EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
2025, Vol. 18, Issue 3, Article Number 6165
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

On r-Bell-Based Apostol-Frobenius-Type Poly-Euler
Polynomials

Roberto B. Corcino1,2, Cristina B. Corcino1,2, Rodin Paspasan1,2, Ronald Alambra1,2

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. This paper introduces a novel variation of Frobenius-Euler polynomials derived from
Bell numbers and Apostol-type functions, incorporating the polylogarithm concept. We explore
their structure and properties through various analytical methods, with a particular emphasis
on generating functions designed for higher-order Apostol-Frobenius-Type Poly-Euler polynomials
based on Bell numbers. These functions facilitate the derivation of both explicit and implicit sum-
mation formulas. Furthermore, we establish symmetric identities that unveil intricate polynomial
relationships. This integration provides a new framework that deepens the understanding and
expands the applicability of Poly-Euler polynomials. Our findings contribute to combinatorial and
algebraic mathematics, fostering further research in related areas.

2020 Mathematics Subject Classifications: 05A15, 11B68; 11B73, 26C05, 33B10

Key Words and Phrases: Bell Polynomials, Apostol-Type Frobenius-Euler Polynomials, Bell-
Based Apostol-Type Frobenius-Euler Polynomials, Stirling Numbers, Polylogarithm

1. Introduction

In recent years, a growing number of authors [1-4] have explored the use of generat-
ing functions to introduce new families of special polynomials, including two-parameter
versions of well-known polynomials such as Bernoulli, Euler, and Genocchi polynomials.

This approach enables researchers to uncover new properties for these polynomial fam-
ilies, which often involve relationships between trigonometric functions and other paramet-
ric forms of special polynomials. By applying partial differentiation to these generating
functions, additional derivative formulas can be derived, as well as finite combinatorial
sums associated with these polynomials and their related numerical sequences. Further-
more, these special polynomials provide an accessible means to derive various useful math-
ematical identities.

DOI: https://doi.org/10.29020/nybg.ejpam.v18i3.6165

Email addresses: rcorcino@yahoo.com (R. Corcino), corcinoc@cnu.edu.ph (C. Corcino),
paspasanr@cnu.edu.ph (R. Paspasan), alamrar@cnu.edu.ph (R. Alambra)

https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0)



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A notable example is the Apostol-type Frobenius-Euler polynomials, which are signifi-
cant in combinatorial mathematics and hold an essential place in mathematical theory and
applications. These polynomials have inspired a wide range of theoretical developments
and are the subject of extensive research in combinatorics, yielding numerous intriguing
findings (see, for example, [5-10] and references therein).

The Apostol-type Frobenius-Euler polynomials E
(α)
j (ξ;u;λ) of order α are defined by

(see[7, 8]): (
1− u

λez − u

)α

eξz =
∞∑
j=0

E
(α)
j (ξ;u;λ)

zj

j!
, (1.1)

(
α, ξ, λ ∈ C, u ∈ C \{1} , λ ̸= u, |z| <

∣∣∣Log(u
λ
)
∣∣∣)

and eξz is an entire function of z for any ξ ∈ C. At the point ξ = 0, E
(α)
j (u;λ) =

E
(α)
j (0;u;λ) are called the Apostol-type Frobenius-Euler numbers of order α. From (1),

we find

E
(α)
j (ξ;u;λ) =

j∑
v=0

(
j

v

)
E(α)

v (u;λ)ξj−v (2)

and
Eα

j (ξ;−1;λ) = E
(α)
j (ξ;λ) (3)

where E
(α)
J (ξ;λ) are the jth Apostol-Euler polynomials of order α. When u = −1, (1.1)

will reduce to the Apostol-type Euler polynomials:(
2

λez + 1

)α

eξz =

∞∑
j=0

E
(α)
j (ξ;λ)

zj

j!
, (1.2)

