EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
Vol. 13, No. 3, 2020, 403-413
ISSN 1307-5543 – www.ejpam.com
Published by New York Business Global

Identities on Generalized Apostol-Genocchi Numbers
and Polynomials Involving Binomial Coefficients

Nestor Acala1,∗, Edward Rowe Aleluya2

1 Mathematics Department, College of Natural Sciences and Mathematics, Mindanao State
University, Marawi City, Lanao del Sur, Philippines
2 Department of Physical Sciences and Mathematics, College of Science and Environment,
Mindanao State University-Naawan, Misamis Oriental, Philippines

Abstract. In [11], Jolany et al. defined generalizations of Apostol-Genocchi numbers and poly-
nomials. Most identities on classical or generalized Apostol-Genocchi numbers and polynomials
are related to the well-known Bernoulli and Euler numbers and polynomials. However, in this
paper, identities on generalized Apostol-Genocchi numbers and polynomials which are not associ-
ated with the Bernoulli- and Euler-types are introduced. Specifically, identities involving binomial
coefficients and some integral identities which only relate generalized Apostol-Genocchi numbers
and polynomials are established.

2020 Mathematics Subject Classifications: 11B65, 05A10, 11B83

Key Words and Phrases: Genocchi number, Genocchi polynomial, Apostol-Genocchi number,
Apostol-Genocchi polynomial, Binomial coefficient, Generalized Apostol-Genocchi polynomials,
Binomial inversion

1. Introduction

The long history of the Genocchi numbers and polynomials can be traced back to
Angelo Genocchi (1817-1889). The classical Genocchi numbers are a sequence of integers
that satisfy the exponential generating function

2t

et + 1
=
∞∑
n=0

Gn
tn

n!
, |t| < π.

The first few Genocchi numbers are

G0 = 0, G1 = 1, G2 = −1, G3 = 0, G4 = 1, G5 = 0, G6 = −3.

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v13i3.3705

Email addresses: nestor.acala@gmail.com (N. Acala), eraleluya@gmail.com (E. Aleluya)

https://www.ejpam.com 403 c© 2020 EJPAM All rights reserved.



N. Acala, E. Aleluya / Eur. J. Pure Appl. Math, 13 (3) (2020), 403-413 404

The classical Genocchi polynomials are usually defined by means of the exponential gen-
erating function

2t

et + 1
· ext =

∞∑
n=0

Gn(x)
tn

n!
, |t| < π.

It can be seen that Gn(0) = Gn. Nowadays, Genocchi numbers and kinds of Genocchi
polynomials have been widely studied and extensive studies have linked these numbers
and polynomials in many branches of mathematics such as in analytic number theory,
p-adic number theory, special functions and mathematical analysis, numerical analysis,
combinatorics [1–7], and others.

Many researchers introduced generalizations to the classical Genocchi numbers and
polynomials. For instance, Araci et.al [7] and Kim et al. [12] explored the Genocchi
polynomials of higher order arising from Genocchi basis, which were defined by(

2t

et + 1

)k
· ext =

∞∑
n=0

G(k)
n (x)

tn

n!
, (|t| < π, k ∈ N ∪ {0})

and established interesting identities. Moreover, He et al.[9] defined the Apostol-Genocchi
polynomials as an extension of the classic Genocchi polynomials, which were given by

2t

λet + 1
· ext =

∞∑
n=0

Gλn(x)
tn

n!
(|t+ log λ| < π, λ 6= 0).

In [11], Jolany et al. generalized Apostol-Genocchi numbers and polynomials using the
following generating functions: For a, b, c > 0 and λ 6= 0,

2t

λbt + at
=

∞∑
n=0

Gλn(a, b)
tn

n!
(|t log(b/a) + log λ| < π) , (1)

2t

λbt + at
ext =

∞∑
n=0

Gλn(x; a, b)
tn

n!
(|t log(b/a) + log λ| < π) , (2)

2t

λbt + at
cxt =

∞∑
n=0

Gλn(x; a, b, c)
tn

n!
(|t log(b/a) + log λ| < π) . (3)

For similar generalizations and applications of Genocchi polynomials and other type of
polynomials involving parameters a, b and c, see [8, 13, 15].

