EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
2025, Vol. 18, Issue 4, Article Number 5203
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

On Apostol-Type Multi Poly-Genocchi Polynomials
with Parameters a, b, and c

Mark P. Laurente1,∗, AZ D. Ababa2

1 Department, Faculty, Davao De Oro State College – Main Campus (Compostela),
Compostela City, Davao de Oro, Philippines
2 Institute of Mathematics, Arts and Sciences, Faculty, Davao del Sur State College,
Digos City, Davao del Sur, Philippines

Abstract. In this paper, we investigate and analyze the Apostol numbers and polynomials, ex-
tending these properties by integrating them with the multi-polylogarithm function. Through this
approach, we establish new properties and introduce a novel concept, which we refer to as the
Apostol-type multi-poly Genocchi polynomials with parameters a, b, and c. Several properties of
these polynomials are established including identities, the relation to Bernoulli polynomials, in-
cluding some recurrence relations, addition and explicit formulas which are parallel on generalized
poly-Genocchi polynomials.

2020 Mathematics Subject Classifications: 11B68, 11M35, 33C45

Key Words and Phrases: Genocchi numbers and polynomials, apostol Genocchi numbers and
polynomials, poly-Genocchi numbers and polynomials, multi poly-Genocchi numbers and polyno-
mials, apostol-type multi poly-Genocchi numbers and polynomials, poly Bernoulli and generating
function

1. Introduction

The Genocchi numbers, denoted by Gn, are an important sequence of integers that
arise in various areas of mathematics, including combinatorics, number theory, and graph
theory. They were named after the Italian mathematician Angelo Genocchi (1817–1889),
who studied it extensively. The Genocchi polynomials are related to the well-known
Bernoulli and Euler polynomials a famous work of Jakob Bernoulli (1654-1705) when
he studied the sums of the pth power of the first n − 1 integers 1p + 2p + 3p + (n − 1)p.
One of the outgrowths on the generalization of classical Bernoulli polynomials Bn(x) is

the development of the poly-Bernoulli numbers B
(k)
n and polynomials B

(k)
n (x). Many

researchers in recent decades provided relation to Euler numbers En and polynomials

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v18i4.5203

Email addresses: mark.laurente@ddosc.edu.ph (M. Laurente), az.ababa@dssc.edu.ph (AZ D. Ababa)

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



M. Laurente, AZ D. Ababa / Eur. J. Pure Appl. Math, 18 (4) (2025), 5203 2 of 19

En(x), Genocchi numbers Gn and polynomials Gn(x), poly-Euler numbers E
(k)
n and poly-

Euler polynomials E
(k)
n (x), poly-Genocchi numbers G(k)

n and poly-Genocchi polynomials

G(k)
n (x).
In 1970 J.M. Gandhi [1] presents the conjectured of Genocchi Numbers by

G2N = (−1)N
∑

12
∑

22
∑

32···
∑

(N−1)2

where
∑

notation is defined by∑
k2 = k2 − (k − 1)2

∑
k2 (k − 1)2

∑
k2
∑

(k − 1)2 = k2
∑

(k + 1)2 − (k − 1)2
∑

k2

= k2
{
(k + 1)2 − k2

}
− (k − 1)2

{
k2 − (k − 1)2

}
which generalizes to the recurrence form

∑
k2
∑

(k − 1)2 · · ·
∑

(k +N)2 = k2
∑

(k + 1)2
∑

(k + 2)2 · · ·
∑

(k +N)2−(k − 1)2
∑

k2
∑

(k + 1)2

The recurrence relation was proved by Riordan and Stien [2] by means of Analytic Cal-
culus. There are several ways to define the Genocchi numbers. In this paper, we adopt
the definition of [3–8]which are a sequence of integers that are defined by the exponential
generating function

2t

et + 1
=

∞∑
n=0

Gn
tn

n!
, |t| < π. (1)

with the usual convention about replacing Gn by Gn, is used. These are few Genocchi
numbers: G0 = 0, G1 = 1, G2 = −1, G3 = 0, G4 = 1, G5 = 0, G6 = −3, G7 = 0,
G8 = 17,and etc.

The classical definition of Genocchi Polynomials, denoted by Gn(x), is usually defined
by means of the exponential generating function

2t

et + 1
ext =

∞∑
n=0

Gn(x)
tn

n!
, |t| < π (2)

where Gn(x) is the Genocchi polynomials of degree n and is given by

Gn(x) =

n∑
k=0

(
n
k

)
Gkx

n−k.

The first few Genocchi polynomials are [9]: G1(x) = 1, G2(x) = 2x− 1, G3(x) = 3x2 − 3x,
G4(x) = 4x3 − 6x2 + 1, G5(x) = 5x4 − 10x3 + 5x



M. Laurente, AZ D. Ababa / Eur. J. Pure Appl. Math, 18 (4) (2025), 5203 3 of 19

2. Preliminaries

2.1. Generating Function and Polylogarithm and Genocchi Polynomials

The geometric series [10]
1

1− x
= 1+x+x2+x3+ . . . , has formal power series

∑
n≥0

xn,

say for example the generating function of a+ ab+ ab2 + ab3 + . . . =
∑
n≥0

abnxn is
a

1− bx
.

