EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
Vol. 15, No. 4, 2022, 1549-1565
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

Generalized Laguerre-Apostol-Frobenius-Type
Poly-Genocchi Polynomials of Higher Order with

Parameters a, b and c

Roberto B. Corcino1,2,∗, 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. In this paper, the generalized Laguerre-Apostol-Frobenius-type poly-Genocchi polyno-
mials of higher order with parameters a, b and c are defined using the concept of polylogarithm,
Laguerre, Apostol and Frobenius polynomials. These polynomials possess numerous properties
including recurrence relations, explicit formulas and certain differential identity. Moreover, some
connections of these higher order generalized Laguerre-Apostol-Frobenius-type poly-Genocchi poly-
nomials to Stirling numbers of the second kind and different variations of higher order Euler and
Bernoulli-type polynomials are obtained.
2020 Mathematics Subject Classifications: 11B68, 11B73, 05A15
Key Words and Phrases: Poly-Genocchi polynomials, Laguerre polynomials, Apostol polyno-
mials, Frobenius numbers, polylogarithm function, Appell polynomials, Euler polynomials, Bernoulli
polynomials

1. Introduction

The Genocchi numbers Gn are defined by means of the following generating function

∞∑
n=0

Gn
tn

n!
=

2t

et + 1
, |t| < π.

These numbers have been generalized in different ways [2, 3, 8, 9, 11, 14, 19, 21–23, 26–28].
Most of the generalizations are done by mixing the Genocchi numbers with the concept
of some known polynomials. For instance, mixing with exponential polynomials yields
the Genocchi polynomials and Genocchi polynomials of higher order, which are given as
follows:

∞∑
n=0

Gn(x)
tn

n!
=

2t

et + 1
ext, |t| < π, (1)

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v15i4.4505

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

https://www.ejpam.com 1549 © 2022 EJPAM All rights reserved.



R. Corcino, C. Corcino / Eur. J. Pure Appl. Math, 15 (4) (2022), 1549-1565 1550

∞∑
n=0

G(k)
n (x)

tn

n!
=

(
2t

et + 1

)k

ext. (2)

These polynomials are well-studied and two of the most recent studies are the works of
Corcino-Corcino [5, 7] on asymptotic approximations.

Now, mixing with the Apostol polynomials yields the Apostol-Genocchi polynomi-
als, and Apostol-Genocchi polynomials of higher order, which are respectively defined as
follows:

∞∑
n=0

Gn(x, λ)
tn

n!
=

2t

λet + 1
ext, (3)

∞∑
n=0

G(k)
n (x, λ)

tn

n!
=

(
2t

λet + 1

)k

ext, (4)

where |t| < π when λ = 1 and |t| < log(−λ) when λ ̸= 1, λ ∈ C. These polynomials were
given Fourier series expansion in [6]. Also, mixing with Frobenius polynomials yields the
so-called Frobenius-Genocchi polynomials, which are given by

∞∑
n=0

GF
n (x;u)

tn

n!
=

(1− u)t

et − u
ext, (5)

(see [3, 11–13, 22, 25, 27, 28]). Moreover, mixing the Genocchi numbers with the concept
of polylogarithm Lik(z)

Lik(z) =

∞∑
n=1

zn

nk
, k ∈ Z, (6)

yields the poly-Genocchi polynomials, which are defined as follows

∞∑
n=0

G(k)
n (x)

xn

n!
=

2Lik(1− et)

et + 1
ext. (7)

Furthermore, with a slight modification of the generating function, another generalization,

denoted by G
(k)
n,2(x), was defined by Kim et al. [26] as follows

∞∑
n=0

G
(k)
n,2(x)

xn

n!
=

Lik(1− e−2t)

et + 1
ext. (8)

These polynomials are called modified poly-Genocchi polynomials. Note that, when k = 1,
equations (7) and (8) give the Genocchi polynomials in (1). That is,

G(1)
n (x) = G

(1)
n,2(x) = Gn(x).

Kim et. al [26] obtained several properties of these polynomials.



