EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
2025, Vol. 18, Issue 3, Article Number 6167
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

The Type 2 Degenerate Hermite-Based
Apostol-Frobenius-type Poly-Genocchi Polynomials

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
3 Research Institute for Tropical Biology and Pharmacological Biotechnology, Cebu Normal
University, 6000 Cebu City, Philippines

Abstract. This paper introduces a novel variation of poly-Genocchi polynomials by combining
elements from the modified degenerate polyexponential function, Hermite-Based Apostol-Genocchi
polynomials, and Frobenius polynomials. These newly formulated polynomials, termed as the
degenerate Hermite-Based Apostol-Frobenius-type poly-Genocchi polynomials with parameters a,
b, and c, are explored to derive various identities and formulas. These include recurrence relations,
explicit formulas, and specific differential identities. Furthermore, the paper establishes connections
between these polynomials and degenerate Stirling numbers of the first and second kind, higher-
order degenerate Bernoulli polynomials, and higher-order degenerate Frobenius-Euler polynomials.

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

Key Words and Phrases: Genocchi Polynomials, Bell Polynomials, Apostol-Type Frobenius-
Genocchi Polynomials, Hermite-Based Apostol-Type Frobenius-Genocchi Polynomials, Degenerate
Stirling Numbers

1. Introduction

The theory of special polynomials is vibrant area of modern mathematics, owing to
its fundamental role in number theory, combinatorics, mathematical analysis, and math-
ematical physics. Among these, the Bernoulli, Euler, and Genocchi polynomials occupy a
central place due to their deep connections with zeta functions, generating functions, and
summation formulas [1–6]. Classical Genocchi numbers Gn are defined by the generating
function:

∞∑
n=0

Gn
tn

n!
=

2t

et + 1
, |t| < π. (1)

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v18i3.6167

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

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



R. B. Corcino, C. B. Corcino / Eur. J. Pure Appl. Math, 18 (3) (2025), 6167 2 of 20

Several results involving these numbers, such as recurrence formulas and identities, can
be found in [7–9].

Through multiplication with exponential functions, various generalizations of the Genoc-
chi numbers have been constructed, including:

∞∑
n=0

Gn(x)
tn

n!
=

2t

et + 1
ext, (2)

∞∑
n=0

G(k)
n (x)

tn

n!
=

(
2t

et + 1

)k

ext, (3)

∞∑
n=0

Gn(x, λ)
tn

n!
=

2t

λet + 1
ext, (4)

∞∑
n=0

G(k)
n (x, λ)

tn

n!
=

(
2t

λet + 1

)k

ext, (5)

where the case λ = 1 retrieves the classical forms. The first two of these have been
extensively studied, with asymptotic approximations and Fourier expansions discussed in
[10–14].

Further generalization has been achieved by introducing the Frobenius-Genocchi poly-
nomials, defined as:

∞∑
n=0

GF
n (x;u)

tn

n!
=

(1− u)t

et − u
ext, (6)

as considered in [15–23].

Another prominent line of generalization involves the use of the polylogarithm function

Lik(z) =

∞∑
n=1

zn

nk
.

When applied to Genocchi polynomials, this yields the poly-Genocchi polynomials:

∞∑
n=0

G(k)
n (x)

xn

n!
=

2Lik(1− et)

et + 1
ext, (7)

∞∑
n=0

G
(k)
n,2(x)

xn

n!
=

Lik(1− e−2t)

et + 1
ext, (8)

as studied in [24–26]. When k = 1, these reduce to the classical Genocchi polynomials in
(2).