Also the exponential generating function for the sequence of numbers (ar) is defined to

be the power series a0 + a1
x
1! + a2

x
2! + a3

x
3! + . . . + ar

x
r! + . . . =

∞∑
r=0

ar
xr

r!
. Consider the

sequence (ar) = (1, k, k2, k3, . . . , kr, . . .) where k is a nonzero constant. The exponential

generating function for (ar) is 1 +
kx
1! +

k2x2

2! + . . . =
∞∑
r=0

(kx)r

r!
= ekx.

The classical polylogarithmic function [11] is defined by

Lik(z) =

∞∑
n=1

zn

nk
(3)

which is the k-th polylogarithm if k ≥ 1 and a rational function if k ≤ 0.
The multiple polylogarithms [12] are defined by

Lik1,k2,k3,··· ,kr
(z) =

∑
0<m1<m2...<mr

zmr

mk1
1 mk2

2 · · ·mkr
r

. (4)

where k1, k2, · · · , kr are positive integers and z a complex number in the unit disk. For
r = 1, this is the classical polylogarithm Lik(z).

2.2. Variation of Genocchi Polynomial

Different authors who studied the work of Apostol, used generating functions with a
twist—inserting a parameter λ—which gave them a greater flexibility and allowed connec-
tions to new types of zeta functions, when they were studying in the generalization the
Bernoulli polynomials. Then several mathematician was intrigue of this grand work and
further do more research on it which now resulted to the closed relation of and several
variation of Bernoulli, Euler, and Genocchi polynomials. One of the interesting result that
arise from those researches is the Apostol-Genocchi polynomial.

∞∑
n=0

G(x, λ) t
n

n!
=

2t

λet + 1
ext, (5)

Other variations of Genocchi polynomials that appeared in the literature is the Apostol-
Genocchi polynomials of higher order, which defined by

∞∑
n=0

G(k)(x, λ)
tn

n!
=

(
2t

λet + 1

)k

ext, (6)



M. Laurente, AZ D. Ababa / Eur. J. Pure Appl. Math, 18 (4) (2025), 5203 4 of 19

(see [13], [14], [3, 15, 16],[8]).
It is also important to note that the notation of Araci in equation (7) of higher order

denoted by k has a different meaning in the notation of k the poly-Logarithm other re-
searchers like Corcino, Lou, Srivastava, who study the higher order of Genocchi polynomial
is using α to denote the higher order.

∞∑
n=0

G(k)
n (x)

tn

n!
=

(
2t

et + 1

)k

ext, (7)

Araci [15] and Kim et al. [14] did some research on equation (7), the Genocchi polyno-
mials of higher order arising from Genocchi basis. The main objective of their studies is
to derive interesting identities on (5) using a new method constructed by Kim et al. [17].

Moreover, Araci and He [3, 15, 16] introduced (7) an extension of the classic Genocchi
polynomials called the Apostol-Genocchi polynomials from which Araci [8] introduced (6)
the higher order of such polynomials as the generalized Apostol-Genocchi polynomials

G(k)
n (x, λ) of order k ∈ C,
Another variation of Genocchi polynomials, is also known as poly-Genocchi polynomi-

als, was introduced by Kim et al. [18] using the concept of kth poly-logarithm, denoted
by Lik(z), which is given by

∞∑
n=0

G(k)
n (x)

tn

n!
=

2Lik(1− e−t)

et + 1
ext. (8)

Moreover, a modified poly-Genocchi polynomials, denoted by G(k)
n,2(x), were defined by

Kim et al. [19] as follows

∞∑
n=0

G(k)
n,2(x)

tn

n!
=

Lik(1− e−2t)

et + 1
ext. (9)

Note that, when k = 1, equations (8) and (9) give the Genocchi polynomials in (2). That
is,

G(1)
n (x) = G(1)

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

Kim et. al [18] obtained several properties of these polynomials.
On the other hand, Kurt [20] defined two forms of generalized poly-Genocchi polyno-

mials 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)

tn

n!

(
|t| < π

| ln a+ ln b|

)
(10)

2Lik(1− (ab)−2t)

a−t + bt
ext =

∞∑
n=0

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

tn

n!
,

(
|t| < π

| ln a+ ln b|

)
(11)



M. Laurente, AZ D. Ababa / Eur. J. Pure Appl. Math, 18 (4) (2025), 5203 5 of 19

In [21], Lim defined the degenerate Genocchi polynomials G(r)
n (λ, x) of order r as(

2t

(1 + λt)1/λ

)r

(1 + λt)x/λ =

∞∑
n=0

G(r)
n (λ, x)

tn

n!
.

Besides those generalizations, Araci [5], Duran et al. [22] and Agyuz et al. [23] also
introduced the q-analogue of the Genocchi polynomials Gn,q(x) as follows:

∞∑
n=0

Gn,q(x)
tn

n!
= t

∫
Zp

q−yet[y+x]qdµ−q(y),

where

[x]q =
1− qx

1− q
, [x]−q =

1− (−q)x

1 + q
.