R. Corcino, C. Corcino / Eur. J. Pure Appl. Math, 15 (4) (2022), 1549-1565 1551

Kurt [14] defined two forms of generalized poly-Genocchi polynomials with parameters
a, b, and c as follows

2Lik(1− (ab)−t)

a−t + bt
ext =

∞∑
n=0

G(k)
n (x; a, b, c)

xn

n!
(9)

2Lik(1− (ab)−2t)

a−t + bt
ext =

∞∑
n=0

G
(k)
n,2(x; a, b, c)

xn

n!
. (10)

These were motivated by the generalizations introduced in (7) and (8), respectively. Note
that, when x = 0, (7) reduces to

2Lik(1− et)

et + 1
=

∞∑
n=0

G(k)
n

xn

n!
, (11)

where G
(k)
n are called the poly-Genocchi numbers. Recently, a new variation of poly-

Genocchi polynomials with parameters a, b and c was defined in [8] by mixing the defi-
nitions of Apostol and Frobenius polynomials, namely, the Apostol-Frobenius-type poly-
Genocchi polynomials of higher order with parameters a, b and c. More precisely, the said
polynomials, denoted by

∞∑
n=0

G(k,α)
n (x;λ, u, a, b, c)

tn

n!
=

(
Lik(1− (ab)−(1−u)t)

λbt − ua−t

)α

cxt. (12)

It is worth-mentioning that, using multi-polylogarithm, the generalized poly-Genocchi
polynomials in (9) and (10) have been extended further in [19].

On the other hand, a generalization of Laguerre polynomials, denoted by Ln(x, y), was
defined in [10] by means of the following generating function

eytC0(xt) =
∞∑
n=0

Ln(x, y)
tn

n!
, (13)

where C0(x) is the 0− th order Tricomi function [20]

C0(x) =
∞∑
r=0

(−1)rxr

(r!)2
, C0(0) := 1. (14)

This 2-variable generalization of Laguerre polynomials possessed the following explicit
formula

Ln(x, y) =
n∑

s=0

n!(−1)syn−sxs

(n− s)!(s!)2
.

Also, the 2-variable generalization of Hermite polynomials were defined by Kampe de
Feriet [1] as follows

ext+yt2 =
∞∑
n=0

Hn(x, y)
tn

n!
, (15)



R. Corcino, C. Corcino / Eur. J. Pure Appl. Math, 15 (4) (2022), 1549-1565 1552

which reduces to the ordinary Hermite polynomials Hn(x) when taking y = −1 and x is
replaced by 2x. These generalized Hermite polynomials possessed the following explicit
formula

Hn(x, y) = n!
n∑

r=0

yrxn−2r

r!(n− 2r)!
.

This can further be generalized through the following polynomials, denoted byHn,L(x;u, v)
as follows

evt+wt2C0(xt) =

∞∑
n=0

Hn,L(x;u, v)
tn

n!
. (16)

We call these polynomials as generalized Laguerre-Hermite polynomials.

In this paper, a new variation of poly-Genocchi polynomials with parameters a, b
and c is constructed by mixing the concepts of Laguerre, Apostol and Frobenius poly-
nomials. These polynomials are called the generalized Laguerre-Apostol-Frobenius-type
poly-Genocchi polynomials of higher order with parameters a, b and c. Some special cases
of these polynomials are enumerated and some identities that contain a number of relations
of this new variation with some Genocchi-type polynomials are provided. One section of
the paper devotes its discussion on some identities that link the generalized Laguerre-
Apostol-Frobenius-type poly-Genocchi polynomials of higher order with parameters a, b
and c to Appell polynomials. Finally, some connections of these higher order generalized
Laguerre-Apostol-Frobenius-type poly-Genocchi polynomials to Stirling numbers of the
second kind and different variations of higher order Euler and Bernoulli-type polynomials
are discussed.

2. Definition and Some Preliminary Results

Let us formally define the generalized Laguerre-Apostol-Frobenius-type poly-Genocchi
polynomials of higher order with parameters a, b and c.

Definition 2.1. The Generalized Laguerre-Apostol-Frobenius-type poly-Genocchi poly-

nomials of higher order with parameters a, b and c, denoted by G(k,α)
n (x;λ, u, v, w, a, b, c),

are defined as coefficients of the following generating function:

∞∑
n=0

G(k,α)
n,L (x;λ, u, v, w, a, b, c)

tn

n!
=

(
Lik(1− (ab)−(1−u)t)

λbt − ua−t

)α

cvt+wt2C0(xt), (17)

where |t| <
√

(ln(λ
u))2+4π2

| ln a+ln b| .

Now, let us consider some preliminary results of this paper. It is important to note
that, using (15),

cxt+yt2 = cxtcyt
2
= ext ln ceyt

2 ln c = e(x ln c)te(y ln c)t2



R. Corcino, C. Corcino / Eur. J. Pure Appl. Math, 15 (4) (2022), 1549-1565 1553

=

∞∑
n=0

Hn(x ln c, y ln c)
tn

n!
.

By making use of (17), we can easily establish the following relation.