Generalized versions with parameters a, b, and c were introduced by Kurt in [27],
yielding:

2Lik(1− (ab)−t)

a−t + bt
cxt =

∞∑
n=0

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

xn

n!
, (9)



R. B. Corcino, C. B. Corcino / Eur. J. Pure Appl. Math, 18 (3) (2025), 6167 3 of 20

2Lik(1− (ab)−2t)

a−t + bt
cxt =

∞∑
n=0

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

xn

n!
. (10)

These forms provide a unifying framework for further generalization. When x = 0, (7)
simplifies to:

2Lik(1− et)

et + 1
=

∞∑
n=0

G(k)
n

xn

n!
, (11)

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

By extending these forms further through multi-polylogarithms and Hermite struc-
tures, Corcino et al. [28, 29] introduced the polynomials:

∞∑
n=0

Ĝ(k,α)
n (x;λ, ρ, u, a, b)

tn

n!
=

(
Lik,ρ(1− (ab)−(1−u)t)

λbt − ua−t

)α

cxt, (12)

which encapsulate multiple layers of generalization, including Apostol-type, Frobenius-
type, and polylogarithmic components.

Degeneracy offers yet another lens of generalization. The degenerate exponential func-
tion introduced by Carlitz [6, 30] is given by:

exλ(t) = (1 + λt)x/λ =
∞∑
n=0

(x)n,λ
tn

n!
, (13)

where (x)n,λ denotes the degenerate falling factorial. This function is central to construct-
ing degenerate versions of classical polynomials. It can easily be seen that

ex+y
λ (t) = (1 + λt)(x+y)/λ = (1 + λt)(x/λ)+(y/λ)

= (1 + λt)(x/λ)(1 + λt)(y/λ)

= exλ(t)e
y
λ(t), (14)

and
d

dx
exλ(t) = log(1 + λt)1/λexλ(t). (15)

The degenerate Bernoulli polynomials Bn,λ(x) and degenerate Euler polynomials En,λ(x)
were defined by Carlitz [30] by means of the following generating functions

t

eλ(t) + 1
exλ(t) =

t

(1 + λt)1/λ + 1
(1 + λt)x/λ =

∞∑
n=0

Bn,λ(x)
tn

n!

2

eλ(t) + 1
exλ(t) =

2

(1 + λt)1/λ + 1
(1 + λt)x/λ =

∞∑
n=0

En,λ(x)
tn

n!
.



R. B. Corcino, C. B. Corcino / Eur. J. Pure Appl. Math, 18 (3) (2025), 6167 4 of 20

Parallel to these, Lim [31] defined the degenerate Genocchi polynomials as follows

2t

eλ(t) + 1
exλ(t) =

2t

(1 + λt)1/λ + 1
(1 + λt)x/λ =

∞∑
n=0

Gn,λ(x)
tn

n!
. (16)

Kim and Kim [32] introduced the degenerate Frobenius-Euler polynomials as coefficients
of the following generating function

1− u

eλ(t)− u
exλ(t) =

∞∑
n=0

hn,λ(x|u)
tn

n!
(17)

and Kim et al. [33] derived formulas that express any polynomial in terms of hn,λ(x|u). In
their separate paper, Kim and Kim [34] defined the generalized degenerate Euler-Genocchi

polynomials, denoted by A
(r)
n,λ(x), as coefficients of the following generating function

2tr

eλ(t) + 1
exλ(t) =

∞∑
n=0

A
(r)
n,λ(x)

tn

n!
,

which reduce to the degenerate Genocchi polynomials in (26) when r = 1. That is,

A
(1)
n,λ(x) = Gn,λ(x).

The growing literature on degenerate versions includes significant contributions from
Kim and collaborators [35–42], who explored degenerate Bernoulli, Euler, and Genocchi-
type polynomials using probabilistic and algebraic techniques.