This definition is constructed by p-adic fermionic q-integral on Zp with respect to µ−q. It
can also be defined by

∞∑
n=0

Gn,q(x)
tn

n!
= [2]qt

∞∑
m=0

(−1)met[m+x]q ,

which becomes Gn,q(0) := Gn,q when x = 0, and referred to as the nth q-Genocchi number.
Also, Corcino C.B. and Corcino R.B [24]study regarding Higher Order Apostol-Type

Poly-Genocchi Polynomials with Parameters a, b and c, they define this by

∞∑
n=0

G(k,α)
n (x)

tn

n!
=

(
Lik(1− (ab)−2t)

a−t + bt
ext
)α

, |t| <
√
(lnλ)2 + π2

|lna+ lnb|
,

they also study the other variation [25], which the concept is from modified degenerate
polylexponential function.

3. Close Relation of Bernoulli polynomial and Euler polynomial to
Genocchi Polynomial

A generalization of Bernoulli polynomials introduced by Kaneko [26] is defined in terms
of the following kth polylogarithm Lik(z):

Lik(z) =

∞∑
n=1

zn

nk
, (|z| < 1) (12)

where k ∈ Z. When z = 1, the kth polylogarithm gives the Riemann zeta function. That
is,

Lik(1) = ζ(k) =

∞∑
n=1

1

nk
.



M. Laurente, AZ D. Ababa / Eur. J. Pure Appl. Math, 18 (4) (2025), 5203 6 of 19

Also, when k = 1, the 1st polylogarithm yields the natural logarithmic function as
follows:

Li1(z) = − ln(1− z).

This special case of the polylogarithm motivates the construction of poly-
Bernoulli numbers in the sense that

Li1(1− e−x) = x.

The poly-Bernoulli polynomials and numbers were defined in [19] by means of the
following generating functions, respectively:

Lik(1− e−t)

et − 1
ext =

∞∑
n=0

B(k)
n (x)

tn

n!
(|t| < 2π) (13)

Lik(1− e−t)

et − 1
=

∞∑
n=0

B(k)
n

tn

n!
, (|t| < 2π) (14)

If k = 1, from equations (13) and (14), respectively, we have

B(1)
n (x) = Bn(x), B

(1)
n = Bn. (15)

We present again equation (8), since this a poly-Genoccchi Polynomials and this work

of Kim et al. [19] defined poly-Genocchi polynomials G(k)
n (x) as follows

2Lik(1− e−t)

et + 1
ext =

∞∑
n=0

G(k)
n (x)

tn

n!
, (|t| < π).

Note that when x = 0, (8) reduces to

2Lik(1− e−t)

et + 1
=

∞∑
n=0

G(k)
n

tn

n!
, (|t| < π). (16)

where G(k)
n are called the poly-Genocchi numbers.

Note also that when k = 1, equation (8) gives the Genocchi polynomials

∞∑
n=0

G(1)
n (x)

tn

n!
=

2Li1(1− e−t)

et + 1
ext

=
2t

et + 1
ext

=

∞∑
n=0

Gn(x)
tn

n!
.



M. Laurente, AZ D. Ababa / Eur. J. Pure Appl. Math, 18 (4) (2025), 5203 7 of 19

Moreover, they defined a modified poly-Genocchi polynomials, denoted by G(k)
n,2(x), as

follows:
Lik(1− e−2t)

et + 1
ext =

∞∑
n=0

G(k)
n,2(x)

tn

n!
. (17)

Equation (17) also gives the Genocchi polynomials (??)

Li1(1− e−2t)

et + 1
ext =

∞∑
n=0

G(1)
n,2(x)

tn

n!

=
2t

et + 1
ext

=
∞∑
n=0

Gn(x)
tn

n!
.

Consequently, Kim et al. [19] obtained several properties of the poly-Genocchi poly-
nomials.

On the other hand Kurt [20] ivestigated most creative stage of poly-Genocchi of equa-
tion (10) and (11) which are following

2Lik(1− (ab)−t)

a−t + bt
cxt =

∞∑
n=0

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

tn

n!

2Lik(1− (ab)−2t)

a−t + bt
cxt =

∞∑
n=0

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

tn

n!

which are motivated by the definitions in (8) and (9), respectively.
If we put x = 0, a = 1, b = c = e in (10), we have

G(k)
n (0; 1, e, e)

tn

n!
=

2Lik(1− e−t)

et + 1

which gives the definition of Kurt [20] for the poly-Genocchi numbers G(k)
n which is also

the definition of Kim et al. in [19] the poly-Genocchi numbers.
Further, in [12] they defined the Generalized Poly-Genocchi Polynomials with Param-

eters a, b, and c by

∞∑
n=0

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

tn

n!
=

Lik(1− (ab)−2t)

a−t + bt
cxt, |t| < π

| ln a+ ln b|
. (18)



M. Laurente, AZ D. Ababa / Eur. J. Pure Appl. Math, 18 (4) (2025), 5203 8 of 19

4. On Apostol-Type Multi Poly-Genocchi Polynomials with Parameters
a, b and c

Definition 4.1. An Apostol-type of multi poly Genocchi polynomials with pa-
rameters a, b and c is defined by

∞∑
n=0

G(k1,k2,k3,··· ,kr)
n (x;λ, a, b, c)

tn

n!
=

Lik1,k2,k3,··· ,kr
(1− (ab)−2t)

(a−t + λbt)r
crxt, (19)

where x is any real number with k1, k2, . . . , kr ∈ Z+, λ ∈ C and a, b, c are any positive real
numbers.