The generating function of the Apostol-Genocchi numbers (polynomials) is similar to
those of the Bernoulli numbers(polynomials) and the Euler numbers(polynomials), so it
may be expected that the Apostol-Genocchi numbers(polynomials) satisfy similar identi-
ties as those established for Euler and Bernoulli numbers and polynomials. In fact, most
literature on Apostol-Genocchi numbers and polynomials provide the associations of these
three kinds of numbers (polynomials) (e.g.[10]). In [14], Ozden unified the generating
functions of the Bernoulli, Euler and Genocchi numbers and polynomials and gave some
new relations on these numbers.



N. Acala, E. Aleluya / Eur. J. Pure Appl. Math, 13 (3) (2020), 403-413 405

In [16], Zou obtained identities which associate only the classical Genocchi numbers Gn
and polynomials Gn(x). This motivates us to establish identities which concern only the
multiparameter generalized Apostol-Genocchi numbers Gλn(a, b) and generalized Apostol-
Genocchi polynomials Gλn(x; a, b, c). Obviously, Gλn(a, b) and Gλn(x; a, b, c) reduce to Gn
and Gn(x) when λ = 1, b = c = e, and a = 1. Hence, results here are generalizations of
the results obtained in [16].

2. Identities on Generalized Apostol-Genocchi Numbers and
Polynomials

In this section, we establish some identities involving the generalized Apostol-Genocchi
numbers and generalized Apostol-Genocchi polynomials using their generating functions
with the aid of binomial inversion formula and summation transform techniques.

Theorem 1. For n ≥ 2,

(i)
1

2

n∑
k=0

(
n

k

)
Gλk(x; a, b, c)

[
λ ln b ·Gλn−k+1(ln b; a, b) + ln a ·Gλn−k+1(ln a; a, b)

]
n− k + 1

= x ln c ·Gλn(x; a, b, c)− n

n+ 1
Gλn+1(x; a, b, c).

(ii)
1

2

n∑
k=0

(
n

k

)
Gλk+1(x; a, b, c)

[
λ ln b ·Gλn−k(ln b; a, b) + ln a ·Gλn−k(ln a; a, b)

]
k + 1

= x ln c ·Gλn(x; a, b, c)− n

n+ 1
Gλn+1(x; a, b, c).

Proof. Taking the partial derivatives of the left side of equation (3) with respect to t
yields

∂

∂t

(
2t

λbt + at
cxt
)

=
2cxt

λbt + at
+
x ln c · 2tcxt

λbt + at
− 2tcxt(λ ln b · bt + ln a · at)

(λbt + at)2

=
1

t
· 2tcxt

λbt + at
+
x ln c · 2tcxt

λbt + at
− 2tcxt

λbt + at
· 1

2t

[
λ ln b · 2tet ln b + ln a · 2tet ln a

λbt + at

]
(4)

=
∞∑
n=0

Gλn(x; a, b, c)
tn−1

n!
+ x ln c ·

∞∑
n=0

Gλn(x; a, b, c)
tn

n!

−1

2

∞∑
n=0

Gλn(x; a, b, c)
tn

n!
·
∞∑
n=0

[
λ ln b ·Gλn(ln b; a, b) + ln a ·Gλn(ln a; a, b)

] tn−1
n!

=
∞∑
n=1

Gλn(x; a, b, c)
tn−1

n!
+ x ln c ·

∞∑
n=0

Gλn(x; a, b, c)
tn

n!

−1

2

∞∑
n=0

Gλn(x; a, b, c)
tn

n!
·
∞∑
n=1

[
λ ln b ·Gλn(ln b; a, b) + ln a ·Gλn(ln a; a, b)

] tn−1
n!
.

The last equation follows from the fact that Gλ0(x; a, b) = Gλ0(x; a, b, c) = 0. Reindexing
and using Cauchy product for series, we obtain

∂

∂t

(
2t

λbt + at
cxt
)

=
∞∑
n=0

[
Gλn+1(x; a, b, c)

n+ 1
+ x ln c ·Gλn(x; a, b, c)

−1

2
·
n∑
k=0

(
n

k

)
Gλk(x; a, b, c)

(
λ ln b ·Gλn−k+1(ln b; a, b) + ln a ·Gλn−k+1(ln a; a, b)

n− k + 1

)]
tn

n!
. (5)



N. Acala, E. Aleluya / Eur. J. Pure Appl. Math, 13 (3) (2020), 403-413 406

On the other hand, taking the partial derivative of the right hand side of equation (3)
gives us

∂

∂t

[ ∞∑
n=0

Gλn(x; a, b, c)
tn

n!