Theorem 2.2. The generalized Laguerre-Apostol-Frobenius-type poly-Genocchi polynomi-
als of higher order with parameters a, b, c satisfy the relation

G(k,α)
n,L (x;λ, u, v + y, w + z, a, b, c)

=

n∑
m=0

(
n

m

)
G(k,α)
n−m,L(x;λ, u, v, w, a, b, c)Hm(y ln c, z ln c). (18)

Proof. We can write (17) as follows:

∞∑
n=0

G(k,α)
n,L (x;λ, u, v + y, w + z, a, b, c)

tn

n!

=

(
Lik(1− (ab)−(1−u)t)

λbt − ua−t

)α

cvt+wt2C0(xt)c
yt+zt2

=

( ∞∑
n=0

G(k,α)
n,L (x;λ, u, v, w, a, b, c)

tn

n!

)( ∞∑
n=0

Hn(y ln c, z ln c)
tn

n!

)

=
∞∑
n=0

(
n∑

m=0

(
n

m

)
G(k,α)
n−m,L(x;λ, u, v, w, a, b, c)Hm(y ln c, z ln c)

)
tn

n!

Comparing the coefficients of tn

n! completes the proof of the theorem.

The next result is a kind of addition formula for G(k,α)
n (x;λ, u, v, w, a, b, c).

Theorem 2.3. The generalized Laguerre-Apostol-Frobenius-type poly-Genocchi polynomi-
als of higher order with parameters a, b, c satisfy the relation

G(k,α)
n,L (x;λ, u, v + y, w, a, b, c)

=
n∑

m=0

(
n

m

)
(y ln c)mG(k,α)

n−m,L(x;λ, u, v, w, a, b, c). (19)

Proof. We can write (17) as follows:

∞∑
n=0

G(k,α)
n,L (x;λ, u, v + y, w, a, b, c)

tn

n!

=

(
Lik(1− (ab)−(1−u)t)

λbt − ua−t

)α

cvt+wt2C0(xt)e
yt ln c



R. Corcino, C. Corcino / Eur. J. Pure Appl. Math, 15 (4) (2022), 1549-1565 1554

=

( ∞∑
n=0

G(k,α)
n,L (x;λ, u, v, w, a, b, c)

tn

n!

)( ∞∑
n=0

(yt ln c)n

n!

)

=
∞∑
n=0

(
n∑

m=0

(
n

m

)
G(k,α)
n−m,L(x;λ, u, v, w, a, b, c)(y ln c)

m

)
tn

n!

Comparing the coefficients of tn

n! completes the proof of the theorem.

By giving special values to the parameters involved, the polynomials G(k,α)
n (x;λ, u, v, w, a, b, c)

reduce to some interesting Genocchi-type polynomials.

(i) When c = e, equation (17) reduces to

∞∑
n=0

G(k,α)
n,L (x;λ, u, v, w, a, b, e)

tn

n!
=

(
Lik(1− (ab)−(1−u)t)

λbt − ua−t

)α

evt+wt2C0(xt). (20)

For convenience, we use G(k,α)
n,L (x;λ, u, v, w, a, b) to denote G(k,α)

n,L (x;λ, u, v, w, a, b, e).
That is,

∞∑
n=0

G(k,α)
n,L (x;λ, u, v, w, a, b)

tn

n!
=

(
Lik(1− (ab)−(1−u)t)

λbt − ua−t

)α

evt+wt2C0(xt). (21)

(ii) When a = 1, b = e, (21) will reduce to

∞∑
n=0

G(k,α)
n,L (x;λ, u, v, w, 1, e)

tn

n!
=

(
Lik(1− e−(1−u)t)

λet − u

)α

evt+wt2C0(xt). (22)

We may use the notations

G(k,α)
n,L (x;λ, u, v, w) = G(k,α)

n,L (x;λ, u, v, w, 1, e)

G(k)
n,L(x;λ, u, v, w) = G(k)

n,L(x;λ, u, v, w, 1, e)

and call them generalized Laguerre-Apostol-Frobenius-type poly-Genocchi polyno-
mials of higher order and generalized Laguerre-Apostol-Frobenius-type poly-Genocchi
polynomials, respectively.

(iii) When λ = 1, (22) gives

∞∑
n=0

G(k,α)
n,L (x;u, v, w, 1, e)

tn

n!
=

(
Lik(1− e−(1−u)t)

et − u

)α

evt+wt2C0(xt). (23)

which is the higher order version of equation (8) and are called the higher or-

der Laguerre-poly-Genocchi polynomials. We may use G(k,α)
n,L (x;u, v, w) to denote

G(k,α)
n,L (x;u, v, w, 1, e).