When c = e equation(19) reduces to

∞∑
n=0

G(k1,k2,k3,...,kr)
n (x;λ, a, b, e)

tn

n!
=

Lik1,k2,k3,··· ,kr
(1− (ab)−2t)

(a−t + λbt)r
erxt (20)

We use G(k1,k2,k3,··· ,kr)
n (x;λ, a, b) to denote G(k1,k2,k3,··· ,kr)

n (x;λ, a, b, e). That is,

∞∑
n=0

G(k1,k2,k3,··· ,kr)
n (x;λ, a, b)

tn

n!
=

Lik1,k2,k3,··· ,kr
(1− (ab)−2t)

(a−t + λbt)r
erxt. (21)

For instance, if we put a = 1, b = e in equation (21), then this will reduce to

∞∑
n=0

G(k1,k2,k3,··· ,kr)
n (x;λ, 1, e)

tn

n!
=

Lik1,k2,k3,··· ,kr
(1− e−2t)

(1 + λet)r
erxt. (22)

We may use the notation G(k1,k2,k3,...,kr)
n (x;λ) = G(k1,k2,k3,...,kr)

n (x;λ, 1, e) and call them
Apostol-type multi poly-Genocchi polynomials. When λ = 1 equation (22) gives

∞∑
n=0

G(k1,k2,k3,...,kr)
n (x; 1)

tn

n!
=

Lik1,k2,k3,··· ,kr
(1− e−2t)

(1 + et)r
erxt. (23)

When x = 0 equation (21) gives

∞∑
n=0

G(k1,k2,k3,··· ,kr)
n (0;λ, a, b)

tn

n!
=

Lik1,k2,k3,··· ,kr
(1− (ab)−2t)

(a−t + λbt)r
. (24)

We use G(k1,k2,k3,...,kr)
n (a, b) to denote G(k1,k2,k3,...,kr)

n (0; a, b) the Apostol-type multi poly-
Genocchi Numbers.



M. Laurente, AZ D. Ababa / Eur. J. Pure Appl. Math, 18 (4) (2025), 5203 9 of 19

Theorem 4.2. An Apostol-type of multi poly-Genocchi polynomials with parameters a, b,
and c satisfy the relation.

G(k1,k2,k3,...,kr)
n (x;λ, a, b, c) = (ln a+ ln b)nG(k1,k2,k3,...,kr)

n

(
rx ln c+ r ln a

ln ab
;λ

)
, ab ̸= 1.

(25)

Proof. By definition of an Apostol-type of multi poly-Genocchi polynomials in equation
(19) we have

∞∑
n=0

G(k1,k2,k3,...,kr)
n (x;λ, a, b, c)

tn

n!
=

Lik1,k2,k3,··· ,kr
(1− (ab)−2t)

(a−t + λbt)r
crxt

=
Lik1,k2,k3,...,kr

(1− (ab)−2t)(
(a−t)(1 + λ bt

a−t )

)r crxt

=
Lik1,k2,k3,...,kr

(1− (ab)−2t)

(1 + λ(ab)t)r
crxtart

=
Lik1,k2,k3,...,kr

(1− (e)−2t ln ab)

(1 + λet ln ab)r
erxt ln cert ln a

=
Lik1,k2,k3,...,kr

(1− (e)−2t ln ab)

(1 + λet ln abt)r
et(rx ln c+r ln a)

Let z = tlnab, then t=
z

lnab
. Thus we have,

∞∑
n=0

G(k1,k2,k3,··· ,kr)
n (x;λ, a, b, c)

tn

n!
=

Lik1,k2,k3,...,kr
(1− (e)−2z)

(1 + λez)r
e

z
ln ab

(rx ln c+r ln a)

=
Lik1,k2,k3,...,kr

(1− (e)−2z)

(1 + λez)r
e(

rx ln c+r ln a
ln ab

)z

=
∞∑
n=0

G(k1,k2,k3,...,kr)
n

(
rx ln c+ r ln a

ln ab
;λ

)
zn

n!

=

∞∑
n=0

G(k1,k2,k3,...,kr)
n

(
rx ln c+ r ln a

ln ab
;λ

)
(t ln ab)n

n!

=
∞∑
n=0

G(k1,k2,k3,...,kr)
n

(
rx ln c+ r ln a

ln ab
;λ

)
(t)n(ln ab)n

n!

=

∞∑
n=0

(ln a+ ln b)nG(k1,k2,k3,...,kr)
n

(
rx ln c+ r ln a

ln ab
;λ

)
(t)n

n!

Comparing the coeffiecient of
tn

n!
we obtain the desired result. Therefore,

G(k1,k2,k3,...,kr)
n (x;λ, a, b, c) = (ln a+ ln b)nG

(k1,k2,k3,...,kr)
n

(
rx ln c+ r ln a

ln ab
;λ

)
.



M. Laurente, AZ D. Ababa / Eur. J. Pure Appl. Math, 18 (4) (2025), 5203 10 of 19

The next theorem contains a kind of recurrence relation of an Apostol-type of multi
poly-Genocchi polynomials.