It is worth-mentioning that (1.1) is a special case of the generalized Apostol type
Frobenius-Euler polynomials of Kurt and Simsek [35]. Moreover, mixing the Frobenius-
Euler polynomials with the concept of polylogarithm Lik(z) [14]

Lik(z) =
∞∑
n=0

zn

nk
, k ∈ Z, (1.3)

yields the Frobenius-type poly-Euler polynomials, which are defined as follows

∞∑
n=0

E(k)
n (x;u, λ)

tn

n!
=

Lik(1− e(1−u))

λet − u
ext, (1.4)

such that when k = 1, Li1(1 − e(1−u)) = −ln(1 − (1 − e−(1−u))) = −ln(e−(1−u)) = 1 − u
and so (1.4) gives (1.1).



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For j ≥ 0, the Stirling numbers of the first kind are defined by

(ξ)j =

j∑
p=0

S1(j, p)ξ
p, (4)

where (ξ)0 = 1, and (ξ)j = ξ(ξ − 1) · · · (ξ − j + 1),(j ≥ 1). From (4), we obtain

1

r!
(Log(1 + z))r =

∞∑
j=r

S1(j, r)
zj

j!
, (r ≥ 0). (5)

For j ≥ 0, the Stirling numbers of the second kind are defined by

ξj =

j∑
q=0

S2(j, q)(ξ)q (6)

From (6), we see that

1

r!
(ez − 1)r =

∞∑
j=r

S2(j, r)
zj

j!
(7)

For any nonnegative integerr, the r-Stirling numbers Sr(j, k) of the second kind are defined
by (see[11])

1

k!
erz(ez − 1)k =

∞∑
j=k

Sr(j + r, k + r)
zj

j!
. (8)

For any positive integer m, the r-Whitney numbers Wm,r(j, k) of the second kind are
defined by (see[12, 12])

1

mkk!
erz(emz − 1)k =

∞∑
j=k

Wm,r(j, k)
zj

j!
. (9)

The Bell polynomials Bj(ξ) are defined by the generating function(see[14,15])

eξ(e
z−1) =

j∑
k=0

Bj(ξ)
zj

j!
. (13)

When ξ = 1, Bj = Bj(1), (j ≥ 0) are called the Bell numbers. From (7) and (13), we note
that

Bj(ξ) =

j∑
k=1

S2(j, k)ξ
k(j ≥ 0) (14)

Recently, Duran et al. [12], introduced the Bell polynomials Bj(ξ; η) of two variable
defined by the generating function

eξz+η(ez−1) =

∞∑
j=0

Bj(ξ; η)
zj

j!
. (1.5)



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This is exactly the exponential generating of r-Bell polynomials of Mezo [25, 26], which is
given by

erz+x(ez−1) =
∞∑
n=0

Bn,r(x)
zn

n!
,

where

Bn,r(x) =
n∑

j=0

Bn,rx
j , Bn,r =

n∑
i=0

{
n

i

}
r

,

with Bn,r and

{
n

i

}
r

are called the r-Bell numbers and r-Stirling numbers of the second

kind, respectively. By taking ξ = r, we have

Bn(r; η) = Bn,r(η).

From (1.5), Alam et al. [3] defined bivariate Bell-based Apostol-Frobenius-type Euler
Polynomials denoted by BellHn(x, y;µ, λ) as follows

∞∑
n=0

BellHn(x, y;u, λ)
tn

n!
=

(
1− u

λet − u

)α

ext+y(et−1) (1.6)

The manuscript of this paper is arranged as follows: In Section 2, we introduce r-
Bell-based Apostol-type Frobenius-Euler numbers and polynomials and investigate some
properties of these numbers and polynomials. In Section 3, we derive summation formulas
of Apostol-type Frobenius-Euler numbers and polynomials, connected with Apostol-type
Bernoulli, Euler, and Genocchi polynomials. In Section 4, we prove several identities of
Apostol-type Frobenius-Euler polynomials by using different analytical means and apply-
ing generating functions.

