EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
2025, Vol. 18, Issue 1, Article Number 5559
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

Some Properties of the Bell-Based Apostol-Frobenius-
Type Tangent Polynomials

Jay Ontolan1,∗, Jeneveb Malusay1, Edward Kiunisala1, Joris Buloron1

1 Mathematics Department, Cebu Normal University, Cebu City, 6000, Philippines

Abstract. In this paper, we introduce a new class of Frobenius-Tangent polynomials, derived
from the Bell numbers and Apostol-type functions. We conduct a detailed investigation into the
properties of these polynomials, utilizing various analytical techniques. By employing generating
functions for Bell-based Apostol-Frobenius-Type Tangent polynomials of higher order, we obtain
both explicit and implicit summation formulas and its relation to Appell polynomials.

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

Key Words and Phrases: Tangent polynomials, Bell polynomials, Apostol-Frobenius- Type
poly-Tangent polynomials, Bell-based Apostol-Frobenius-Type poly-Tangent polynomials, Stirling
numbers, Appell polynomials

1. Introduction

In mathematical analysis, special polynomials play a pivotal role due to their extensive
applications across various domains. Among these, the Frobenius-Tangent polynomials
have garnered significant attention. These polynomials, denoted as Tn(x), are defined by
the generating function ([11]),[12])

∞∑
n=0

Tn(x)
zn

n!
=

(
2

e2z + 1
exz
)
, (1)

where Tn(0) = Tn, the Tangent numbers defined coefficient of the following series expansion
of the tangent function ([13])

tan z =
∞∑
n=0

(−1)n+1T2n+1
z2n+1

(2n+ 1)!
, (2)

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v18i1.5559

Email addresses: ontolanj@cnu.edu.ph (J. Ontolan), malusayj@cnu.edu.ph (J. Malusay),
kiunisalae@cnu.edu.ph (E. Kiunisala), buloronj@cnu.edu.ph (J. Buloron)

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



J. Ontolan et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5559 2 of 16

with T0 = 1 and T2n = 0, n ∈ N.
They emerge naturally in the study of differential equations and have applications in

numerical analysis and approximation theory.
Parallel to this, the Apostol polynomials, particularly the Apostol-Bernoulli and Apostol-

Euler polynomials, have been extensively studied. The Apostol-Bernoulli polynomials

B
(a)
n (x) are defined via the generating function:

∞∑
n=0

B(a)
n (x)

tn

n!
=

text

aet − 1
, (3)

where a is a non-zero parameter. These polynomials generalize the classical Bernoulli
polynomials and have applications in number theory and combinatorics.

Bell polynomials, denoted as Bn(x), are another important class, defined by the gen-
erating function:

∞∑
n=0

Bn(x)
tn

n!
= ex(e

t−1). (4)

They are instrumental in the study of combinatorial structures and have applications
in the theory of partitions and moments of probability distributions.

In the framework of orthogonal polynomials, it is noteworthy that certain classes of
Apostol and Bell polynomials exhibit orthogonality properties under specific weight func-
tions. For instance, the study by Luo and Srivastava [8] looks into some generalizations of
Apostol-Bernoulli and Apostol-Euler polynomials, exploring their orthogonality and other
properties. Similarly, the work by Kurt [7] introduces new families of polynomials asso-
ciated with the Bell numbers and polynomials, discussing their potential orthogonality
under certain conditions. Additionally, recent research by Khan and Riaz [6] investigates
certain subclasses of Apostol-type polynomials, providing insights into their structural
properties within the orthogonal polynomial framework. Further studies have expanded
the landscape of these polynomials. Dattoli et al. [5] introduced a family of hybrid poly-
nomials that exhibit characteristics of both Hermite and Laguerre polynomials, enriching
the theory of special functions. Ramı́rez and Cesarano [9] explored new classes of degen-
erated generalized Apostol-Bernoulli, Apostol-Euler, and Apostol-Genocchi polynomials,
deriving explicit expressions and recurrence relations. In a subsequent work, Ramı́rez et
al. [10] presented new results for these degenerated polynomials, establishing algebraic
relationships and recurrence formulas.