Theorem 4.3. An Apostol-type of multi poly-Genocchi polynomials satisfy the
recurrence relation

G(k1,k2,...,kr)
n (x+ 1;λ, a, b, c) =

n∑
m=0

(
n

m

)
(r ln c)mG(k1,k2,...,kr)

n−m (x;λ, a, b, c).

Proof. By the equation (19) in Definition (4.1)

∞∑
n=0

G(k1,k2,...,kr)
n (x+ 1;λ, a, b, c)

tn

n!
=

Lik1,k2,...,kr
(1− (ab)−2t)

(a−t + λbt)r
crt(x+1)

=
Lik1,k2,...,kr

(1− (ab)−2t)

(a−t + λbt)r
crxtcrt

=
Lik1,k2,...,kr

(1− (ab)−2t)

(a−t + λbt)r
erxt ln cert ln c

Rewriting ert ln c =
∞∑
n=0

(rt ln c)n

n!
as exponential generating function form so we have,

∞∑
n=0

G(k1,k2,...,kr)
n (x+ 1;λ, a, b, c)

tn

n!
=

( ∞∑
n=0

G(k1,k2,...,kr)
n (x;λ, a, b, c)

tn

n!

)( ∞∑
n=0

(rt ln c)n

n!

)
Using the product of two generating function, so we have,

∞∑
n=0

G(k1,k2,...,kr)
n (x+ 1;λ, a, b, c)

tn

n!
=

∞∑
n=0

n∑
m=0

G(k1,k2,...,kr)
n−m (x;λ, a, b, c)

tn−m

(n−m)!

(r ln c)mtm

m!

=

∞∑
n=0

n∑
m=0

G(k1,k2,...,kr)
n−m (x;λ, a, b, c)

tn−m

(n−m)!

(r ln c)mtm

m!

n!

n!

=
∞∑
n=0

( n∑
m=0

(
n

m

)
(r ln c)mG(k1,k2,...,kr)

n−m (x;λ, a, b, c)

)
tn

n!
.

Comparing the coeffiecient of
tn

n!
we obtain the desired result. Therefore,

G(k1,k2,...,kr)
n (x+ 1;λ, a, b, c) =

∑n
m=0

(
n
m

)
(r ln c)mG(k1,k2,...,kr)

n−m (x;λ, a, b, c).

Theorem 4.4. An Apostol-type of multi poly-Genocchi polynomials satisfy the
relation

G(k1,k2,...,kr)
n (x;λ, a, b, c) =

n∑
i=0

(
n

i

)
(r ln c)n−iG(k1,k2,...,kr)

i (λ, a, b)xn−i, x ̸= 0.



M. Laurente, AZ D. Ababa / Eur. J. Pure Appl. Math, 18 (4) (2025), 5203 11 of 19

Proof. Using equation (19) of Definition (4.1) can be written as

∞∑
n=0

G(k1,k2,...,kr)
n (x;λ, a, b, c)

tn

n!
=

Lik1,k2,...,kr
(1− (ab)−2t)

(a−t + λbt)r
erxt ln c

= erxt ln c
∞∑
n=0

G(k1,k2,...,kr)
n (λ, a, b)

tn

n!

=

∞∑
n=0

(rxt ln c)n

n!

∞∑
n=0

G(k1,k2,...,kr)
n (λ, a, b)

tn

n!

=
∞∑
n=0

n∑
i=0

(rxt ln c)n−i

(n− i)!
G(k1,k2,...,kr)
i (λ, a, b)

ti

i!

=
∞∑
n=0

( n∑
i=0

(xr ln c)n−i tn−i

(n− i)!
G(k1,k2,...,kr)
r (λ, a, b)

ti

i!

)

=
∞∑
n=0

( n∑
i=0

(xr ln c)n−i tn−i

(n− i)!
G(k1,k2,...,kr)
r (λ, a, b)

ti

i!

n!

n!

)

=
∞∑
n=0

(
n∑

i=0

(
n

i

)
(r ln c)n−iG(k1,k2,...,kr)

i (λ, a, b)xn−i

)
tn

n!
.

Comparing the coefficients of
tn

n!
, we obtain the desired result. Therefore,

G(k1,k2,...,kr)
n (x;λ, a, b, c) =

∑n
i=0

(
n
i

)
(r ln c)n−iG(k1,k2,...,kr)

i (λ, a, b)xn−i.

The following theorem contains a differential equation that can be used to classify an
Apostol-type of multi poly-Genocchi polynomials as Appell polynomials [12].

Theorem 4.5. An Apostol-type of multi poly-Genocchi polynomials satisfy the
relation

d

dx
G(k1,k2,...,kr)
n+1 (x;λ, a, b, c) = (n+ 1)(r ln c)G(k1,k2,...,kr)

n (x;λ, a, b, c). (26)

Proof. By applying the first derivative to equation (4.1), with respect to x we have

∞∑
n=0

d

dx
G(k1,k2,...,kr)
n (x;λ, a, b, c)

tn

n!
= rt(ln c)

Lik1,k2,...,kr
(1− (ab)−2t)

(a−t + λbt)r
erxt ln c

this means that,

∞∑
n=0

d

dx

G(k1,k2,...,kr)
n (x;λ, a, b, c)

r ln c

tn−1

n!
=

Lik1,k2,...,kr
(1− (ab)−2t)

(a−t + λbt)r
erxt ln c(∗)



M. Laurente, AZ D. Ababa / Eur. J. Pure Appl. Math, 18 (4) (2025), 5203 12 of 19

Rewriting the left hand side of(*) we get

∞∑
n=0

d

dx

G(k1,k2,...,kr)
n (x;λ, a, b, c)

r ln c

tn−1

n!
=

d

dx

G(k1,k2,...,kr)
0 (x;λ, a, b, c)

r ln c

t−1

0!