R. Corcino, C. Corcino / Eur. J. Pure Appl. Math, 15 (4) (2022), 1549-1565 1555

(iv) When k = 1, (22) gives

∞∑
n=0

G(1,α)
n,L (x;λ, u, v, w)

tn

n!
=

(
(1− u)t

λet − u

)α

evt+wt2C0(xt), (24)

and when λ = 1, (24) gives

∞∑
n=0

G(1,α)
n,L (x; 1, u, v, w)

tn

n!
=

(
(1− u)t

et − u

)α

evt+wt2C0(xt),

where G(1,α)
n,L (x;λ, u, v, w) = G(α)

n,L(x;λ, u, v, w) and G(1,α)
n,L (x; 1, u, v, w) = G(α)

n,L(x;u, v, w)
are called the Generalized Laguerre-Apostol-Frobenius-type Genocchi polynomials
and Laguerre-Frobenius-Genocchi polynomials of higher order in (4) and (2), respec-
tively. Furthermore, when α = 1, we have

∞∑
n=0

Gn,L(x;λ, u, v, w)
tn

n!
=

(1− u)t

λet − u
evt+wt2C0(xt), (25)

and
∞∑
n=0

Gn,L(x;u, v, w)
tn

n!
=

(1− u)t

et − u
evt+wt2C0(xt),

where Gn,L(x;λ, u, v, w) and Gn,L(x;u, v, w) are called the Generalized Laguerre-
Apostol-Frobenius-type Genocchi polynomials and Laguerre-Frobenius-Genocchi poly-
nomials in (4) and (2), respectively.

(v) When v = w = 0, (17) reduces to

∞∑
n=0

G(k,α)
n,L (x;λ, u, a, b, c)

tn

n!
= C0(xt)

(
Lik(1− (ab)−(1−u)t)

λbt − ua−t

)α

. (26)

Consider a special case of (21) by taking x = 0. This gives

∞∑
n=0

G(k,α)
n,L (0;λ, u, v, w, a, b, c)

tn

n!
=

(
Lik(1− (ab)−(1−u)t)

λbt − ua−t

)α

cvt+wt2 . (27)

We use the notation G(k,α)
n,L (λ, u, v, w, a, b, c) = G(k,α)

n,L (0;λ, u, v, w, a, b, c) and call them
the generalized Laguerre-Apostol-Frobenius-type poly-Genocchi numbers of higher order
with parameters a, b and c. The following theorem contains an identity that expresses

G(k,α)
n,L (x;λ, u, v, w, a, b, c) as polynomial in x with G(k,α)

n,L (λ, u, v, w, a, b, c) as coefficients.

Theorem 2.4. The generalized Laguerre-Apostol-Frobenius-type poly-Genocchi polynomi-
als of higher order with parameters a, b, c satisfy the relation,

G(k,α)
n,L (x;λ, u, v, w, a, b, c) =

n∑
i=0

(
n

i

)
(−1)i

i!
G(k,α)
n−i,L(λ, u, v, w, a, b, c)x

i. (28)



R. Corcino, C. Corcino / Eur. J. Pure Appl. Math, 15 (4) (2022), 1549-1565 1556

Proof. Equation (17) can be written as

∞∑
n=0

G(k,α)
n,L (x;λ, u, v, w, a, b, c)

tn

n!
=

(
Lik(1− (ab)−(1−u)t)

λbt − ua−t

)α

cvt+wt2C0(xt)

=

( ∞∑
n=0

G(k,α)
n,L (λ, u, v, w, a, b, c)

tn

n!

)( ∞∑
n=0

(−1)n(xt)n

(n!)2

)

=
∞∑
n=0

n∑
i=0

(−xt)n−i

[(n− i)!]2
G(k,α)
i,L (λ, u, v, w, a, b, c)

ti

i!

=
∞∑
n=0

(
n∑

i=0

(
n

i

)
(−1)n−i

(n− i)!
G(k,α)
i,L (λ, u, v, w, a, b, c)xn−i

)
tn

n!

Comparing the coefficients of tn

n! , we obtain the desired result.

The next identity gives the relation between

G(k,α)
n,L (x;λ, u, v, w, a, b, c)) and G(k,α)

n,L (x;λ, u, v, w).

Theorem 2.5. The generalized Laguerre-Apostol-Frobenius-type poly-Genocchi polynomi-
als of higher order with parameters a, b, c satisfy the relation,

G(k,α)
n,L (x;λ, u, v, w, a, b, c) = (ln ab)nG(k,α)

n,L

(
x

ln ab
;λ, u,

v ln c+ α ln a

ln ab
,
w ln c

(ln ab)2

)
. (29)

Proof. Using (17), we have

∞∑
n=0

G(k,α)
n,L (x;λ, u, v, w, a, b, c)

tn

n!