The study of special numbers such as the tangent numbers, Bernoulli numbers, Euler
numbers, and Genocchi numbers has become an interesting area for many mathemati-
cians ([2],[4],[14]). Tangent numbers and polynomials possess many significant properties
that can be found in mathematics and physics. Analogues and symmetric properties for
tangent polynomials are derived in [12] and [11]. Building upon these foundational stud-
ies, this paper aims to explore higher-order bivariate Bell-based Apostol-Frobenius-type
poly-Tangent polynomials. We will derive explicit representations and investigate their
structural properties.



J. Ontolan et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5559 3 of 16

The Bell-based Apostol-Frobenius-Type Tangent Polynomials BTn(x, y, u, λ)
is defined by the generating function

∞∑
n=0

BTn(x, y, u, λ)
tn

n!
=

(
1− u

λe2t − u

)
ext+y(et−1).

This paves way to our working definition of the Bell-based Apostol-Frobenius-Type
Tangent Polynomials of higher order BT

r
n(x, y, u, λ) defined by the generating func-

tion

∞∑
n=0

BT
(r)
n (x, y;u, λ)

tn

n!
=

(
1− u

λe2t − u

)r

ext+y(et−1). (5)

The next three functions are special cases of (5); when x = 0 and y ̸= 0, we obtain

BT
(r)
n (y, u, λ) defined by

∞∑
n=0

BT
(r)
n (y, u, λ)

tn

n!
=

(
1− u

λe2t − u

)r

ey(e
t−1), (6)

known as Bell-based Apostol-Frobenius-Type Tangent numbers of higher order.
When y = 0 and x ̸= 0 we have the polynomial T r

n(x, u, λ) defined by

∞∑
n=0

T r
n(x, u, λ)

tn

n!
=

(
1− u

λe2t − u

)r

ext, (7)

known as Apostol-Frobenius-Type Tangent polynomials of higher order. Lastly,

when both x = y = 0 we have the polynomial T
(r)
n (u, λ) defined by

∞∑
n=0

T (r)
n (u, λ)

tn

n!
=

(
1− u

λe2t − u

)r

, (8)

known as Apostol-Frobenius-Type Tangent numbers of higher order.
Meanwhile, in [1] we define a sequence of polynomials {Pn(x)}∞0 satisfying

P ′
n(x) = nPn−1(x), n ≥ 1, (9)

as Appell polynomials. Moreover, [3], [15], and [16] established an important characteri-
zation of Appell polynomials in the following equivalent conditions:

(a) {Pn(x)}∞0 is a sequence of Appell polynomials.



J. Ontolan et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5559 4 of 16

(b) {Pn(x)}∞0 has a generating function of the form

A(t)ext =
∞∑
n=0

Pn(x)
tn

n!
, (10)

where A(t) is a formal power series independent of x with A(0) ̸= 0.

(c) There exists a sequence {an}∞n=0 with a0 ̸= 0 such that

Pn(x) =
n∑

k=0

(
n

k

)
an−kx

k. (11)

(d) There exists a sequence {an}∞n=0 with a0 ̸= 0 such that

Pn(x) =

( ∞∑
k=0

ak
k!

Dk

)
xn, (12)

where D = d
dx .

The next lemma will be useful in some of our results.

Lemma 1. Let f be a function and {f(N)}∞0 a sequence and coefficients of the power
series

∞∑
N=0

f(N)
(x+ y)N

N !
.

There exists a pair of integers n and m such that

∞∑
N=0

f(N)
(x+ y)N

N !
=

∞∑
m=0

∞∑
n=0

f(m+ n)
xmyn

m!n!
, (13)

where N = n+m.

Proof. For each N ∈ N, we write

f(N)
(x+ y)N

N !
=

f(N)
N∑
i=0

(
N

i

)
xiyN−i

N !

=

N∑
i=0

f(N)
(
N
i

)
xiyN−i

N !

=

N∑
i=0

f(N) N !
(N−i)!i!x

iyN−i

N !



J. Ontolan et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5559 5 of 16

=
N∑
i=0

f(N)
xiyN−i

(N − i)!i!

=
N∑
i=0

f((N − i) + i)
xiyN−i

(N − i)!i!

=

∞∑
N=0

N∑
i=0

f((N − i) + i)
xiyN−i

(N − i)!i!

∞∑
N=0

f(N)
(x+ y)N

N !
=

∞∑
m=0

∞∑
n=0

f(m+ n)
xmyn

m!n!
.