+
∞∑
n=1

d

dx

G(k1,k2,...,kr)
n (x;λ, a, b, c)

r ln c

tn−1

n!

=
∞∑
n=1

d

dx

G(k1,k2,...,kr)
n (x;λ, a, b, c)

r ln c

tn−1

n!

Changing the index,

Lik1,k2,...,kr
(1− (ab)−2t)

(a−t + λbt)r
erxt ln c =

∞∑
n=0

d

dx

G(k1,k2,...,kr)
n (x;λ, a, b, c)

r ln c

tn

(n+ 1)!

=
∞∑
n=0

d

dx

G(k1,k2,...,kr)
n (x;λ, a, b, c)

r ln c

tn

(n+ 1)n!

so we have,

∞∑
n=0

d

dx
G(k1,k2,...,kr)
n (x;λ, a, b, c)

tn

n!
=

∞∑
n=0

d

dx

G(k1,k2,...,kr)
n (x;λ, a, b, c)

r ln c

tn

(n+ 1)n!

∞∑
n=0

(r ln c)(n+ 1)
d

dx
G(k1,k2,...,kr)
n (x;λ, a, b, c)

tn

n!
=

∞∑
n=0

d

dx
G(k1,k2,...,kr)(x;λ,a,b,c)
n

tn

n!

Comparing the coefficients of
tn

n!
, we obtain the desired result.Therefore,

d
dxG

(k1,k2,...,kr)
n+1 (x;λ, a, b, c) = (n+ 1)(r ln c)G(k1,k2,...,kr)

n (x;λ, a, b, c).

When c = e1/r, the next corollary is obtained from (4.5)

Corollary 4.6. An Apostol-type multi poly-Genocchi polynomials satisfy the
relation

d

dx
G(k1,k2,...,kr)
n+1 (x;λ, a, b, e1/r) = (n+ 1)G(k1,k2,...,kr)

n (x;λ, a, b, e1/r).

Corollary (4.6) is one the properties to be classified as Appell polynomial. Being
classified as Appell polynomials, the Apostol-type multi poly-Genocchi polynomials must
possess the following properties

G(k1,k2,...,kr)
n (λ, a, b) =

∞∑
i=0

(
n

i

)
cix

n−i



M. Laurente, AZ D. Ababa / Eur. J. Pure Appl. Math, 18 (4) (2025), 5203 13 of 19

G(k1,k2,...,kr)
n (λ, a, b) =

( ∞∑
i=0

ci
i!
Di

)
xn

for some scalar ci ̸= 0 and where D is the operator
d

dx
. It is necessary to find the sequence

(cn). However, using Theorem (4.4) with c = e,

G(k1,k2,...,kr)
n (λ, a, b) =

∞∑
i=0

(
n

i

)
G(k1,k2,...,kr)
n (λ, a, b)xn−i

And letting ci = G(k1,k2,...,kr)
n (λ, a, b) we obtain the next corollary.

Corollary 4.7. An Apostol-type of multi poly-Genocchi polynomials satisfy the
following formula:

G(k1,k2,...,kr)
n (x;λ, a, b, e1/r) =

∞∑
i=0

G(k1,k2,...,kr)
i (λ, a, b)

i!
Dixn.

The following theorem contains the addition formula for G(k1,k2,...,kr)
n (x;λ, a, b, c).

Theorem 4.8. An Apostol of type multi poly-Genocchi polynomials satisfy the
following addition formula:

G(k1,k2,...,kr)
n (x+ y;λ, a, b, c) =

n∑
i=0

(
n

i

)
(r ln c)n−iG(k1,k2,...,kr)

i (x;λ, a, b, c)yn−i, y ̸= 0.

Proof. Using equation 19 of Definition 4.1,

∞∑
n=0

G(k1,k2,...,kr)
n (x+ y;λ, a, b, c)

tn

n!
=

Lik1,k2,...,kr
(1− (ab)−2t)

(a−t + λbt)r
c(x+y)rt

=
Lik1,k2,...,kr

(1− (ab)−2t)

(a−t + bt)r
cxrtcyrt

=

( ∞∑
n=0

G(k1,k2,...,kr)
n (x;λ, a, b, c)

tn

n!

)
eyrt ln c

Since eyrt ln c =

∞∑
n=0

(yrt ln c)n

n!
=

∞∑
n=0

(yr ln c)n
(t)n

n!
, so have,

∞∑
n=0

G(k1,k2,...,kr)
n (x+ y;λ, a, b, c)

tn

n!
=

( ∞∑
n=0

G(k1,k2,...,kr)
n (x;λ, a, b, c)

tn

n!

)( ∞∑
n=0

(yr ln c)n
tn

n!