The degenerate Stirling numbers of the first and second kind, denoted by S1,ρ(n, k)
and S2,ρ(n, k), were defined in ([43–45])

(logρ(1 + t))k

k!
=

∞∑
n=0

S1,ρ(n, k)
tn

n!
,

(eρ(t)− 1)k

k!
=

∞∑
n=0

S2,ρ(n, k)
tn

n!
, (18)

where
logρ(eρ(t)) = eρ(logρ(t)) = t. (19)

When ρ → 0,

lim
ρ→0

S1,ρ(n, k) = S1(n, k), lim
ρ→0

S2,ρ(n, k) = S2(n, k)

where S1(n, k) and S2(n, k) are the classical Stirling numbers of the first and second kind.
Also, the degenerate Bernoulli polynomials of the second kind are defined by

(1 + t)x

logρ(1 + t)
t =

∞∑
n=0

bn,ρ(x)
tn

n!
. (20)



R. B. Corcino, C. B. Corcino / Eur. J. Pure Appl. Math, 18 (3) (2025), 6167 5 of 20

The degenerate Stirling numbers of the second kind appeared in the probability distribu-
tion of the random variable given as the sum of a finite number of random variables with
degenerate zero-truncated Poisson distributions and a random variable with degenerate
Poisson distribution, all having the same parameter (see [45]).

The polyexponential functions are defined by the following generating functions [43, 46–
48]

Eik(x) =
∞∑
n=1

xn

nk(n− 1)!
, k ∈ Z. (21)

For k = 1, Ei1(x) = ex − 1. The modified degenerate polyexponential function are given
by ([43, 46–48])

Eik,ρ(x) =

∞∑
n=1

(1)n,ρx
n

nk(n− 1)!
, ρ ∈ R. (22)

Note that

Ei1,ρ(x) =
∞∑
n=1

(1)n,ρ
xn

n!
= eρ(x)− 1, ρ ∈ R. (23)

Also, we have

d

dx
Eik,ρ(logρ(1 + x)) =

(1 + x)ρ−1

logρ(1 + x)
Eik−1,ρ(logρ(1 + x)), (24)

which implies

Eik,ρ(logρ(1 + x)) =

∫ x

0

(1 + t)ρ−1

logρ(1 + t)

∫ y

0
. . .

(1 + x)ρ−1

logρ(1 + x)

∫ y

0

(1 + x)ρ−1

logρ(1 + x)
xdx . . . dx

=

∞∑
m=0

∑
m1+m2+...+mk−1=m

(
m

m1, . . . ,mk−1

)

× bm1,ρ(ρ− 1)

m1 + 1

bm2,ρ(ρ− 1)

m1 +m2 + 1
. . .

bmk−1,ρ(ρ− 1)

m1 + . . .+mk−1 + 1

xm+1

m!
. (25)

The degenerate poly-Euler polynomials were defined in [49] by means of the following
generating function

2
Eik,ρ(logρ(1 + t))

teρ(t) + 1
exρ(t) =

∞∑
n=0

E(k)
n,ρ(x)

tn

n!
, (26)

where k ∈ Z.
Building upon these foundations, this paper introduces a novel variant of poly-Genocchi

polynomials, created by integrating concepts from the modified degenerate polyexponential
function, Apostol-Genocchi polynomials, and Frobenius polynomials. Termed as the type
2 degenerate hermite-based Apostol-Frobenius-type poly-Genocchi polynomials of higher
order with parameters a and b, special cases of these polynomials are outlined, along



R. B. Corcino, C. B. Corcino / Eur. J. Pure Appl. Math, 18 (3) (2025), 6167 6 of 20

with several identities showcasing their relations with various Genocchi-type polynomi-
als. Additionally, connections of these degenerate Apostol-Frobenius-type poly-Genocchi
polynomials with degenerate Stirling numbers of the first and second kind, higher-order de-
generate Bernoulli polynomials, and higher-order degenerate Frobenius-Euler polynomials
are discussed.

Furthermore, this paper establishes significant links between the new polynomials and
several well-studied degenerate sequences, including the degenerate Stirling numbers of the
first and second kinds, higher-order degenerate Bernoulli polynomials [35, 36, 38, 41], and
higher-order degenerate Frobenius-Euler polynomials [37]. These interrelations highlight
the rich algebraic structure of the new family and position them within the expanding
domain of probabilistic and degenerate special functions [39, 40, 42].