In this study, the authors are interested to explore some properties of Bell-based

Apostol-Frobenius-Type Tangent polynomials of higher order BT
(r)
n (x, y;u, λ) in terms

of the three aforementioned cases (6), (7), and (8).

2. Higher Order Bivariate Bell-Based Apostol-Frobenius-Type Tangent
Polynomials

The following theorems contain identities for the bivariate Bell-based ApostolFrobenius-
Type Tangent polynomials of higher order expressed in terms of (6), (7), and (8) and the
Bell polyomials.

Theorem 1. The Bell-based Apostol-Frobenius-Type Tangent Polynomials of higher order

BT
(r)
n (x, y;u, λ) satisfies the equation

BT
(r)
n (x, y;u, λ) =

n∑
k=0

(
n

k

)
T (r)
n (x;u, λ)Bn−k(y) (14)

where Bn(y) is the Bell Polynomial defined by the generating function

∞∑
n=0

Bn(y)
tn

n!
= ey(e

t−1).

Proof. We write

∞∑
n=0

BT
(r)
n (x, y;u, λ)

tn

n!
=

(
(1− u)

λe2t − u

)r

ext+y(et−1)

=

{(
(1− u)

λe2t − u

)r

ext
}
ey(e

t−1)

=

( ∞∑
n=0

T (r)
n (x;u, λ)

tn

n!

)( ∞∑
n=0

Bn(y)
tn

n!

)



J. Ontolan et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5559 6 of 16

=

∞∑
n=0

{
n∑

k=0

(
n

k

)
T
(r)
k (x;u, λ)Bn−k(y)

}
tn

n!
.

Comparing coefficients, we obtain the desired result

BT
(r)
n (x, y;u, λ) =

n∑
k=0

(
n

k

)
T
(r)
k (x;u, λ)Bn−k(y).

Theorem 2. The function BT
(r)
n (x, y;u, λ) satisfies the equation

BT
(r)
n (x, y;u, λ) =

n∑
k=0

(
n

k

)
T (r)
n (u, λ)Bn−k(x, y) (15)

where Bn(x, y) is the Bivariate Bell Polynomial defined by the generating function

∞∑
n=0

Bn(x, y)
tn

n!
= ext+y(et−1).

Proof. We start with

∞∑
n=0

BT
(r)
n (x, y;u, λ)

tn

n!
=

(
(1− u)

λe2t − u

)r

ext+y(et−1)

=

( ∞∑
n=0

T (r)
n (u, λ)

tn

n!

)( ∞∑
n=0

Bn(x, y)
tn

n!

)

=

∞∑
n=0

{
n∑

k=0

(
n

k

)
T
(r)
k (u, λ)Bn−k(x, y)

}
tn

n!
.

Comparing coefficients,

BT
(r)
n (x, y;u, λ) =

n∑
k=0

(
n

k

)
T
(r)
k (u, λ)Bn−k(x, y).

as desired.

In view of (5), we can express the bivariate Bell Polynomial Bn(x, y) as

Bn(x, y) = BT
(0)
n (x, y;u, λ), (16)

which allows as to write the result from Theorem 1.2 into

BT
(r)
n (x, y;u, λ) =

n∑
k=0

(
n

k

)
T
(r)
k (u, λ)BT

(0)
n−k(x, y;u, λ).



J. Ontolan et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5559 7 of 16

Theorem 3. The function BT
(r)
n (x, y;u, λ) satisfies the equation

BT
(r)
n (x, y;u, λ) =

n∑
k=0

(
n

k

)
BT

(r)
n−k(y, u, λ)x

k. (17)

Proof. Writing

∞∑
n=0

BT
(r)
n (x, y;u, λ)

tn

n!
=

(
(1− u)

λe2t − u

)r

ey(e
t−1)ext

=

( ∞∑
n=0

BT
(r)
n (y;u, λ)

tn

n!

)( ∞∑
n=0

(xt)n

n!

)

=
∞∑
n=0

{
n∑

k=0

(
n

k

)
BT

(r)
k (y;u, λ)xn−k

}
tn

n!
.

Comparing coefficients,

BT
(r)
n (x, y;u, λ) =

n∑
k=0

(
n

k

)
BT

(r)
k (y;u, λ)xn−k

=
n∑

k=0

(
n

k

)
BT

(r)
n−k(y;u, λ)x

k

as desired.