)
Using the product of two generating function,

=
∞∑
n=0

( n∑
i=0

G(k1,k2,...,kr)
n−i (x;λ, a, b, c)

tn−i

(n− i)!

)
(ln c)i

ti

i!



M. Laurente, AZ D. Ababa / Eur. J. Pure Appl. Math, 18 (4) (2025), 5203 14 of 19

and we multiply it by
i!

i!
, so we have,

=

∞∑
n=0

( n∑
i=0

G(k1,k2,...,kr)
r (x;λ, a, b, c)

tn−i

(n− i)!

)
(ln c)i

ti

i!

i!

i!

=
∞∑
n=0

(
n∑

i=0

(
n

i

)
(yr ln c)n−iG(k1,k2,...,kr)

i (x;λ, a, b, c)

)
tn

n!

=
∞∑
n=0

(
n∑

i=0

(
n

i

)
(r ln c)n−iG(k1,k2,...,kr)

i (x;λ, a, b, c)yn−i

)
tn

n!

Comparing the coefficients of
tn

n!
yields the desired result.

When c = e1/r, the next corollary is obtained.

Corollary 4.9. An Apostol-type of multi poly-Genocchi polynomials satisfy the
following addition formula:

G(k1,k2,...,kr)
n (x+ y;λ, a, b, e1/r) =

n∑
i=0

(
n

i

)
G(k1,k2,...,kr)
i (x;λ, a, b, e1/r)yn−i, y ̸= 0.

Theorem 4.10. An Apostol-type multi of poly-Genocchi polynomials satisfy the following
explicit formulas:

G(k1,k2,...,kr)
n (x;λ, a, b, c) =

∞∑
m=0

n∑
l=m

{
l
m

}(
n

l

)
(r ln c)lG(k1,k2,...,kr)

n−l (−m ln c;λ, a, b)(x)(m)

(27)

G(k1,k2,...,kr)
n (x;λ, a, b, c) =

∞∑
m=0

n∑
l=m

{
l
m

}(
n

l

)
(r ln c)lG(k1,k2,...,kr)

n−l (λ, a, b)(x)m (28)

G(k1,k2,...,kr)
n (x;λ, a, b, c) =

n∑
l=0

n−l∑
m=0

(
n

l

){
l + s

s

}(n−l
m

)(
l+s
s

)G(k1,k2,...,kr)
n−l−m (λ, a, b)B(s)

m (rx ln c)

(29)

where (x)(n) = x(x+ 1) · · · (x+ n− 1), (x)n = x(x− 1) · · · (x− n+ 1),
and (

t

et − 1

)s

ext =

∞∑
n=0

B(s)
n (x)

tn

n!

Proof. First, we prove 27. Using equation 19 of Definition (4.1) and rewrite erxt ln c =

(e−rt ln c)−x =
[
1− (1− e−rt ln c)

]−x
we have,

∞∑
n=0

G(k1,k2,...,kr)
n (x;λ, a, b, c)

tn

n!
=

Lik(1− (ab)−2t)

(a−t + bt)r

[
1− (1− e−rt ln c)

]−x
.



M. Laurente, AZ D. Ababa / Eur. J. Pure Appl. Math, 18 (4) (2025), 5203 15 of 19

Using generalized Binomial Theorem and the definition of Stirling numbers of second kind
we have,

∞∑
n=0

G(k1,k2,...,kr)
n (x;λ, a, b, c)

tn

n!
=

Lik1,k2,...,kr
(1− (ab)−2t)

(a−t + λbt)r

∞∑
m=0

(
x+m− 1

m

)
(1− e−rt ln c)m

=
∞∑

m=0

(x)(m) (e
rt ln c − 1)m

ert ln cm!

Lik1,k2,...,kr
(1− (ab)−2t)

(a−t + λbt)r

Note that (x)(m) = x(x+ 1) · · · (x+m− 1)
(x− 1)!

(x− 1)!
=

(x+m− 1)!

(x− 1)!
, so we have

∞∑
n=0

G(k1,k2,...,kr)
n x;λ, a, b, c)

tn

n!
=

∞∑
m=0

(x+m− 1)!

(x− 1)!m!

(
ert ln c − 1

ert ln c

)m
Lik(1− (ab)−2t)

(a−t + λbt)r

=
∞∑

m=0

(x+m− 1)!

(x− 1)!

(et ln c − 1)m

m!

Lik(1− (ab)−2t)

(a−t + λbt)r
e−rmt ln c

=

∞∑
m=0

(x)(m)
∞∑
n=0

{
n
m

}
(r ln c)n

tn

n!

( ∞∑
n=0

G(k)
n (−mr ln c;λ, a, b)

tn

n!

)

=
∞∑

m=0

(x)(m)

( ∞∑
n=0

{
n
m

}
(tr ln c)n

n!

) ( ∞∑
n=0

G(k1,k2,...,kr)
n (−m ln c;λ, a, b)

tn

n!

)

=

∞∑
m=0

(x)(m)
∞∑
n=0

n∑
l=0

{
l
m

}
(r ln c)l

tl

l!
G(k1,k2,...,kr)
n−l (−m ln c;λ, a, b)

tn−l

(n− l)!