2. r-Bell-Based Apostol-Frobenius-Type Poly-Euler Polynomials

BE
(α)
j (r, y;u;λ)

In this section, we introduce the r-Bell-based Apostol-type Frobenius-Euler polyno-

mials, denoted as BE
(α)
j (r, y;u;λ), and present an explicit formula for these polynomials.

Additionally, we explore their fundamental properties. We begin with the following defi-
nition.

Definition 2.1. The r-Bell-based Apostol-Frobenius-Type Poly-Euler Polynomials BE
(k,α)
j (r, y;u;λ)

of order α are defined by means of the following generating function:(
Lik(1− e−(1−u))

λet − u

)α

ert+y(et−1) =
∞∑
j=0

BE
(k,α)
j (r, y;u;λ)

tj

j!
, (2.1)

where
(α, r, y, λ ∈ C \ {1} , λ ̸= u, |t|)

∣∣∣Log (u
λ

)∣∣∣ .



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Remark 2.2. When k = 1, (2.1) reduces to

∞∑
j=0

BE
(k,α)
j (r, y;u;λ)

tj

j!
=

(
Li1(1− e−(1−u))

λet − u

)α

ert+y(et−1)

=

(
−ln(1− (1− e−(1−u)))

λet − u

)α

ert+y(et−1)

=

(
1− u

λet − u

)α

ert+y(et−1).

This is exactly the bivariate Bell-Based Apostol-Frobenius-Type Poly-Euler Polynomials
introduced by Alam et al. [3]. That is, by (1.6)

BellHn(r, y;u, λ) = BE
(1,α)
j (r, y;u;λ).

Remark 2.3. On taking r = 0 in (2.1), we obtain new type Bell-based Apostol-Frobenius-

Type Poly-Euler polynomials BE
(k,α)
j (y;u;λ)(

Lik(1− e−(1−u))

λet − u

)α

ey(e
t−1) =

∞∑
j=0

BE
(k,α)
j (y;u;λ)

tj

j!
. (2.2)

We call E
(k,α)
j (y;u;λ) the Bell-based Apostol-Frobenius-type poly-Euler polynomials. More-

over, when k = 1, we have

∞∑
j=0

BE
(k,α)
j (y;u;λ)

tj

j!
=

(
Li1(1− e−(1−u))

λet − u

)α

ey(e
z−1)

=

(
−ln(1− (1− e−(1−u)))

λet − u

)α

ey(e
z−1)

=

(
1− u

λet − u

)α

ey(e
z−1),

where BellHn(y;u, λ) = BE
(1,α)
j (y;u;λ), the Bell-based Apostol-Frobenius-type Euler poly-

nomials.

Remark 2.4. Upon setting y = 0 in (2.1), the r-Bell-based Apostol-Frobenius-type poly-

Euler polynomials BE
(k,α)
j (r, y;u;λ) of order α reduces to familiar Apostol-Frobenius-type

poly-Euler polynomials BE
(k,α)
j (r;u;λ) of order α

Remark 2.5. When y = 0 and α = 1, r-Bell-based Apostol-Frobenius-type poly-Euler

polynomials BE
(k,α)
j (r, y;u;λ) of order α reduce to the usual Frobenius-Euler polynomials

Ej(r;u;λ).

We note that

BE
(k,1)
j (r, 0;u;λ) = BE

(k)
j (r;u;λ). (2.3)



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The following theorem contains a summation formula relating BE
(r)
n (r, y;u, λ) with

E
(r)
k (r;u, λ) and Bn(y)

Theorem 2.6. The r-Bell-based Apostol-Frobenius-Type Poly-Euler polynomials BE
(α)
j (r, y;u;λ)

of order α satisfy the following summation identity

BE
(k,α)
j (r, y;u;λ) =

n∑
j=0

(
n

k

)
E

(k,α)
j (r;u;λ)Bn−j(y).