]
=
∞∑
n=0

Gλn+1(x; a, b, c)
tn

n!
. (6)

Comparing the coefficients of
tn

n!
in equations (5) and (6), we obtain Theorem 1(i).

For the second part, we note that (4) can also be expressed as

∂

∂t

(
2t

λbt + at
cxt
)

=
1

t
· 2tcxt

λbt + at
+
x ln c · 2tcxt

λbt + at
− 1

2t
· 2tcxt

λbt + at

[
λ ln b · 2tet ln b + ln a · 2tet ln a

λbt + at

]

=
∞∑
n=0

Gλn(x; a, b, c)
tn−1

n!
+ x ln c ·

∞∑
n=0

Gλn(x; a, b, c)
tn

n!

−1

2

∞∑
n=0

Gλn(x; a, b, c)
tn−1

n!
·
∞∑
n=0

[
λ ln b ·Gλn(ln b; a, b) + ln a ·Gλn(ln a; a, b)

] tn
n!

Reindexing and grouping, we obtain

∂

∂t

(
2t

λbt + at
cxt
)

=
∞∑
n=0

Gλn+1(x; a, b, c)

n+ 1

tn

n!
+ x ln c ·

∞∑
n=0

Gλn(x; a, b, c)
tn

n!

−1

2

∞∑
n=0

Gλn+1(x; a, b, c)

n+ 1

tn

n!
·
∞∑
n=0

[λ ln b ·Gλn(ln b; a, b) + ln a ·Gλn(ln a; a, b)]
tn

n!

=
∞∑
n=0

[
Gλn+1(x; a, b, c)

n+ 1
+ x ln c ·Gλn(x; a, b, c)

−1

2
·
n∑
k=0

(
n

k

)
Gλk+1(x; a, b, c)

k + 1
[λ ln b ·Gλn−k(ln b; a, b) + ln a ·Gλn−k(ln a; a, b)]

]
tn

n!
. (7)

Comparing the coefficients of
tn

n!
in equations (6) and (7), we obtain Theorem 1(ii).

When k goes from 0 to n, n− k also goes from 0 to n. Hence, replacing k by n− k in
Theorem 1, we have the following remark.

Remark 1. For n ≥ 2,

(i)
1

2

n∑
k=0

(
n

k

)
Gλn−k(x; a, b, c)

[
λ ln b ·Gλk+1(ln b; a, b) + ln a ·Gλk+1(ln a; a, b)

]
k + 1

= x ln c ·Gλn(x; a, b, c)− n

n+ 1
Gλn+1(x; a, b, c).

(ii)
1

2

n∑
k=0

(
n

k

)
Gλn−k+1(x; a, b, c)

[
λ ln b ·Gλk(ln b; a, b) + ln a ·Gλk(ln a; a, b)

]
n− k + 1

= x ln c ·Gλn(x; a, b, c)− n

n+ 1
Gλn+1(x; a, b, c).

In the case when c = 1 or x = 0 in Theorem 1 yields the following corollary.



N. Acala, E. Aleluya / Eur. J. Pure Appl. Math, 13 (3) (2020), 403-413 407

Corollary 1. For n ≥ 2,

(i)
1

2

n∑
k=0

(
n

k

)
Gλk(a, b)

[
λ ln b ·Gλn−k+1(ln b; a, b) + ln a ·Gλn−k+1(ln a; a, b)

]
n− k + 1

= − n

n+ 1
Gλn+1(a, b).

(ii)
1

2

n∑
k=0

(
n

k

)
Gλk+1(a, b)

[
λ ln b ·Gλn−k(ln b; a, b) + ln a ·Gλn−k(ln a; a, b)

]
k + 1

= − n

n+ 1
Gλn+1(a, b).

At this point, we now take a look on the Gλn(ln b; a, b). Differentiating both sides of

equation (2) with respect to t, and evaluating it at t = 0, we obtain Gλ1(x; a, b) =
2

λ+ 1
.

Also, we note that

∞∑
n=0

[
λGλn(ln b; a, b) +Gλn(ln a; a, b)

] tn
n!

=
λ2t

λbt + at
bt +

2t

λbt + at
at = 2t. (8)

Hence, evaluating the nth derivative of (8) at t = 0 for n ≥ 2, we obtain

λGλn(ln b; a, b) +Gλn(ln a; a, b) = 0.