Finally, notable applications of Genocchi and poly-Genocchi polynomials include:

(i) Estimating the number of finite languages accepted by finite automata [50];

(ii) Deriving wavelet-based numerical solutions to fractional Rosenau-Hyman equations
[51];

(iii) Solving fractional differential equations through operational matrix methods using
poly-Genocchi polynomials [52].

2. Definition and Some Explicit Formulas

Similar to the definition of degenerate poly-Euler polynomials outlined in (26), we can
establish the desired variation of Apostol-type poly-Genocchi polynomials by introducing
the parameter u to encompass the concept of Frobenius polynomials, along with the pa-
rameters a and b as well as the degenerate exponential polynomials eyρ(t2). The following
presents the formal definition of these desired polynomials.

Definition 2.1. The type 2 degenerate Hermite-based Apostol-Frobenius-type poly-Genocchi

polynomials of higher order with parameters a and b, denoted by Ĝ(k,α)
n (x, y;λ, ρ, u, a, b),

are defined as coefficients of the following generating function:

∞∑
n=0

Ĝ(k,α)
n (x, y;λ, ρ, u, a, b)

tn

n!
=

(
Eik,ρ(logρ(1 + (1− u)t ln ab))

λbt − ua−t

)α

exρ(t)e
y
ρ(t

2), (27)

where |t| <
√

(ln(λ
u))2+4π2

| ln a+ln b| . When α = 1, (27) yields

∞∑
n=0

Ĝ(k)
n (x, y;λ, ρ, u, a, b)

tn

n!
=

Eik,ρ(logρ(1 + (1− u)t ln ab))

λbt − ua−t
exρ(t)e

y
ρ(t

2). (28)

where Ĝ(k)
n (x, y;λ, ρ, u, a, b) = Ĝ(k,1)

n (x, y;λ, ρ, u, a, b) denotes the degenerate Hermite-based
Apostol-Frobenius-type poly-Genocchi polynomials with parameters a and b.



R. B. Corcino, C. B. Corcino / Eur. J. Pure Appl. Math, 18 (3) (2025), 6167 7 of 20

Now, if x = (1− u)t ln ab, then (25) yields

Eik,ρ(logρ(1 + (1− u)t ln ab))

= t
∞∑

m=0

((1− u) ln ab)m+1
∑

m1+m2+...+mk−1=m

(
m

m1, . . . ,mk−1

)

× bm1,ρ(ρ− 1)

m1 + 1

bm2,ρ(ρ− 1)

m1 +m2 + 1
. . .

bmk−1,ρ(ρ− 1)

m1 + . . .+mk−1 + 1

tm

m!
.

Also,

exρ(t) =
∞∑
n=0

(x)n,ρ
tn

n!

eyρ(t
2) =

∞∑
n=0

(y)n,ρ
(t2)n

n!
=

∞∑
n=0

(y)n,ρ
t2n

n!
=

∞∑
m=0

(y)m
2
,ρ

tm

(m2 )!

(y)m
2
,ρ = (y)(y − ρ) · · ·

(
y −

(m
2

− 1
)
ρ
)

= ρ
m
2

(
y

ρ

)(
y

ρ
− 1

)
· · ·
(
y

ρ
−
(m
2

− 1
))

= ρ
m
2

(
y

ρ

)
m
2

(y)m
2
,ρ(

m
2

)
!

= ρ
m
2

( y
ρ
m
2

)

eyρ(t
2) =

∞∑
m=0

ρ
m
2 m!

( y
ρ
m
2

)
tm

m!
,

(y
ρ

i

)
= 0 , i /∈ N ∪ {0}.

exρ(t) e
y
ρ(t

2) =

( ∞∑
n=0

ρmn!

(x
ρ

n

)
tn

n!

) ( ∞∑
n=0

ρ
n
2
n!

( y
ρ
n
2

)
tn

n!

)

=
∞∑
n=0

n∑
j=0

ρn−j(n− j)!

( x
ρ

n− j

)
tn−j

(n− j)!
ρ

j
2 (j)!