=

(
Lik(1− (ab)−(1−u)t)

a−t(λ(ab)t − u)

)α

e
v ln c
ln ab

t ln ab+ w ln c
(ln ab)2

(t ln ab)2
C0

( x

ln ab
t ln ab

)
=

(
Lik(1− e−(1−u)t ln ab)

λet ln ab − u

)α

e
v ln c+α ln a

ln ab
t ln ab+ w ln c

(ln ab)2
(t ln ab)2

C0

( x

ln ab
t ln ab

)
=

∞∑
n=0

(ln ab)nG(k,α)
n,L

(
x

ln ab
;λ, u,

v ln c+ α ln a

ln ab
,
w ln c

(ln ab)2

)
tn

n!
.

Comparing the coefficients of tn

n! , we obtain the desired result.

3. A Differential Identity and Its Consequences

In this section, we consider G(k,α)
n,L (x;λ, u, v, w, a, b, c) as polynomial in v. Now, applying

the first derivative to equation (17) with respect to v yields

∞∑
n=0

d

dv
G(k,α)
n,L (x;λ, u, v, w, a, b, c)

tn

n!
= t(ln c)

(
Lik(1− (ab)−(1−u)t)

(λbt − ua−t)

)α

evt ln c+wt2 ln cC0(xt)



R. Corcino, C. Corcino / Eur. J. Pure Appl. Math, 15 (4) (2022), 1549-1565 1557

∞∑
n=0

d

dv
G(k,α)
n,L (x;λ, u, v, w, a, b, c)

tn−1

n!
=

∞∑
n=0

(ln c)G(k,α)
n,L (x;λ, u, v, w, a, b, c)

tn

n!
.

It follows that

∞∑
n=0

1

n+ 1

d

dv
G(k,α)
n+1,L(x;λ, u, v, w, a, b, c)

tn

n!
=

∞∑
n=0

(ln c)G(k,α)
n,L (x;λ, u, v, w, a, b, c)

tn

n!
.

Comparing the coefficients of tn

n! yields the following differential identity, which can be
used to classify generalized Laguerre-Apostol-type poly-Genocchi polynomials of higher
order as Appell polynomials [15, 16, 24].

Theorem 3.1. The generalized Laguerre-Apostol-Frobenius-type poly-Genocchi polynomi-
als with parameters a, b, c satisfy the relation,

d

dv
G(k,α)
n+1,L(x;λ, u, v, w, a, b, c) = (n+ 1)(ln c)G(k,α)

n,L (x;λ, u, v, w, a, b, c). (30)

Remark 3.1. When c = e, equation (30) reduces to

d

dv
G(k,α)
n+1,L(x;λ, u, v, w, a, b) = (n+ 1)G(k,α)

n,L (x;λ, u, v, w, a, b), (31)

where G(k,α)
n,L (x;λ, u, v, w, a, b) is the generalized Laguerre-Apostol-Frobenius-type poly-

Genocchi polynomials in (21). Consequently, this makes G(k,α)
n,L (x;λ, u, v, w, a, b) an Appell

polynomial.

Being classified as Appell polynomials, the generalized Laguerre-Apostol-Frobenius-

type poly-Genocchi polynomials G(k,α)
n,L (x;λ, u, v, w, a, b) must possess the following prop-

erties

G(k,α)
n,L (x;λ, u, v, w, a, b) =

n∑
i=0

(
n

i

)
cix

n−i

G(k,α)
n,L (x;λ, u, v, w, a, b) =

( ∞∑
i=0

ci
i!
Di

)
xn,

for some scalar ci ̸= 0. It is then necessary to find the sequence {cn}. However, by using

(28) with c = e, ci = G(k,α)
i,L (λ, u, v, w, a, b). This implies the following corollary.

Corollary 3.2. The generalized Apostol-Frobenius-type poly-Genocchi polynomials with
parameters a, b, c satisfy the formula,

G(k,α)
n,L (x;λ, u, v, w, a, b) =

 ∞∑
i=0

G(k,α)
i,L (λ, u, v, w, a, b)

i!
Di

xn. (32)



R. Corcino, C. Corcino / Eur. J. Pure Appl. Math, 15 (4) (2022), 1549-1565 1558

In particular, when n = 3, (32) gives

G(k,α)
3,L (x;λ, u, v, w, a, b) =

 ∞∑
i=0

G(k,α)
i,L (λ, u, v, w, a, b)

i!
Di

x3

=
G(k,α)
0,L (λ, u, v, w, a, b)

0!
x3 +

G(k,α)
1,L (λ, u, v, w, a, b)

1!
D1x3

+
G(k,α)
2,L (λ, u, v, w, a, b)

2!
D2x3 +

G(k,α)
3,L (λ, u, v, w, a, b)

3!
D3x3

= G(k,α)
0,L (λ, u, v, w, a, b)x3 + 3G(k,α)

1,L (λ, u, v, w, a, b)x2 + 3G(k,α)
2,L (λ, u, v, w, a, b)x

+ G(k,α)
3,L (λ, u, v, w, a, b).

