J. Pure Appl. Math, 16 (2) (2023), 791-805 800

eδ(ba−1)t + 1 = eδet ln(ba
−1) + 1

= 2 exp

(
t ln(ba−1) + δ

2

)
cosh

(
t ln(ba−1) + δ

2

)
,

yields

En(x;λ; a, b, c)

n!
=

e−
δ
2

2πi

∫
C

((ab)−
1
2 cx)t

cosh( t ln(ba
−1)+δ
2 )

dt

tn+1

=
(λ)−

1
2

2πi

∫
C

(ab)−
1
2
tcxt

cosh( t ln(ba
−1)+δ
2 )

dt

tn+1

=
λ− 1

2

2πi

∫
C
f(t)cxt

dt

tn+1
(3.18)

where

f(t) =
(ab)−

1
2
t

cosh( t ln(ba
−1)+δ
2 )

. (3.19)

Taking x 7→ nx and writing cxt = etx ln c, (3.18) will take the form

En(nx;λ, a, b, c)

λ− 1
2

=
n!

2πi

∫
C
f(t)et(nx ln c) dt

tn+1
,

which is of the form (1.8) where z = x ln c.

The saddle-point occurs at

d

dt
(nxt ln c− n log t) = 0

x ln c− n

t
= 0

⇔ t = (x ln c)−1 = z−1 := µ

The function f(t) is defined at t = 0 with f(0) = 1
cosh δ

2

. Also f ′(0) is defined. The

singularities of f(t) are the zeros of cosh t ln(ba−1)+δ
2 , which are computed by solving for t

such that

cosh

(
t ln(ba−1) + δ

2

)
= 0.



C. Corcino, R. Corcino / Eur. J. Pure Appl. Math, 16 (2) (2023), 791-805 801

Let w = t ln(ba−1)+δ
2 . Then

coshw = 0 ⇔ w =

(
k +

1

2

)
πi, (k ∈ Z).

That is,

t ln(ba−1) + δ

2
= (k +

1

2
)πi

t ln(ba−1) = (2k + 1)πi− δ

tk := t =
(2k + 1)πi− δ

ln(ba−1)
, k ∈ Z.

The tk are simple poles of f(t).
Assume that µ = (x ln c)−1 is not a singularity of f(t). Then f(t) can be expanded

about t = µ as follows:

f(t) =
∞∑
k=0

f (k)(µ)

k!
(t− µ)k, |t− µ| < r

where r is the distance from µ to the nearest singularity of f(t).

The first few derivatives of f at t = µ are given below:

f ′(µ) =

(
− ln(ab)

2
− ln(ba−1)

2
tanh ρ

)
(ab)−

t
2

cosh ρ
, (3.20)

f ′′(µ) =

{[
ln(ab)

2
+

ln(ba−1)

2
tanh ρ

]2
− ln2(ba−1)

4
sech2 ρ

}
(ab)−

t
2

cosh ρ
. (3.21)

Applying Theorem 1.3, the result follows.

For the Apostol-Euler type polynomials of order α > 1 see the following theorem.

Theorem 3.4. (Apostol-Euler type polynomials of order α ≥ 2) Let a, b, c ∈ R+\{1}, α ∈
Z+, a ̸= b and µ = (x ln c)−1. For x ∈ C such that

∣∣µ± πi−δ
ln(ba−1)

∣∣ and n ≥ α, the following

formula holds,

E(α)
n (nx;λ; a, b, c) =

(nx ln c)n

λ
α
2

(
(ab)−

µ
2

cosh ρ

)α{
1− αF + α(α− 1)H2

2n(x ln c)2
+O(n−2)

}
, (3.22)

where

H =
− ln(ab)

2
− ln(ba−1)

2
tanh ρ,

and F is given in Theorem 3.3.



C. Corcino, R. Corcino / Eur. J. Pure Appl. Math, 16 (2) (2023), 791-805 802

Proof. Applying the Cauchy - Integral Formula to (1.2) yields

E
(α)
n (x;λ; a, b, c)

n!
=

2α

2πi

∫
C

cxt

(λbt + at)α
dt

tn+1
,

where C is a circle centered at zero with radius <
∣∣∣ πi−δ
ln(ba−1)

∣∣∣.
Write

1

λbt + at
=

a−t

eδ(ba−1)t + 1
,

1

(λbt + at)α
=

a−αt

[eδ(ba−1)t + 1]
α ,

eδ(ba−1)t + 1 = 2 exp

(
t ln(ba−1) + δ

2

)
cosh

(
t ln(ba−1) + δ

2

)
,[

eδ(ba−1)t + 1
]α

=

[
2 exp

(
t ln(ba−1) + δ

2

)]α [
cosh

(
t ln(ba−1) + δ

2

)]α
= 2αλ

α
2 et

α
2
ln(ba−1)

[
cosh

(
t ln(ba−1) + δ

2

)]α
.

Then,

E
(α)
n (x;λ; a, b, c)

n!
=

λ−α
2

2πi

∫
C

(ab)−
α
2
t[

cosh
(
t ln(ba−1)+δ

2

)]α cxt
dt

tn+1
,

Let

h(t) =
(ab)−

α
2
t[

cosh( t ln(ba
−1)+δ
2 )

]α = [f(t)]α .