Thus, we have the following lemma.

Lemma 1.

Gλn(ln b; a, b) =


2

λ+ 1
, if n = 1

− 1

λ
Gλn(ln a; a, b), if n ≥ 2.

(9)

By applying Lemma 1 and using the fact that Gλ0(ln b; a, b) = 0, Theorem 1 reduces to
the next corollary.

Corollary 2. For n ≥ 2,

(i)
1

2
ln

(
b

a

) n−1∑
k=0

(
n

k

)
Gλk(x; a, b, c)Gλn−k+1(ln a; a, b)

n− k + 1
=

(
λ ln b+ ln a

λ+ 1
− x ln c·

)
Gλn(x; a, b, c) +

n

n+ 1
Gλn+1(x; a, b, c).

(ii)
1

2
ln

(
b

a

) n−2∑
k=0

(
n

k

)
Gλk+1(x; a, b, c)Gλn−k(ln a; a, b)

k + 1
=

(
λ ln b+ ln a

λ+ 1
− x ln c·

)
Gλn(x; a, b, c) +

n

n+ 1
Gλn+1(x; a, b, c).

Proof. For the first part, we replaceGλ1(ln a; a, b) andGλ1(ln b; a, b) by 2
λ+1 , Gλn−k+1(ln b; a, b)

by − 1
λG

λ
n−k+1(ln a; a, b) for k 6= n in Theorem 1 (i) to obtain

x ln c ·Gλn(x; a, b, c)− n

n+ 1
Gλn+1(x; a, b, c) = Gλn(x; a, b, c)

(
λ ln b+ ln a

λ+ 1

)
−1

2

n−1∑
k=0

(
n

k

)
Gλk(x; a, b, c)Gλn−k+1(ln a; a, b)[ln b− ln a]

n− k + 1



N. Acala, E. Aleluya / Eur. J. Pure Appl. Math, 13 (3) (2020), 403-413 408

Arranging and grouping the terms will give the desired result.
For the second part, we replace Gλn−k(ln b; a, b) by − 1

λG
λ
n−k(ln a; a, b) for k 6= n and

k 6= n− 1 in Theorem 1(ii) to obtain

x ln c ·Gλn(x; a, b, c)− n

n+ 1
Gλn+1(x; a, b, c) = Gλn(x; a, b)

(
λ ln b+ ln a

λ+ 1

)
−1

2

n−2∑
k=0

(
n

k

)
Gλk+1(x; a, b, c)Gλn−k(ln a; a, b)[ln b− ln a]

k + 1
.

It can be seen that Gλn(ln a; a, b) can be expressed in terms of the generalized Apostol-
Genocchi numbers. Indeed,

∞∑
n=0

Gλn(ln a; a, b)
tn

n!
=

2t · at

λbt + at
=

2t

λ
(
b
a

)t
+ 1t

=

∞∑
n=0

Gλn (1, b/a)
tn

n!
.

Thus, we have
Gλn(ln a; a, b) = Gλn (1, b/a) . (10)

Taking x = 0 in Corollary 2 and using identity (10), we obtain the following identities
involving generalized Apostol-Genocchi numbers only.

Corollary 3. For n ≥ 2,

(i)
1

2
ln

(
b

a

) n−1∑
k=0

(
n

k

)
Gλk(a, b)Gλn−k+1 (1, b/a)

n− k + 1
=

(
λ ln b+ ln a

λ+ 1

)
Gλn(a, b) +

n

n+ 1
Gλn+1(a, b).

(ii)
1

2
ln

(
b

a

) n−2∑
k=0

(
n

k

)
Gλk+1(a, b)G

λ
n−k (1, b/a)

k + 1
=

(
λ ln b+ ln a

λ+ 1

)
Gλn(a, b) +

n

n+ 1
Gλn+1(a, b).

Now, we consider the generalized Apostol-Genocchi polynomials Gλn(x+ y; a, b, c) that
involve sum of two variables.

Theorem 2. For n ≥ 2 and y 6= 0,

Gλn(x+ y; a, b, c) =

n∑
k=0

(
n

k

)
(ln c)n−kGλk(x; a, b, c)yn−k. (11)

Proof. By definition,

∞∑
n=0

Gλn(x+ y; a, b, c)
tn

n!
=

2t

λbt + at
c(x+y)t =

2t

λbt + at
cxt · cyt.