The next theorem contains the addition formula for bivariate Bell-based Apostol-
Frobenius-Type Tangent polynomials of higher order.

Theorem 4. The Bell-based Apostol-Frobenius-Type Tangent Polynomials of higher order

BT
(r)
n (x, y;u, λ) satisfies the equation

BT
(r)
n (x+ y, z;u, λ) =

n∑
k=0

(
n

k

)
T
(r)
k (x;u, λ)Bn−k(y, z) (18)

Proof. We start by writing

∞∑
n=0

BT
(r)
n (x+ y, z;u, λ)

tn

n!
=

(
(1− u)

λe2t − u

)r

e(x+y)t+z(et−1)

=

{(
(1− u)

λe2t − u

)r

ext
}
eyt+z(et−1)

=

( ∞∑
n=0

T (r)
n (x;u, λ)

tn

n!

)( ∞∑
n=0

Bn(y, z)
tn

n!

)

=
∞∑
n=0

{
n∑

k=0

(
n

k

)
T
(r)
k (x;u, λ)Bn−k(y, z)

}
tn

n!
.



J. Ontolan et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5559 8 of 16

Comparing coefficients, we obtain the desired result

BT
(r)
n (x+ y, z;u, λ) =

n∑
k=0

(
n

k

)
T
(r)
k (x;u, λ)Bn−k(y, z).

Implicit Summation Formula

Within this section, we will derive different summation formulas for BT
(r)
n (x+y, z;u, λ),

establishing implicit connections among the variables by considering them as arguments.
The subsequent theorem shows a particular expression of these summation formulas.

Theorem 5. The bivariate Bell-based Apostol-Frobenius-Type Tangent polynomials of

higher order BT
(r)
n (x, y;u, λ) satisfy the summation formula:

BT
(r1+r2)
n (x1 + x2, y2 + y2;u, λ) =

n∑
k=0

(
n

k

)
BT

(r1)
k (x1, y1;u, λ)BG

(r2)
n−k(x2, y2;u, λ) (19)

Proof. we can express the right hand side of (5) as follows:(
(1− u)

λe2t − u

)r1+r2

e(x1+x2)t+(y1+y2)(et−1)

=

{(
(1− u)

λe2t − u

)r1

ex1t+y1(et−1)

}{(
(1− u)

λe2t − u

)r2

ex2t+y2(et−1)

}
∞∑
n=0

BT
(r1+r2)
n (x1 + x2, y2 + y2;u, λ)

tn

n!
=

( ∞∑
n=0

BT
(r1)
n (x1, y1;u, λ)

tn

n!

)( ∞∑
n=0

BT
(r2)
n (x2, y2;u, λ)

tn

n!

)

=
∞∑
n=0

n∑
k=0

BT
(r1)
n (x1, y1;u, λ)BT

(r2)
n−k(x2, y2;u, λ)

(
n

k

)
.

Comparing coefficients, we obtain the desired result

BT
(r1+r2)
n (x1 + x2, y2 + y2;u, λ) =

n∑
k=0

(
n

k

)
BT

(r1)
k (x1, y1;u, λ)BT

(r2)
n−k(x2, y2;u, λ).

Remark 1. When r1 = r, r2 = 0, x1 = x, x2 = 1,y1 = y, y2 = 0, the summation formula
in (19) reduces to

BT
(r)
n (x+ 1, y;u, λ) =

n∑
k=0

(
n

k

)
BT

(r)
k (x, y;u, λ)Bn−k(1, 0)

=

n∑
k=0

(
n

k

)
BT

(r)
k (x, y;u, λ). (20)



J. Ontolan et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5559 9 of 16

On the other hand, when y = 1, (18) gives

BT
(r)
n (x+ 1, z;u, λ) =

n∑
k=0

(
n

k

)
T
(r)
k (x;u, λ)Bn−k(1, z). (21)

Replacing z with y in (21) and compare it to (20) yields

n∑
k=0

(
n

k

)
T
(r)
k (x;u, λ)Bn−k(1, y) =

n∑
k=0

(
n

k

)
BT

(r)
k (x, y;u, λ).