=
∞∑
n=0

{ ∞∑
m=0

n∑
l=m

{
l
m

}(
n

l

)
r ln c)lG(k1,k2,...,kr)

n−l (−m ln c;λ, a, b)(x)(m)

}
tn

n!

Comparing the coefficients of
tn

n!
completes the proof of equation 27.

To prove identity of equation 28, express erxt ln c as[
1 + (ert ln c − 1)

]x
=

∞∑
m=0

(
x

m

)
(ert ln c − 1)m

and take note that

(x)m = x(x− 1) · · · (x−m+ 1) =
x!

(x−m)!

applying it to definition 19, we have



M. Laurente, AZ D. Ababa / Eur. J. Pure Appl. Math, 18 (4) (2025), 5203 16 of 19

∞∑
n=0

G(k1,k2,...,kr)
n (x;λ, a, b, c)

tn

n!
=

Lik1,k2,...,kr
(1− (ab)−2t)

(a−t + λbt)r
erxt ln c

=
Lik1,k2,...,kr

(1− (ab)−2t)

(a−t + λbt)r

∞∑
m=0

(
x

m

)
(ert ln c − 1)m

=
∞∑

m=0

(x)m
(ert ln c − 1)m

m!

Lik1,k2,...,kr
(1− (ab)−2t)

(a−t + λbt)r

=

∞∑
m=0

(x)m

( ∞∑
n=0

{
n
m

}
(rt ln c)n

n!

) ( ∞∑
n=0

G(k1,k2,...,kr)
n (0;λ, a, b)

tn

n!

)

=
∞∑

m=0

(x)m

∞∑
n=0

n∑
l=0

{
l
m

}
(r ln c)l

tl

l!
G(k1,k2,...,kr)
n−l (a, b)

tn−l

(n− l)!

=
∞∑
n=0

{ ∞∑
m=0

n∑
l=m

{
l
m

}(
n

l

)
(r ln c)lG(k1,k2,...,kr)

n−l (a, b)(x)m

}
tn

n!
.

Again,comparing the coefficients of
tn

n!
completes the proof of (28).

For relation (29), from equation 19 of Definition 4.1 can be written as

∞∑
n=0

G(k)
n (x;λ, a, b, e)

tn

n!
=

Lik(1− (ab)−2t)

a−t + bt
ext ln c(∗)

and multiplying the right-hand side by (*)
(et − 1)s

(et − 1)s
ts

ts
s!

s!
, we get

∞∑
n=0

G(k)
n (x;λ, a, b, c)

tn

n!
=

(
(et − 1)s

s!

)(
tsext ln c

(et − 1)s

)(
Lik1,k2,...,kr

(1− (ab)−2t)

(a−t + λbt)r

)
s!

ts

=

( ∞∑
n=0

{
n+ s

s

}
tn+s

(n+ s)!

)( ∞∑
m=0

B(s)
m (x ln c)

tm

m!

)( ∞∑
n=0

Gk1,k2,...,kr
n (λ, a, b)

tm

m!

)
s!

ts

=

( ∞∑
n=0

{
n+ s

s

}
tn+s

(n+ s)!

)( ∞∑
n=0

n∑
m=0

(
n

m

)
B(s)

m (x ln c)Gk1,k2,...,kr
n−m (λ, 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 (x ln c)G(k1,k2,...,kr)
n−l−m (λ, a, b)

tn−l

(n− l)!

)
s!

ts

=

( ∞∑
l=0

∞∑
n=l

n−l∑
m=0

{
l + s

s

}
l!s!

(l + s)!

(
n− l

m

)
B(s)

m (x ln c)G(k1,k2,...,kr)
n−l−m (λ, a, b)

n!

(n− l)!l!

tn

n!

)



M. Laurente, AZ D. Ababa / Eur. J. Pure Appl. Math, 18 (4) (2025), 5203 17 of 19

=

∞∑
n=0

(
n∑

l=0

n−l∑
m=0

(
n

l

){
l + s

s

}(n−l
m

)(
l+s
s

)B(s)
m (x ln c)G(k1,k2,...,kr)

n−l−m (λ, a, b)

)
tn

n!
.

Again,comparing the coefficients of
tn

n!
completes the proof of equation (29).

5. Conclusion

In this paper, we define Apostol-type of multi poly Genocchi polynomials with param-
eters a, b and c through the concept of multi-poly logarithm. We present the special cases
when c = e that gives as Equation (22), and when a = 1, and b = e we obtain as Equation
(23). Furthermore, we establish the relation in Theorem 4.2 and the recurrence relations
given in Theorems 4.3 and 4.4, derived from the concept of poly-Bernoulli polynomials.
We also provide Theorem 4.5 together with corollaries that involve a differential equation,
which allows us to classify the Apostol-type multi poly-Genocchi polynomials as a family
of Appell polynomials. Finally, we derive the addition formula in Theorem 4.8 and the
explicit formulas in Theorem 4.10. The researcher recommends the following for possi-
ble investigation, by letting indices to non-positive. The modified Apostol-type of multi
poly-Genocchi polynomials with parameters a, b, and c.

Acknowledgements

The authors sincerely extend their gratitude to Davao de Oro State College–Main
Campus (Compostela) and Davao del Sur State College for their unwavering support,
encouragement, and valuable contributions, which greatly facilitated the successful com-
pletion of this study.

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