Proof. Using (2.1), we have

∞∑
j=0

BE
(k,α)
j (r, y;u;λ)

tj

j!
=

{(
Lik(1− e−(1−u))

λet − u

)α

ert

}
ey(e

t−1)

=

( ∞∑
n=0

E
(k,α)
j (r;u;λ)

tn

n!

)( ∞∑
n=0

Bn(y)
tn

n!

)

=

∞∑
n=0


n∑

j=0

(
n

k

)
E

(k,α)
j (r;u;λ)Bn−j(y)

 tn

n!
.

Comparing the coefficients of tn/n! completes the proof of the theorem.

The next theorem expresses BE
(k,α)
n (r, y;u, λ) as polynomial in r with BE

(k,α)
n−k (y;u, λ)

as coefficients.

Theorem 2.7. The r-Bell-based Apostol-Frobenius-Type Poly-Euler polynomials BE
(α)
j (r, y;u;λ)

of order α satisfy the following summation identity

BE
(k,α)
n (r, y;u, λ) =

n∑
j=0

(
n

j

)
BE

(k,α)
n−j (y;u;λ)rj

Proof. Using (2.1), we have

∞∑
j=0

BE
(k,α)
j (r, y;u;λ)

tj

j!
=

{(
Lik(1− e−(1−u))

λet − u

)α

ey(e
t−1)

}
ert

=

( ∞∑
n=0

BE
(k,α)
j (y;u;λ)

tn

n!

)( ∞∑
n=0

(rt)n

n!

)

=

∞∑
n=0


n∑

j=0

(
n

j

)
BE

(k,α)
j (y;u;λ)rn−j

 tn

n!
.

Comparing the coefficients of tn/n! completes the proof of the theorem.

The following theorem contains the summation formula for BE
(k,α)
n (r, y;u;λ).



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Theorem 2.8. The r-Bell-based Apostol-Frobenius-Type Poly-Euler polynomials BE
(k,α)
n (r, y;u;λ)

of order α satisfy the following summation identity

BE
(r)
n (r + y, z;u, λ) =

n∑
j=0

(
n

j

)
E

(k,α)
j (y;u;λ)Bn−j(y, z)

Proof. Using (2.1), we have

∞∑
j=0

BE
(k,α)
j (r + y, z;u;λ)

tj

j!
=

{(
Lik(1− e−(1−u))

λet − u

)α

ert

}
eyt+z(et−1)

=

( ∞∑
n=0

E
(k,α)
j (r;u;λ)

tn

n!

)( ∞∑
n=0

Bn(y, z)
tn

n!

)

=

∞∑
n=0


n∑

j=0

(
n

j

)
E

(k,α)
j (y;u;λ)Bn−j(y, z)

 tn

n!
.

Comparing the coefficients of tn/n! completes the proof of the theorem.

3. Implicit Summation Formula

Theorem 3.1. The r-Bell-based Apostol-Frobenius-Type Poly-Euler polynomials BE
(α)
j (r, y;u;λ)

of order α satisfy the following summation identity

BE
(k,α1+α2)
n (r1 + r2, y2 + y2;u, λ)

=
n∑

j=0

(
n

j

)
BE

(k,α1)
j (r1, y1;u, λ)BE

(k,α2)
n−j (r2, y2;u, λ)

Proof. Replacing the parameters α, r and y at the left-hand side of equation (2.1) in
Definition 2.1, with α1 + α2, r1 + r2 and y1 + y2, respectively, we obtain(

Lik(1− e−(1−u))

λet − u

)α1+α2

e(r1+r2)t+(y1+y2)(et−1)

=

(
Lik(1− e−(1−u))

λet − u

)α1

e(r1)t+(y1)(et−1)

(
Lik(1− e−(1−u))

λet − u

)α2

e(r2)t+(y2)(et−1)

∞∑
n=0

BE
(k,α1+α2)
n (r1 + r2, y2 + y2;u, λ)

tn

n!