Hence, for y 6= 0,

∞∑
n=0

Gλn(x+ y; a, b, c)
tn

n!
=

∞∑
n=0

Gλn(x; a, b, c)
tn

n!

∞∑
n=0

(ln c)nyn
tn

n!



N. Acala, E. Aleluya / Eur. J. Pure Appl. Math, 13 (3) (2020), 403-413 409

=

∞∑
n=0

n∑
k=0

(
n

k

)
Gλk(x; a, b, c)(ln c)n−kyn−k

tn

n!
.

Comparing the coefficients of
tn

n!
, we obtain the desired identity.

Theorem 3. For n ≥ 2 and y 6= 0,

Gλn(x; a, b, c) = (−1)n
n∑
k=0

(−1)k
(
n

k

)
(ln c)n−kGλk(x+ y; a, b, c)yn−k. (12)

Proof. In this case, we need the binomial inversion formula

rn =
n∑
k=0

(
n

k

)
(−1)ksk ⇔ sn =

n∑
k=0

(
n

k

)
(−1)krk.

Note that equation (11) can written as

Gλn(x+ y; a, b, c)

(ln c)nyn
=

n∑
k=0

(
n

k

)
Gλk(x; a, b, c)

(ln c)kyk
.

Taking

rk =
Gλk(x+ y; a, b, c)

(ln c)kyk
and (−1)ksk =

Gλk(x; a, b, c)

(ln c)kyk

gives us

(−1)n
Gλn(x; a, b, c)

(ln c)nyn
=

n∑
k=0

(
n

k

)
(−1)k

Gλk(x+ y; a, b, c)

(ln c)kyk
.

That is,

Gλn(x; a, b, c) = (−1)n
n∑
k=0

(−1)k
(
n

k

)
(ln c)n−kGλk(x+ y; a, b, c)yn−k.

Symmetrically, we obtain the following:

Corollary 4. For n ≥ 2 and x 6= 0,

(i) Gλn(x+ y; a, b, c) =
n∑
k=0

(
n

k

)
(ln c)n−kGλk(y; a, b, c)xn−k;

(ii) Gλn(y; a, b, c) = (−1)n
n∑
k=0

(−1)k
(
n

k

)
(ln c)n−kGλk(x+ y; a, b, c)xn−k.

Replacing k by n − k in Theorem 3 and Corollary 4 (ii), we obtain a more beautiful
expressions given in the next corollary.



N. Acala, E. Aleluya / Eur. J. Pure Appl. Math, 13 (3) (2020), 403-413 410

Corollary 5. For n ≥ 2,

(i) Gλn(x; a, b, c) =

n∑
k=0

(−1)k
(
n

k

)
(ln c)kGλn−k(x+ y; a, b, c)yk, y 6= 0;

(ii) Gλn(y; a, b, c) =
n∑
k=0

(−1)k
(
n

k

)
(ln c)kGλn−k(x+ y; a, b, c)xk, x 6= 0.

By taking y = (p − 1)x, equation (11) reduces to the multiplication formula of the
generalized Apostol-Genocchi polynomials as shown in the following corollary.

Corollary 6. For p 6= 1 and x 6= 0,

Gλn(px; a, b, c) =
n∑
k=0

(
n

k

)
(ln c)n−kGλk(x; a, b, c)(p− 1)n−kxn−k.

Theorem 4. For n ≥ 2,

n∑
k=0

(−1)k+1

(
n

k

)
(ln b)1−k(ln c)n−kGλn(1, b/a) = (−1)nλ(ln b)1−nGλn(a, b) + 2n.

Proof. Note that

cxt =
1

2t

[
2tλbtcxt + 2tatcxt

λbt + at

]
=

1

2t

[
λ2tc(x+logc b)t + 2tc(x+logc a)t

λbt + at

]
.

Consequently,

∞∑
n=0

(ln c)nxn
tn

n!
=

∞∑
n=0

[
λGλn+1(x+ logc b; a, b, c) +Gλn+1(x+ logc a; a, b, c)

2(n+ 1)

]
tn

n!
.