In view of (5), notice that we can write

∞∑
n=0

BT
(r)
n (x, y;u, λ)

(t+ v)n

n!
=

(
1− u

λe2(t+v) − u

)r

ex(t+v)+y(et+v−1)

which allows us to express

(
1− u

λe2(t+v) − u

)r

ex(t+v)ey(e
t+v−1) =

∞∑
n=0

BT
(r)
n (x, y;u, λ)

(t+ v)n

n!
,

consequently

(
1− u

λe2(t+v) − u

)r

ey(e
t+v−1) = e−x(t+v)

∞∑
n=0

BT
(r)
n (x, y;u, λ)

(t+ v)n

n!
. (22)

Applying (13), we obtain

(
1− u

λe2(t+v) − u

)r

ey(e
t+v−1) = e−x(t+v)

∞∑
k=0

∞∑
l=0

BT
(r)
k+l(x, y;u, λ)

tk

k!

vl

l!
. (23)

Replacing x with z, equation (23) becomes

(
(1− u)

λe2(t+v) − u

)r

ey(e
t+v−1) = e−z(t+v)

∞∑
k=0

∞∑
l=0

BT
(r)
k+l(z, y;u, λ)

tk

k!

vl

l!(
(1− u)

λe2(t+v) − u

)r

ey(e
t+v−1)ex(t+v) = ex(t+v)e−z(t+v)

∞∑
k=0

∞∑
l=0

BT
(r)
k+l(z, y;u, λ)

tk

k!

vl

l!(
(1− u)

λe2(t+v) − u

)r

ex(t+v)+y(et+v−1) = e(x−z)(t+v)
∞∑
k=0

∞∑
l=0

BT
(r)
k+l(z, y;u, λ)

tk

k!

vl

l!

Thus, using (13) again, we have



J. Ontolan et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5559 10 of 16

∑
k,l≥0

BT
(r)
k+l(x, y;u, λ)

tk

k!

vl

l!
= e(x−z)(t+v)

∑
k,l≥0

BT
(r)
k+l(z, y;u, λ)

tk

k!

vl

l!

=

( ∞∑
N=0

(x− z)N
(t+ v)N

N !

)∑
k,l≥0

BT
(r)
k+l(z, y;u, λ)

tk

k!

vl

l!


=

 ∑
n,m≥0

(x− z)n+m tn

n!

vm

m!

∑
k,l≥0

BT
(r)
k+l(z, y;u, λ)

tk

k!

vl

l!


=
∑
k,l≥0


k,l∑

k,m=0

(
k

n

)(
l

m

)
(x− z)n+m

BT
(r)
k+l(z, y;u, λ)

 tk

k!

vl

l!
.

Comparing coefficients, we then have

∑
k,l≥0

BT
(r)
k+l(x, y;u, λ) =

k,l∑
n,m=0

(
k

n

)(
l

m

)
(x− z)n+m

BTk+l−n−m(z, y;u, λ),

which proves our next theorem.

Theorem 6. The bivariate Bell-based Apostol-Frobenius-Type Tangent polynomials of

higher order BT
(r)
n (x, y;u, λ) satisfy the summation formula

∑
k,l≥0

BT
(r)
k+l(x, y;u, λ) =

k,l∑
n,m=0

(
k

n

)(
l

m

)
(x− z)n+m

BTk+l−n−m(z, y;u, λ). (24)

The next result provides the difference when x in BT
(r)
n (x, y;u, λ) is shifted by 1.

Theorem 7. For n ≥ 1, the difference BT
(r)
n (x+ 1, y;u, λ)−B T

(r)
n (x, y;u, λ) is given by

the difference formula

BT
(r)
n (x+ 1, y;u, λ)− BT

(r)
n (x, y;u, λ) =

n−1∑
k=0

(
n

k

)
BT

(r)
k (x, y;u, λ). (25)

Proof.

∞∑
n=0

BT
(r)
n (x+ 1, y;u, λ)

tn

n!
−

∞∑
n=0

BT
(r)
n (x, y;u, λ)

tn

n!

=

(
(1− u)

λe2t − u

)r

e(x+1)t+y(et−1) −
(

(1− u)

λe2t − u

)r

ext+y(et−1)



J. Ontolan et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5559 11 of 16

=

(
(1− u)

λe2t − u

)r

ext+y(et−1)(et − 1)

=

( ∞∑
n=0

BT
(r)
n (x, y;u, λ)

tn

n!

)∑
n≥0

tn+1

(n+ 1)!