=

( ∞∑
n=0

BE
(k,α1)
n (r1, y1;u, λ)

tn

n!

)( ∞∑
n=0

BE
(k,α2)
n (r2, y2;u, λ)

tn

n!

)



8 of 15

=

∞∑
n=0

n∑
j=0

(
n

j

)
BE

(k,α1)
j (r1, y1;u, λ)BE

(k,α2)
n−j (r2, y2;u, λ)

tn

n!
.

Comparing the coefficients of tn

n! completes the proof of the theorem.

Taking α1 = α,α2 = 0,r1 = r,r2 = 1,y1 = y, y2 = 0, Theorem 3.1 yields

BE
(k,α)
n (r + 1, y;u, λ) =

n∑
j=0

(
n

j

)
BE

(k,α)
j (r, y;u, λ)Bn−j(1, 0)

=
n∑

j=0

(
n

j

)
BE

(kα)
j (r, y;u, λ).

Theorem 3.2. The r-Bell-based Apostol-Frobenius-Type Poly-Euler polynomials BE
(α)
j (r, y;u;λ)

of order α satisfy the following implicit summation identity

BE
(k,α)
q+l (r, y;u, λ) =

q,l∑
j,m=0

(
q

j

)(
l

m

)
(r − z)q−j+i−m

BE
(k,α)
q+l (z, y;u, λ)

Proof. Let us recall the following series manipulation formula:

∞∑
N=0

f(N)
(r + y)N

N !
=

∞∑
n=0

∞∑
m=0

f(n+m)
rn

n!

yn

m!

Note that we can rewrite (2.1) as follows:(
Lik(1− e−(1−u)

λet+v − u

)α

ey(e
t+v−1) = e−r(t+v)

∞∑
n=0

BE
(k,α)
n (r, y;u, λ)

(t+ v)n

n!
.

Applying the above series manipulation formula yields(
Lik(1− e−(1−u)

λet+v − u

)α

ey(e
t+v−1) = e−r(t+v)

∞∑
j=0

∞∑
l=0

BE
(k,α)
j+l (r, y;u, λ)

tj

n!

vl

l!

= e−z(t+v)
∞∑
j=0

∞∑
l=0

BE
(k,α)
j+l (z, y;u, λ)

tj

j!

vl

l!

∑
j,l≥0

BE
(k,α)
j+l (r, y;u, λ)

tj

j!

vl

l!
= e(r−z)(t+v)

∑
j,l≥0

BE
(k,α)
j+l (z, y;u, λ)

tj

j!

vl

l!

=

( ∞∑
N=0

(r − z)N
(t+ v)N

N !

)∑
j,l≥0

BE
(k,α)
j+l (z, y;u, λ)

tj

j!

vl

l!





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=

 ∑
n,m≥0

(r − z)n+m tn

n!

vm

m!

∑
j,l≥0

BE
(k,α)
j+l (z, y;u, λ)

tj

j!

vl

l!


=
∑
q,l≥0


q,l∑

j,m=0

(
q

j

)(
l

m

)
(r − z)q−j+i−m

BE
(k,α)
q+l (z, y;u, λ)

 tq

q!

vl

l!
.

Comparing the coefficients of tq

q!
vl

l! completes the proof of the theorem.

Theorem 3.3. The r-Bell-based Apostol-Frobenius-Type Poly-Euler polynomials BE
(α)
j (r, y;u;λ)

of order α satisfy the following summation identity

BE
(k,α)
n (r, y;u, λ) =

n∑
k=0

∞∑
j=0

(
n

k

)
(r)jS(k, j)BE

(k,α)
n−k (y;u, λ).

Proof.