The next corollary immediately follows from equation (31) and the characterization of
Appell polynomials [15, 16, 24].

Corollary 3.3. The generalized Laguerre-Apostol-Frobenius-type poly-Genocchi polyno-
mials with parameters a, b, c satisfy the addition formula

G(k,α)
n,L (x;λ, u, v + y, w, a, b) =

∞∑
i=0

(
n

i

)
G(k,α)
i,L (x;λ, u, v, w, a, b)yn−i. (33)

Remark 3.2. Corollary 3.3 can also be deduced immediately from Theorem 2.3 by taking
c = e.

4. Connections with Some Special Numbers and Polynomials

In this section, some connections of the higher order generalized Laguerre-Apostol-type

poly-Genocchi polynomials G(k,α)
n (x;λ, a, b, c) with other well-known special numbers and

polynomials will be established.

Recently, Pathan [17, 18] defined the generalized Hermite-Bernoulli polynomials of two

variables, denoted by B
(s)
n,H(v, w), as follows:(

t

et − 1

)s

evt+wt2 =

∞∑
n=0

B
(s)
n,H(v, w)

tn

n!
. (34)

When w = 0, these polynomials simply reduce to Bernoulli polynomials of order s. Here,
we define the generalized Hermite-Apostol-type Frobenius-Euler polynomials, denoted by

E
(s)
n,H(µ, v, w, λ), as follows(

1− µ

λet − µ

)s

evt+wt2 =
∞∑
n=0

E
(s)
n,H(µ, v, w, λ)

tn

n!
. (35)



R. Corcino, C. Corcino / Eur. J. Pure Appl. Math, 15 (4) (2022), 1549-1565 1559

When s = 1, w = 0, (35) gives E
(s)
n,H(µ, v, 0, λ), the Apostol-type Frobenius-Euler poly-

nomials in [23]. Now, if λ = 0, we can define the generalized Hermite-Frobenius-Euler

polynomials, denoted by E
(s)
n,H(µ, v, w), as follows:(
1− µ

et − µ

)s

evt+wt2 =

∞∑
n=0

E
(s)
n,H(µ, v, w)

tn

n!
. (36)

The following theorem contains an identity that relates the generalized Laguerre-
Apostol-Frobenius-type poly-Genocchi polynomials of higher order with parameters a, b

and c to Stirling numbers of the second kind

{
n

m

}
defined in [4] by

∞∑
n=m

{
n

m

}
tn

n!
=

(et − 1)m

m!
. (37)

Here, it is important to note that if (c0, c1, . . . , cj , . . .) is any sequence of numbers and l is
a positive integer, then ∞∑

j=0

cj
tj

j!

l

=
l∏

i=1

( ∞∑
ni=0

cni

ni!
tni

)

=

∞∑
n=0

{ ∑
n1+n2+...+nα=n

l∏
i=1

cni

(
n

n1, n2, . . . , nα

)}
tn

n!
. (38)

(see [4]). Now, we are ready to introduce the following theorem.

Theorem 4.1. The generalized Laguerre-Apostol-Frobenius-type poly-Genocchi polynomi-
als of higher order with parameters a, b and c satisfies the relation,

G(k,α)
n,L (x;λ, u, v, w, a, b, c)

=

n∑
j=0

(
n

j

)
(−1)α(ln ab)n−jG(1,α)

n−j,L

(
x

ln ab
;λ, u,

v ln c+ α ln a

ln ab
,
w ln c

(ln ab)2

)
dj (39)

where

dj =
∑

n1+n2+...+nα=j

α∏
i=1

cni

(
j

n1, n2, . . . , nα

)

cj =

j∑
m=0

(−1)m+j+1

((1− u) ln ab)jm!

{
j + 1

m+ 1

}
(j + 1)(m+ 1)k−1

.