Comparing the coefficients of
tn

n!
, we obtain

λGλn+1(x+ logc b; a, b, c) +Gλn+1(x+ logc a; a, b, c)

2(n+ 1)
= (ln c)nxn,

or equivalently

λGλn(x+ logc b; a, b, c) +Gλn(x+ logc a; a, b, c) = 2n(ln c)n−1xn−1. (13)

Taking x = − logc b in equation (13) yields

λGλn(a, b) +Gλn(logc a− logc b; a, b, c) = 2n(− ln b)n−1. (14)



N. Acala, E. Aleluya / Eur. J. Pure Appl. Math, 13 (3) (2020), 403-413 411

Moreover, letting x = logc a and y = − logc b in Theorem 2 results to

Gλn(logc a− logc b; a, b, c) =
n∑
k=0

(−1)n−k
(
n

k

)
(ln b · ln c)n−kGλk(logc a; a, b, c). (15)

Plugging (18) in (14) and using the fact that Gλn(logc a; a, b, c) = Gλn(1, b/a), we get the
desired result.

Now, we express Gλn(1, b/a) as linear combination of the generalized Apostol-Genocchi
numbers Gλk(a, b).

Corollary 7. For n ≥ 2,

Gλn(1, b/a) = −λ
n∑
k=0

(
n

k

)
(ln b)n−kGλk(a, b).

Proof. Taking x = logc b and y = 0 in Corollary 4 (i),we obtain

Gλn(logc b; a, b, c) =
n∑
k=0

(
n

k

)
(ln b)n−kGλk(a, b) (16)

Utilizing Lemma 1, we get

Gλn(logc b; a, b, c) = Gλn(ln b; a, b) = − 1

λ
Gλn(ln a; a, b).

Combining (10) and (16) proves this corollary.

Now, let us see some identities involving definite integrals of generalized Apostol-
Genocchi polynomials.

Differentiating both sides of the exponential generating function for Gλn(x; a, b, c) in
(3) with respect to x gives

d

dx
Gλn(x; a, b, c) = n ln c ·Gλn−1(x; a, b, c) and degGλn+1(x; a, b, c) = n.

Consequently, ∫ u2

u1

Gλn(x; a, b, c)dx =
Gλn+1(u2; a, b, c)−Gλn+1(u1; a, b, c)

ln c · (n+ 1)
. (17)

Theorem 5.

∫ logc b

logc a
Gλn(x; a, b, c)dx =


0, n = 0

−
(
λ+ 1

λ ln c

)
Gλn+1(ln a; a, b)

(n+ 1)
, n ≥ 1.

(18)



REFERENCES 412

Proof. This follows from (17) and Lemma 1.

Note that when a = 1, b = c = e and λ = 1, (18) reduces to the known identity for
classical Genocci numbers and polynomials,∫ 1

0
Gn(x)dx =

0, n = 0

−2
Gn+1

n+ 1
, n ≥ 1.

The next corollary shows that the definite integral in the left-hand side of equation (18)
can be expressed as linear combination of generalized Apostol-Genocchi numbers Gλk(a, b).

Corollary 8. For n ≥ 2,∫ logc b

logc a
Gλn−1(x; a, b, c)dx =

λ+ 1

n ln c

n∑
k=0

(
n

k

)
(ln b)n−kGλk(a, b).

Proof. This follows from Theorem 5, identity (10), and Corollary 7.

Using (17), we obtain the double integral of Gλn(x+ y; a, b, c) in the next corollary.

Theorem 6.∫ v2

v1

∫ u2

u1

Gλn(x+ y; a, b, c)dxdy =
Gλn+2(u2 + v2; a, b, c)−Gλn+2(u2 + v1; a, b, c)

(ln c)2(n+ 1)(n+ 2)

−

[
Gλn+2(u1 + v2; a, b, c)−Gλn+2(u1 + v1; a, b, c)

(ln c)2(n+ 1)(n+ 2)

]
.

Remark 2. In the case when a = 1, b = c = e and and λ = 1, the obtained results here
reduce to old (or new) identities of classical Genocchi polynomials.

Conclusion

A significant result of this paper is that we have established relationships between
generalized Apostol-Genocchi numbers and generalized Apostol-Genocchi polynomials in-
volving binomial coefficients even without associating these numbers (polynomials) to the
Bernoulli, Euler and Stirling-type numbers (polynomials). However, combining these new
identities with the existing identities between Genocchi, Bernoulli and Euler numbers
(polynomials), one can obtain other further identities.

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