(y
ρ
j
2

)
tj

j!

=
∞∑
n=0

n∑
j=0

n! ρn−
j
2

( x
ρ

n− j

)(y
ρ
j
2

)
tn

n!

exρ(t) e
y
ρ(t

2) =

∞∑
n=0

 n∑
j=0

n! ρn−
j
2

( x
ρ

n− j

)(y
ρ
j
2

) tn

n!



R. B. Corcino, C. B. Corcino / Eur. J. Pure Appl. Math, 18 (3) (2025), 6167 8 of 20

1

λbt − ua−t
=

−1

ua−t
(
1− λ

u(ab)
t
) =

−at

u

1

1− λ
u(ab)

t

=
−at

u

∞∑
n=0

[
λ

u
(ab)t

]n
=

−et log a

u

∞∑
n=0

(
λ

u

)n

ent log ab

=
−1

u

( ∞∑
n=0

(log a)n
tn

n!

) ( ∞∑
n=0

(
λ

u

)n ∞∑
m=0

nm(log ab)m
tm

m!

)

=
−1

u

∞∑
m=0

 m∑
j=0

(log a)j
tj

j!

∞∑
n=0

(
λ

u

)n

nm−j (log ab)m−j t(m−j)

(m− j)!


=

−1

u

∞∑
m=0

 m∑
j=0

∞∑
n=0

(
m

j

)(
λ

u

)n

(log a)j (n log ab)m−j

 tm

m!

=

∞∑
m=0

−1

u

m∑
j=0

∞∑
n=0

(
m

j

)(
λ

u

)n

(log a)j (n log ab)m−j

 tm

m!

exρ(t) e
y
ρ(t2)

λbt − ua−t
=

∞∑
m=0

m∑
i=0

i∑
s

i!ρi−
s
2

( x
ρ

i− s

)(y
ρ
s
2

)
ti

i!

(
−1

u

)m−i∑
j=0

∞∑
n=0

(
m− i

j

)
(
λ

u

)n

(log a)j (n log ab)m−j tm−j

(m− j)!

exρ(t)e
y
ρ(t2)

λbt − ua−t
=

∞∑
m=0

{ m∑
i=0

(
m

i

) i∑
s=0

m−i∑
j=0

∞∑
n=0

i!ρi−
s
2

( x
ρ

i− s

)(y
ρ
s
2

)
(
−1

u

)(
m− i

j

)
×
(
λ

u

)n

(log a)j (n log ab)m−i−j
} tm
m!

exρ(t)

λbt − ua−t
=

∞∑
m=0

m∑
j=0

∞∑
n=0

(
m

j

)
(x)j,ρ

(u
λ

)n
(−n log ab)m−j t

m

m!
.

With

Bm1,m2,...,mk−1
(m, ρ− 1)

= ((1− u) ln ab)m+1
∑

m1+m2+...+mk−1=m

(
m

m1, . . . ,mk−1

)

× bm1,ρ(ρ− 1)

m1 + 1

bm2,ρ(ρ− 1)

m1 +m2 + 1
. . .

bmk−1,ρ(ρ− 1)

m1 + . . .+mk−1 + 1
, (29)



R. B. Corcino, C. B. Corcino / Eur. J. Pure Appl. Math, 18 (3) (2025), 6167 9 of 20

we have
∞∑

m=0

Ĝ(k)
m (x, y;λ, ρ, u, a, b)

tm

m!
=

Eik,ρ(logρ(1 + (1− u)t ln ab))

λbt − ua−t
exρ(t)e

y
ρ(t

2)

= t
∞∑

m=0

m∑
l=0

(
m

l

)
Bm1,m2,...,mk−1

(l, ρ− 1)
{m−l∑

i=0

(
m− l

i

) i∑
s=0

m−l−i∑
j=0

∞∑
n=0

i!ρi−
s
2

( x
ρ

i− s

)(y
ρ
s
2

)
(
−1

u

)(
m− i

j

)
×
(
λ

u

)n

(log a)j (n log ab)m−i−j
} tm
m!