∞∑
n=0

BE
(k,α)
n (r, y;u, λ)

tn

n!
=

(
Lik(1− e−(1−u)

λet − u

)r

ert+y(et−1)

=

(
Lik(1− e−(1−u)

λet − u

)α

ey(e
t−1)ert

=

(
Lik(1− e−(1−u)

λet − u

)α

ey(e
t−1)(1 + et − 1)r

=

( ∞∑
n=0

BE
(k,α)
n (y;u, λ)

tn

n!

) ∞∑
j=0

(
r

j

)
(et − 1)j


=

( ∞∑
n=0

BE
(k,α)
n (y;u, λ)

tn

n!

) ∞∑
j=0

(r)j
(et − 1)j

j!


=

( ∞∑
n=0

BE
(k,α)
n (y;u, λ)

tn

n!

) ∞∑
j=0

(r)j

∞∑
n=0

S(n, j)
tn

n!


=

( ∞∑
n=0

BE
(k,α)
n (y;u, λ)

tn

n!

) ∞∑
n=0


∞∑
j=0

(r)jS(n, j)

 tn

n!


=

∞∑
n=0

n∑
k=0

(
n

k

)
∞∑
j=0

(r)jS(k, j)BE
(k,α)
n−k (y;u, λ)

 tn

n!



10 of 15

=
∞∑
n=0


n∑

k=0

(
n

k

) ∞∑
j=0

(r)jS(k, j)BE
(k,α)
n−k (y;u, λ)

 tn

n!
.

Comparing the coefficients of tn/n! completes the proof of the theorem.

The next result that we are going to obtain is to express the r-Bell polynomials in
terms of the difference of r-Bell-based Apostol-Frobenius-Type Poly-Euler polynomials

BE
(α)
j (r, y;u;λ) of order α.

Theorem 3.4. The r-Bell-based Apostol-Frobenius-Type Poly-Euler polynomials BE
(k,1)
j (r, y;u;λ)

satisfy the following relation

Bn(r, y) =
λ · BE(k,1)

n+1 (r + 1, y;u, λ)− u · BE(k,1)
n+1 (r, y;u, λ)

Lik(1− e−(1−u))(n+ 1)

Proof. By rewriting the definition of r-Bell polynomials, we obtain

∞∑
n=0

Bn(r, y)
tn

n!
= ert+y(et−1)

=

(
λet − u

Lik(1− e−(1−u))

)(
Lik(1− e−(1−u))

λet − u
ert+y(et−1)

)

=
1

Lik(1− e−(1−u))

(
λ

(
Lik(1− e−(1−u))

λet − u
e(r+1)t+y(et−1)

)

−u

(
Lik(1− e−(1−u))

λet − u
ert+y(et−1)

))

=
1

Lik(1− e−(1−u))

(
λ

∞∑
n=0

BE
(r)
n (r + 1, y;u, λ)

tn

n!

−u
∞∑
n=0

BE
(k,1)
n (r, y;u, λ)

tn

n!

)

=
1

Lik(1− e−(1−u))

(
λ

∞∑
n=0

BE
(k,1)
n (r + 1, y;u, λ)

tn−1

n!

−u

∞∑
n=0

BE
(k,1)
n (r, y;u, λ)

tn−1

n!

)

=
λ

Lik(1− e−(1−u))

∞∑
n=−1

1

n+ 1
BE

(k,1)
n+1 (r + 1, y;u, λ)

tn

n!

− u

Lik(1− e−(1−u))

∞∑
n=−1

1

n+ 1
BE

(k,1)
n+1 (r, y;u, λ)

tn

n!
.



11 of 15

Comparing the coefficients of tn

n! yields the theorem.

The next theorem contains the derivative formula for BE
(α)
j (r, y;u;λ) of order α wit

respect to r.

Theorem 3.5. The r-Bell-based Apostol-Frobenius-Type Poly-Euler polynomials BE
(k,α)
j (r, y;u;λ)

of order α satisfy the derivative formula

∂

∂r
BE

(k,α)
n (r, y;u, λ) = nBE

(k,α)
n−1 (r, y;u, λ).