=

∞∑
n=0

{
n∑

k=0

(
n+ 1

k

)
BT

(r)
k (x, y;u, λ)

}
tn+1

(n+ 1)!

=
∞∑
n=1

{
n−1∑
k=0

(
n

k

)
BT

(r)
k (x, y;u, λ)

}
tn

n!
.

Comparing coefficients,

BT
(r)
n (x+ 1, y;u, λ)− BT

(r)
k (x, y;u, λ) =

n−1∑
k=0

(
n

k

)
BT

(r)
k (x, y;u, λ)

as desired.

Stirling Number of Second Kind and Bivariate Bell Polynomials

In this subsection, we derive some formulas displaying relationship of BT
(r)
n (x, y;u, λ)

with the Stirling numbers of second kind and bivariate Bell polynomials.

Theorem 8. The bivariate Bell-based Apostol-Frobenius-Type Tangent polynomials of

higher order BT
(r)
n (x, y;u, λ) satisfy the summation formula

BT
(r)
n (x, y;u, λ) =

n∑
k=0

k∑
j=0

(
n

k

)
(x)jS(k, j)BT

(r)
n−k(y;u, λ). (26)

Proof. Using (5), we write

∞∑
n=0

BT
(r)
n (x, y;u, λ)

tn

n!
=

(
(1− u)

λe2t − u

)r

ext+y(et−1)

=

(
(1− u)

λe2t − u

)r

ey(e
t−1)ext

=

(
(1− u)

λe2t − u

)r

ey(e
t−1)(1 + et − 1)x

=

( ∞∑
n=0

BT
(r)
n (y;u, λ)

tn

n!

) ∞∑
j=0

(
x

j

)
(et − 1)j





J. Ontolan et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5559 12 of 16

=

( ∞∑
n=0

BT
(r)
n (y;u, λ)

tn

n!

) ∞∑
j=0

x!

(x− j)!

(et − 1)j

j!


=

( ∞∑
n=0

BT
(r)
n (y;u, λ)

tn

n!

) ∞∑
j=0

(x)j
(et − 1)j

j!


=

( ∞∑
n=0

BT
(r)
n (y;u, λ)

tn

n!

) ∞∑
j=0

(x)j

∞∑
n=0

S(n, j)
tn

n!


=

( ∞∑
n=0

BT
(r)
n (y;u, λ)

tn

n!

) ∞∑
n=0


∞∑
j=0

(x)jS(n, j)

 tn

n!


=

∞∑
n=0

n∑
k=0

(
n

k

)
∞∑
j=0

(x)jS(k, j)BT
(r)
n−k(y;u, λ)

 tn

n!

=

∞∑
n=0


n∑

k=0

(
n

k

) ∞∑
j=0

(x)jS(k, j)BT
(r)
n−k(y;u, λ)

 tn

n!
.

Comparing coefficients of tn

n! we obtain,

BT
(r)
n (x, y;u, λ) =

n∑
k=0

∞∑
j=0

(
n

k

)
(x)jS(k, j)BT

(r)
n−k(y;u, λ)

=
n∑

k=0

k∑
j=0

(
n

k

)
(x)jS(k, j)BT

(r)
n−k(y;u, λ),

which proves the theorem.

The next result expresses the bivariate Bell polynomials in terms of bivariate Bell-based
Apostol-Frobenius-Type Tangent polynomials.

Theorem 9. The bivariate Bell polynomials follows the relation

Bn(x, y) =
λBTn(x+ 2, y;u, λ)− uBTn(x, y;u, λ)

(1− u)
. (27)

Proof. From (16), we can write

∞∑
n=0

Bn(x, y)
tn

n!
= ext+y(et−1)

=

(
λe2t − u

(1− u)

)(
(1− u)

λe2t − u
ext+y(et−1)

)



J. Ontolan et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5559 13 of 16

=
1

(1− u)

(
λ

(
(1− u)

λe2t − u
e(x+2)t+y(et−1)

)
− u

(
(1− u)

λe2t − u
ext+y(et−1)

))
=

1

(1− u)

(
λ

∞∑
n=0

BTn(x+ 2, y;u, λ)
tn

n!
− u

∞∑
n=0

BTn(x, y;u, λ)
tn

n!