= t
∞∑

m=0

m∑
l=0

m−l∑
i=0

i∑
s=0

m−l−i∑
j=0

∞∑
n=0

(
m

l

)(
m− l

i

)
Bm1,m2,...,mk−1

(l, ρ− 1)i!ρi−
s
2

( x
ρ

i− s

)(y
ρ
s
2

)
(
−1

u

)(
m− i

j

)(
λ

u

)n

(log a)j (n log ab)m−i−j t
m

m!

Note that the right-hand side of the preceding equation has no constant term. Hence,

when m = 0, Ĝ(k)
0 (x, y;λ, ρ, u, a, b) = 0. Moreover,

∞∑
m=0

1

m
Ĝ(k)
m (x, y;λ, ρ, u, a, b)

tm−1

(m− 1)!

=
∞∑

m=0

m∑
l=0

m−l∑
i=0

i∑
s=0

m−l−i∑
j=0

∞∑
n=0

(
m

l

)(
m− l

i

)
Bm1,m2,...,mk−1

(l, ρ− 1)i!ρi−
s
2

( x
ρ

i− s

)(y
ρ
s
2

)
(
−1

u

)(
m− i

j

)(
λ

u

)n

(log a)j (n log ab)m−i−j t
m

m!
.

By comparing the coefficients of tm

m! , we obtain the following explicit formula:

1

m+ 1
Ĝ(k)
m+1(x, y;λ, ρ, u, a, b)

=
m∑
l=0

m−l∑
i=0

i∑
s=0

m−l−i∑
j=0

∞∑
n=0

(
m

l

)(
m− l

i

)
Bm1,m2,...,mk−1

(l, ρ− 1)i!ρi−
s
2

( x
ρ

i− s

)(y
ρ
s
2

)
(
−1

u

)(
m− i

j

)(
λ

u

)n

(log a)j (n log ab)m−i−j .

Furthermore, using the arithmetic-geometric series formula in [53, p.245], we can write

∞∑
n=0

(u
λ

)n
nm−i−j =

Am−i−j

(
u
λ

)(
1− u

λ

)m−i−j+1
,

where An(u) is the Eulerian polynomial

An(u) =
n∑

k=0

A(n, k)uk (30)



R. B. Corcino, C. B. Corcino / Eur. J. Pure Appl. Math, 18 (3) (2025), 6167 10 of 20

with A(n, k), the Eulerian number, satisfying

A(n, k) = (n− k + 1)A(n− 1, k − 1) + kA(n− 1, k).

Thus,

1

m+ 1
Ĝ(k)
m+1(x, y;λ, ρ, u, a, b)

=
m∑
l=0

m−l∑
i=0

i∑
s=0

m−l−i∑
j=0

(
m

l

)(
m− l

i

)
Bm1,m2,...,mk−1

(l, ρ− 1)Am−i−j

(
λ

u

)

i!ρi−
s
2

( x
ρ

i− s

)(y
ρ
s
2

)(
−1

u

)(
m− i

j

)
(log a)j (log ab)m−i−j(

1− u
λ

)m−i−j+1
.

To state formally this result, we have the following theorem.

Theorem 2.2. The type 2 degenerate Hermite-based Apostol-Frobenius-type poly-Genocchi
polynomials with parameters a and b are equal to

1

m+ 1
Ĝ(k)
m+1(x, y;λ, ρ, u, a, b)

=
m∑
l=0

m−l∑
i=0

i∑
s=0

m−l−i∑
j=0

(
m

l

)(
m− l

i

)
Bm1,m2,...,mk−1

(l, ρ− 1)Am−i−j

(
λ

u

)

i!ρi−
s
2

( x
ρ

i− s

)(y
ρ
s
2

)(
−1

u

)(
m− i

j

)
(log a)j (log ab)m−i−j(

1− u
λ

)m−i−j+1
,

where Ĝ(k,α)
0 (x, y;λ, ρ, u, a, b) = 0, An(u) is the Eulerian polynomial and Bm1,m2,...,mk−1

(i, ρ−
1) satisfies (29).