Proof. . By applying the first derivative to both sides of (2.1) with respect to r, we
have

∂

∂r

∞∑
n=0

BE
(k,α)
n (r, y;u, λ)

tn

n!
=

∂

∂r

(
Lik(1− e−(1−u))

λet − u

)α

ert+y(et−1)

∞∑
n=0

∂

∂r
BE

(k,α)
n (r, y;u, λ)

tn

n!
=

(
Lik(1− e−(1−u))

λet − u

)α

ert+y(et−1) t

= t
∞∑
n=0

BE
(k,α)
n (r, y;u, λ)

tn

n!

=

∞∑
n=0

BE
(k,α)
n (r, y;u, λ)

tn+1

n!

=
∞∑
n=1

nBE
(k,α)
n−1 (r, y;u, λ)

tn

n!
.

Comparing the coefficients of tn

n! completes the proof of the theorem.

Remark 3.6. This relation shows that BE
(k,α)
n (r, y;u, λ) can be classified as an Apell

Polynomial.

The next theorem contains the derivative formula for BE
(k,α)
j (r, y;u;λ) of order α with

respect to y.

Theorem 3.7. The r-Bell-based Apostol-Frobenius-Type Poly-Euler polynomials BE
(k,α)
j (r, y;u;λ)

of order α satisfy the derivative formula

∂

∂y
BE

(k,α)
n (r, y;u, λ) = n

(
BE

(k,α)
n−1 (r + 1, y;u, λ)− BE

(k,α)
n−1 (r, y;u, λ)

)
.

Proof. By applying the first derivative to both sides of (2.1) with respect to y, we have

∞∑
n=0

∂

∂y
BE

(k,α)
n (r, y;u, λ)

tn

n!
=

(
Lik(1− e−(1−u))

λet − u

)α

ert+y(et−1)(et − 1)



12 of 15

=

(
Lik(1− e−(1−u))

λet − u

)α

e(r+1)t+y(et−1) −

(
Lik(1− e−(1−u))

λet − u

)α

ert+y(et−1)

=

∞∑
n=0

BE
(k,α)
n (r + 1, y;u, λ)

tn

n!
−

∞∑
n=0

BE
(k,α)
n (r, y;u, λ)

tn

n!

=
∞∑
n=0

{
BE

(k,α)
n (r + 1, y;u, λ)− BE

(k,α)
n (r, y;u, λ)

} tn+1

n!

=

∞∑
n=1

n
(
BE

(k,α)
n−1 (r + 1, y;u, λ)− BE

(k,α)
n−1 (r, y;u, λ)

) tn

n!
.

Comparing the coefficients of tn

n! completes the proof of the theorem.

4. Conclusion

This study introduces a novel class of Frobenius-Euler polynomials by leveraging the
mathematical structures of Bell numbers, Apostol-type functions, and the polylogarithm
concept. Through rigorous analytical methods, we have successfully derived generating
functions that serve as powerful tools in understanding higher-order Apostol-Frobenius-
Type Poly-Euler polynomials. These generating functions facilitate the formulation of
both explicit and implicit summation formulas, which further contribute to the mathe-
matical characterization of these polynomials.

Additionally, the study establishes symmetric identities that reveal deep interconnec-
tions among these polynomials, highlighting their structural complexity and mathematical
significance. These identities not only enhance the understanding of polynomial relation-
ships but also provide a unified framework for exploring new properties and potential
generalizations.

The integration of these mathematical components offers fresh insights into combi-
natorial and algebraic mathematics, broadening the scope of applications for Poly-Euler
polynomials in various domains, such as number theory, discrete mathematics, and com-
putational algebra. The results of this study serve as a foundation for further theoretical
developments, inviting future research to extend these findings into new mathematical
territories, including special functions, recurrence relations, and their computational ap-
plications.

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