)

=
λ

1− u

∞∑
n=−1

BTn(x+ 2, y;u, λ)
tn

n!
− u

1− u

∞∑
n=−1

BTn(x, y;u, λ)
tn

n!
.

Comparing coefficients,

Bn(x, y) =
λBTn(x+ 2, y;u, λ)− uBTn(x, y;u, λ)

(1− u)
.

Derivative Formulas

Derivative formulas for special polynomials are fundamental tools in mathematics
and its applications to physics, engineering, and other scientific fields. These formulas
facilitate the analysis of the behavior and properties of special polynomials by quanti-
fying their rates of change, a central aspect in calculus and mathematical analysis for
understanding function dynamics. Moreover, they are instrumental in the manipulation
of generating functions, which encode sequences of polynomial coefficients. Generating
functions, in turn, play a crucial role in combinatorics, number theory, and discrete math-
ematics, particularly for problems involving counting and enumeration.

The next theorem contains the derivative formula for BT
(r)
n (x, y;u, λ) with respect to

the variable x.

Theorem 10. The following derivative formula holds

∂

∂x
BT

(r)
n (x, y;u, λ) = nBT

(r)
n−1(x, y;u, λ). (28)

Proof. Applying ∂
∂y to (5),

∂

∂x

∞∑
n=0

BT
(r)
n (x, y;u, λ)

tn

n!
=

∂

∂x

(
(1− u)

λe2t − u

)r

ext+y(et−1)

∞∑
n=0

∂

∂x
BT

(r)
n (x, y;u, λ)

tn

n!
=

(
(1− u)

λe2t − u

)r

ext+y(et−1) t

= t
∞∑
n=0

BT
(r)
n (x, y;u, λ)

tn

n!

=
∞∑
n=0

BT
(r)
n (x, y;u, λ)

tn+1

n!



J. Ontolan et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5559 14 of 16

=

∞∑
n=1

nBT
(r)
n−1(x, y;u, λ)

tn

n!
.

Consequently,

∂

∂x
BT

(r)
n (x, y;u, λ) = nBT

(r)
n−1(x, y;u, λ).

Remark 2. The relation in (28) shows that the sequence of polynomials BT
(r)
n (x, y;u, λ)

satisfy (9), thus BT
(r)
n (x, y;u, λ) is a sequence of Appell polynomials. The polynomials

BT
(r)
n (x, y;u, λ) are anticipated to exhibit the following characteristics:

(1) Equation (5) reflects (10), that is,(
1− u

λe2t − u

)r

ey(e
t−1)ext =

∞∑
n=0

BT
(r)
n (x, y;u, λ)

tn

n!

where A(t) =
(

1−u
λe2t−u

)r
ey(e

t−1) is independent of x with A(0) ̸= 0.

(2) Result in (17) demonstrates (11) and (12),

BT
(r)
n (x, y;u, λ) =

n∑
j=0

(
n

j

)
cjx

n−j

BT
(r)
n (x, y;u, λ) =

 n∑
j=0

cj
j!
Dj

xn

where cj = BT
(r)
j (y;u, λ) and D = d

dx .

The last result shows the derivative of BT
(r)
n (x, y;u, λ) with respect to y.

Theorem 11. The derivative formula given by

∂

∂y
BT

(r)
n (x, y;u, λ) = BT

(r)
n (x+ 1, y;u, λ)− BT

(r)
n (x, y;u, λ) (29)

holds for BT
(r)
n (x, y;u, λ).

Proof. Applying ∂
∂x to both sides of (5)

∞∑
n=0

∂

∂y
BT

(r)
n (x, y;u, λ)

tn

n!
=

(
(1− u)

λe2t − u

)r

ext+y(et−1)(et − 1)



J. Ontolan et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5559 15 of 16

=

(
(1− u)

λe2t − u

)r

e(x+1)t+y(et−1) −
(

(1− u)

λe2t − u

)r

ext+y(et−1)

=
∞∑
n=0

BT
(r)
n (x+ 1, y;u, λ)

tn

n!
−

∞∑
n=0

BT
(r)
n (x, y;u, λ)

tn

n!

=

∞∑
n=0

{
BT

(r)
n (x+ 1, y;u, λ)− BT

(r)
n (x, y;u, λ)

} tn

n!

∂

∂y
BT

(r)
n (x, y;u, λ) = BT

(r)
n (x+ 1, y;u, λ)− BT

(r)
n (x, y;u, λ).