R. Corcino, C. Corcino / Eur. J. Pure Appl. Math, 15 (4) (2022), 1549-1565 1560

Proof. Now, (17) can be written as

∞∑
n=0

G(k,α)
n,L (x;λ, u, v, w, a, b, c)

tn

n!
=

cvt+wt2C0(xt)

(λbt − ua−t)α

( ∞∑
m=1

(1− e−(1−u)t ln ab)m

mk

)α

=
cvt+wt2C0(xt)

(λbt − ua−t)α

( ∞∑
m=0

(1− e−(1−u)t ln ab)m+1

(m+ 1)k

)α

=
cvt+wt2C0(xt)

(λbt − ua−t)α

( ∞∑
m=0

m!

(m+ 1)k−1

(1− e−(1−u)t ln ab)m+1

(m+ 1)!

)α

=
cvt+wt2C0(xt)

(λbt − ua−t)α

 ∞∑
m=0

(−1)m+1m!

(m+ 1)k−1

∞∑
j=m+1

{
j

m+ 1

}
(−(1− u)t ln ab)j

j!

α

= (−1)αcvt+wt2C0(xt)

(
(1− u)t ln ab

λbt − ua−t

)α
 ∞∑

j=0

cj
tj

j!

α

,

where

cj =

j∑
m=0

(−1)m+j+1

((1− u) ln ab)jm!

{
j + 1

m+ 1

}
(j + 1)(m+ 1)k−1

.

Using the fact that Li1(z) = − ln(1− z), we get

Li1(1− (ab)−(1−u)t) = − ln(1− (1− (ab)−(1−u)t)) = (1− u)t ln ab.

Hence,

∞∑
n=0

G(k,α)
n,L (x;λ, u, v, w, a, b, c)

tn

n!
=

( ∞∑
n=0

(−1)αG(1,α)
n,L (x;λ, u, v, w, a, b, c)

tn

n!

) ∞∑
j=0

cj
tj

j!

α

.

Note that, using (38),
(∑∞

j=0 cj
tj

j!

)α
can be expressed as ∞∑

j=0

cj
tj

j!

α

=
∞∑
n=0

dn
tn

n!
,

where

dn =
∑

n1+n2+...+nα=n

α∏
i=1

cni

(
n

n1, n2, . . . , nα

)
.

It follows that

∞∑
n=0

G(k,α)
n,L (x;λ, u, v, w, a, b, c)

tn

n!
=

∞∑
n=0


n∑

j=0

(
n

j

)
(−1)αG(1,α)

n−j,L(x;λ, u, v, w, a, b, c)dj

 tn

n!
.



R. Corcino, C. Corcino / Eur. J. Pure Appl. Math, 15 (4) (2022), 1549-1565 1561

Comparing the coefficients and using equation (29) complete the proof of the theorem.

Remark 4.1. When α = 1, dj = cj .

The identities in the following theorem are derived using the fact that the polynomials

G(k,α)
n,L (x;λ, u, v, w, a, b, ) with parameters a and b satisfy the relation in (20).

Theorem 4.2. The generalized Laguerre-Apostol-type poly-Genocchi polynomials of higher

order G(k,α)
n,L (x;λ, u, v, w, a, b, c) with parameters a, b, c satisfy the following explicit formu-

las:

G(k,α)
n,L (x;λ, u, v, w, a, b, c)

=
n∑

l=0

n−l∑
m=0

(
n

l

){
l + s

s

}(n−l
m

)(
l+s
s

)G(k,α)
n−l−m,L(x;λ, u, a, b)B

(s)
m,H(v ln c, w ln c), (40)

G(k,α)
n,L (x;λ, u, v, w, a, b, c)

=

n∑
m=0

(
n
m

)
(1− µ)s

s∑
j=0

(
s

j

)
(−µ)s−jG(k,α)

n−m,L(x;λ, u, j, a, b)E
(s)
m,H(µ, v ln c, w ln c). (41)

Proof. Using (34), (17) may be expressed as

∞∑
n=0

G(k,α)
n,L (x;λ, u, v, w, a, b, c)

tn

n!

=

(
(et − 1)s

s!

)(
tsevt ln c+wt2 ln c

(et − 1)s

)
C0(xt)

(
Lik(1− (ab)−(1−u)t)

λbt − ua−t

)α
s!

ts

=

( ∞∑
n=0

{
n+ s

s

}
tn+s

(n+ s)!

)( ∞∑
m=0

B
(s)
m,H(v ln c, w ln c)

tm

m!

)( ∞∑
n=0

G(k,α)
n,L (x;λ, u, a, b)

tn

n!

)
s!

ts

=

( ∞∑
n=0

{
n+ s

s

}
tn+s

(n+ s)!

)( ∞∑
n=0

n∑
m=0

(
n

m

)
B

(s)
m,H(v ln c, w ln c)G(k,α)

n−m,L(λ, u, a, b)
tn

n!