Then
E

(α)
n (nx;λ; a, b, c)

n!
=

λ−α
2

2πi

∫
C
h(t) cnxt

dt

tn+1
, (3.23)

still with saddle point at µ = (x ln c)−1. The poles of h(t) are at t = 0 of order n+ 1 and

at tk = (2k+1)πi−δ
ln(ba−1)

, k ∈ Z each of order α. Assuming that µ is not a singularity of h(t) ,

h(t) can be expanded about µ given by

h(t) =
∞∑
k=0

h(k)(µ)

k!
(t− µ)k.

It follows from Theorem 1.3 that

E
(α)
n (nx;λ; a, b, c)

λ−α
2

= (nx ln c)n
∞∑
k=0

h(k)(µ)

k!

pk(n)

(nx ln c)k



C. Corcino, R. Corcino / Eur. J. Pure Appl. Math, 16 (2) (2023), 791-805 803

= (nx ln c)n
{
h(µ)− h′′(µ)

n(x ln c)2
+O(n−2)

}
. (3.24)

Computing the derivatives h(k)(t), k = 0, 1, 2 with h0(t) = h(t) and evaluating at t = µ
will give

h(µ) = [f(µ)]α =
(ab)−

α
2
µ

coshα ρ

h′(µ) = α [f(µ)]α−1 f ′(µ)

h′′(µ) = α
{
[f(µ)]α−1f ′′(µ) + (α− 1)[f(µ)]α−2[f ′]2

}
,

where f(µ) is obtained from (3.19) , f ′(µ), and f ′′(µ) are given in (3.20), and (3.21),
respectively. Substitution to (3.24) will give the desired result.

To obtain an asymptotic formula for the Apostol-Genocchi type polynomials the fol-
lowing lemma will be used.

Lemma 3.5. Let a, b, c ∈ R+\{1}, α ∈ Z+, λ ∈ C\{1}, a ̸= b. For x ∈ C,

G
(α)
n+α(x;λ; a, b, c) = (n+ α)αE

(α)
n (x;λ; a, b, c),

where
(n)α = n(n− 1)(n− 2)....(n− (α− 1).

Proof. Dividing both sides of (1.3) by tα yields,(
2

λbt + at

)α

cxt =
∞∑
n=0

G(α)
n (x;λ; a, b, c)

tn−α

n!

=

∞∑
n=α

(n− α)!

n!
G(α)

n (x;λ; a, b, c)
tn−α

(n− α)!

=
∞∑

n=α

G
(α)
n (x;λ; a, b, c)

(n)α

tn−α

(n− α)!

Let s = n− α. Then n = s+ α and(
2

λbt + at

)α

cxt =
∞∑
s=0

G
(α)
s+α(x;λ; a.b.c)

(s+ α)α

ts

s!

=

∞∑
n=0

G
(α)
n+α(x;λ; a.b.c)

(n+ α)α

tn

n!

∞∑
n=0

E(α)
n (x;λ; a, b, c)

tn

n!
=

∞∑
n=0

G
(α)
n+α(x;λ; a.b.c)

(n+ α)α

tn

n!



C. Corcino, R. Corcino / Eur. J. Pure Appl. Math, 16 (2) (2023), 791-805 804

Comparing coefficients yields

G
(α)
n+α(x;λ; a.b.c) = (n+ α)αE

(α)
n (x;λ; a, b, c). (3.25)

Taking α = 1, it follows from Lemma 3.5 that

Gn+1(x;λ; a.b.c) = (n+ 1)En(x;λ; a, b, c). (3.26)

Corollary 3.6. Let a, b, c ∈ R\{1}, a ̸= b and µ = (x ln c)−1. For λ, x ∈ C\{0}, λ ̸= 1

such that |µ| <
∣∣∣µ± πi−δ

ln(ba−1)

∣∣∣,
Gn+1(nx;λ; a, b, c) = (n+ 1)

(nx ln c)n(ab)
−µ
2 λ

−1
2

cosh ρ

{
1− F

2n(x ln c)2
+O(n−2)

}
, (3.27)

where F is given in Theorem 3.3.

Proof. This follows from (3.26) and Theorem 3.3.

Corollary 3.7. Let a, b, c ∈ R\{1}, α ∈ Z+, a ̸= b and µ = (x ln c)−1.
For λ, x ∈ C\{0}

G
(α)
n+α(nx;λ; a, b, c) = (n+α)α

(nx ln c)n

λ
α
2

(
(ab)−

µ
2

cosh ρ

)α{
1− αF − α(α− 1)H2

2n(x ln c)2
+O(n−2)

}
,

(3.28)
where

H =
− ln(ab)

2
− ln(ba−1)

2
tanh ρ,

and F is given in Theorem 3.3.

Proof. This follows from Lemma 3.5 and Theorem 3.4.

4. Conclusion and Recommendation

The formulas obtained in the paper are valid for nonzero complex numbers x such that
the distance of (x ln c)−1 from the origin is smaller than its distance to the pole of the
generating function nearest to the origin. This validity can be enlarged by isolating the
contribution of the poles. This method was done in [4], [5]. The authors recommend to
obtain approximation formulas with enlarged region of validity for the polynomials studied
here.



REFERENCES 805

Acknowledgements

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

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