Remark 3. Combining the results from (25) and (29), we obtain the equation

∂

∂y
BT

(r)
n (x, y;u, λ) =

n−1∑
k=0

(
n

k

)
BT

(r)
k (x, y;u, λ). (30)

To see this, consider the example below. We use (5) to get the following polynomials

BT
(r)
0 (x, y;u, λ) =

(
u− 1

u− λ

)r

,

BT
(r)
1 (x, y;u, λ) =

(
u−1
u−λ

)r
(λ(2r − x− y) + u(x+ y))

u− λ
,

BT
(r)
2 (x, y;u, λ) = (x+ y)2

(
1− u

λ− u

)r

+ y

(
1− u

λ− u

)r

+
8λ2r(1− u)

(
1−u
λ−u

)r−1

(λ− u)3
+

4λ2(r − 1)r(1− u)2
(

1−u
λ−u

)r−2

(λ− u)4

−
2λr(1− u)(x+ y)

(
1−u
λ−u

)r−1

(λ− u)2
−

2λr(1− u)(x+ y + 2)
(

1−u
λ−u

)r−1

(λ− u)2
.

One can verify using the above polynomials that

∂

∂y
BT

(r)
2 (x, y;u, λ) =

(
2

0

)
BT

(r)
0 (x, y;u, λ) +

(
2

1

)
BT

(r)
1 (x, y;u, λ).

Acknowledgements

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



J. Ontolan et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5559 16 of 16

References

[1] Paul Appell. Sur une classe de polynômes. In Annales scientifiques de l’École normale
supérieure, volume 9, pages 119–144, 1880.

[2] Serkan Araci, Mehmet Acikgoz, and E Sen. A note on the p-adic interpolation function
for multiple generalized genocchi numbers. Turkish Journal of Analysis and Number
Theory, 1(1):17–22, 2013.

[3] Theodore S Chihara. An introduction to orthogonal polynomials. Courier Corporation,
2011.

[4] CB Corcino, B Damgo, and RB Corcino. Fourier expansions for genocchi polynomials
of higher order. J. Math. Comput. Sci, 22:59–72, 2020.

[5] G. Dattoli, S. Lorenzutta, P.E. Ricci, and C. Cesarano. On a family of hybrid poly-
nomials. Integral Transforms and Special Functions, 15(6):485–490, 2004.

[6] W.A. Khan and M. Riaz. Some subclasses of apostol-type polynomials and their
properties. International Journal of Mathematical Analysis, 21(69):3457–3470, 2022.

[7] V. Kurt. New families of polynomials associated with the bell numbers and polyno-
mials. Carpathian Mathematical Publications, 14(2):354–363, 2007.

[8] Q.M. Luo and H.M. Srivastava. Some generalizations of apostol-bernoulli and apostol-
euler polynomials. Journal of Mathematical Analysis and Applications, 308(1):290–
302, 2005.

[9] W. Ramı́rez and C. Cesarano. Some new classes of degenerated generalized apostol-
bernoulli, apostol-euler and apostol-genocchi polynomials. Carpathian Mathematical
Publications, 14(2):354–363, 2022.

[10] W. Ramı́rez, C. Cesarano, and S. Dı́az. New results for degenerated generalized apos-
tol–bernoulli, apostol–euler and apostol–genocchi polynomials. WSEAS Transactions
on Mathematics, 21:604–608, 2022.

[11] CS Ryoo. A note on the tangent numbers and polynomials. Adv. Studies Theor.
Phys, 7(9):447–454, 2013.

[12] CS Ryoo. On the analogues of tangent numbers and polynomials associated with
p-adic integral on zp. Applied Mathematical Sciences, 7(64):3177–3183, 2013.

[13] CS Ryoo. A numerical investigation on the zeros of the tangent polynomials. Journal
of applied mathematics & informatics, 32(3 4):315–322, 2014.

[14] CS Ryoo. Differential equations associated with tangent numbers. Journal of applied
mathematics & informatics, 34(5 6):487–494, 2016.

[15] James Shohat. The relation of the classical orthogonal polynomials to the polynomials
of appell. American Journal of Mathematics, 58(3):453–464, 1936.

[16] G. Szegő. Orthogonal Polynomials. American Math. Soc: Colloquium publ. American
Mathematical Society, 1975.