)
s!

ts

=

( ∞∑
n=0

n∑
l=0

{
l + s

s

}
tl+s

(l + s)!

n−l∑
m=0

(
n− l

m

)
B

(s)
m,H(v ln c, w ln c)G(k,α)

n−l−m,L(λ, u, a, b)
tn−l

(n− l)!

)
s!

ts
.

This can further be written as

∞∑
n=0

G(k,α)
n,L (x;λ, u, v, w, a, b, c)

tn

n!

=

( ∞∑
l=0

∞∑
n=l

n−l∑
m=0

{
l + s

s

}
l!s!

(l + s)!

(
n− l

m

)
B

(s)
m,H(v ln c, w ln c)G(k,α)

n−l−m,L(x;λ, u, a, b)
n!

(n− l)!l!

tn

n!

)



R. Corcino, C. Corcino / Eur. J. Pure Appl. Math, 15 (4) (2022), 1549-1565 1562

=

∞∑
n=0

(
n∑

l=0

n−l∑
m=0

(
n

l

){
l + s

s

}(n−l
m

)(
l+s
s

)B(s)
m,H(v ln c, w ln c)G(k,α)

n−l−m,L(x;λ, u, a, b)

)
tn

n!
.

Comparing the coefficients of tn

n! gives (40).

Now, to prove relation (41), (17) may be expressed as

∞∑
n=0

G(k,α)
n,L (x;λ, u, v, w, a, b, c)

tn

n!

=

(
(1− µ)s

(et − µ)s
evt ln c+wt2 ln c

)(
(et − µ)s

(1− µ)s

)
C0(xt)

(
Lik(1− (ab)−(1−u)t)

λbt − ua−t

)α

=
1

(1− µ)s

( ∞∑
n=0

E
(s)
n,H(µ, v ln c, w ln c)

tn

n!

) s∑
j=0

(
s

j

)
(−µ)s−j×

×C0(xt)

(
Lik(1− (ab)−(1−u)t)

λbt − ua−t

)α

ejt

)

=
1

(1− µ)s

s∑
j=0

(
s

j

)
(−µ)s−j

( ∞∑
n=0

E
(s)
n,H(µ, v ln c, w ln c)

tn

n!

)
×

×

( ∞∑
n=0

G(k,α)
n (x;λ, u, j, a, b)

tn

n!

)

=
1

(1− µ)s

s∑
j=0

(
s

j

)
(−µ)s−j

∞∑
n=0

(
n∑

m=0

(
n

m

)
G(k,α)
n−m,L(x;λ, u, j, a, b)×

×E
(s)
m,H(µ, v ln c, w ln c)

) tn

n!

=
∞∑
n=0

 n∑
m=0

(
n
m

)
(1− µ)s

s∑
j=0

(
s

j

)
(−µ)s−jG(k,α)

n−m,L(x;λ, u, j, a, b)×

×E
(s)
m,H(µ, v ln c, w ln c)

) tn

n!
.

Comparing the coefficients of tn

n! gives (41).

5. Conclusion and Recommendations

In this paper, a certain variation of poly-Genocchi polynomials, called the generalized
Laguerre-Apostol-Frobenius-type poly-Genocchi polynomials of higher order has been in-
troduced using the concept of polylogarithm, Laguerre, Apostol and Frobenius polynomi-
als. Some interesting properties and identities of these polynomials were explored parallel



REFERENCES 1563

to those of the poly-Euler polynomials and poly-Bernoulli polynomials. Using their dif-
ferential identities, the generalized Laguerre-Apostol-Frobenius-type poly-Genocchi poly-
nomials in (21) were classified as Appell polynomials, which, consequently, gave some
interesting relations. The paper was concluded by expressing these generalized Laguerre-
Apostol-Frobenius-type poly-Genocchi polynomials of higher order in terms of Stirling
numbers of the second kind, generalized Hermite-Frobenius-Bernoulli polynomials and
generalized Hermite-Frobenius-Euler polynomials of higher order.

For future research work, one may try to define other variation of Apostol-Frobenius-
type poly-Genocchi polynomials with parameters a, b and c by mixing these polynomials
with the degenerate exponential polynomials. Moreover, it is also interesting to construct
a q-analogue of these generalized Laguerre-Apostol-Frobenius-type poly-Genocchi polyno-
mials using the method employed in [30]. Parallel to the construction of certain mixed
type special polynomials in [29], it would also be interesting to construct another variation
of poly-Genocchi polynomials by mixing these polynomials with Appell polynomials.

Acknowledgements

The authors would also like to thank Cebu Normal University (CNU) for funding this
research project through its Center for Research and Development (CRD).

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