Now, let us extend the explicit formula to higher order degenerate Apostol-Frobenius-
type poly-Genocchi polynomials with parameters a and b. First, we have to find the
expansion of the following function:

Eik,ρ(logρ(1 + (1− u)t ln ab))

λbt − ua−t

= t

∞∑
m=0


m∑
j=0

∞∑
n=0

1

m!

(
m

j

)
Bm1,m2,...,mk−1

(j, ρ− 1)
(u
λ

)n
(−n log ab)m−j

 tm.

Raising this to power α gives(
Eik,ρ(logρ(1 + (1− u)t ln ab))

λbt − ua−t

)α



R. B. Corcino, C. B. Corcino / Eur. J. Pure Appl. Math, 18 (3) (2025), 6167 11 of 20

= tα
∞∑

m=0

∑
k1+k2+...+kα=m

α∏
i=1


ki∑
j=0

∞∑
n=0

1

ki!

(
ki
j

)
Bm1,m2,...,mk−1

(j, ρ− 1)

(u
λ

)n
(−n log ab)ki−j

}
tm.

Hence,

∞∑
m=0

Ĝ(k,α)
m (x, y;λ, ρ, u, a, b)

tm

m!

=

(
Eik,ρ(logρ(1 + (1− u)t ln ab))

λbt − ua−t

)α

exρ(t)e
y
ρ(t

2)

= tα
∞∑

m=0

m∑
q=0

(
m

q

) m∑
j=0

q! ρq−
j
2

( x
ρ

q − j

)(y
ρ
j
2

)
(m− q)!

×
∑

k1+k2+...+kα=m−q

α∏
i=1


ki∑
j=0

∞∑
n=0

1

ki!

(
ki
j

)

Bm1,m2,...,mk−1
(j, ρ− 1)

(u
λ

)n
(−n log ab)ki−j

} tm

m!
.

Note that, when 0 ≤ m ≤ α− 1, Ĝ(k,α)
m (x, y;λ, ρ, u, a, b) = 0. Now, we can further rewrite

the preceding equation as follows:

∞∑
m=−α

Ĝ(k,α)
m+α(x, y;λ, ρ, u, a, b)

tm

(m+ α)!

=

(
Eik,ρ(logρ(1 + (1− u)t ln ab))

λbt − ua−t

)α

exρ(t)e
y
ρ(t

2)

=
∞∑

m=0

m∑
q=0

m∑
j=0

(
m

q

)
q! ρq−

j
2

( x
ρ

q − j

)(y
ρ
j
2

)
(m− q)!

×
∑

k1+k2+...+kα=m−q

α∏
i=1


ki∑
j=0

∞∑
n=0

1

ki!

(
ki
j

)

Bm1,m2,...,mk−1
(j, ρ− 1)

(u
λ

)n
(−n log ab)ki−j

} tm

m!
.

Comparing the coefficients of tm

m! and using the arithmetic-geometric formula yield the
following explicit formula.

Theorem 2.3. The type 2 degenerate Hermite-based Apostol-Frobenius-type poly-Genocchi
polynomials of higher order with parameters a and b are equal to

1

(m+ α)α
Ĝ(k,α)
m+α(x, y;λ, ρ, u, a, b)



R. B. Corcino, C. B. Corcino / Eur. J. Pure Appl. Math, 18 (3) (2025), 6167 12 of 20

=

m∑
q=0

m∑
j=0

(
m

q

)
q! ρq−

j
2

( x
ρ

q − j

)(y
ρ
j
2

)
(m− q)!

×
∑

k1+k2+...+kα=m−q

α∏
i=1


ki∑
j=0

1

ki!

(
ki
j

)

Bm1,m2,...,mk−1
(j, ρ− 1)Aki−j

(u
λ

) (− log ab)ki−j(
1− u

λ

)ki−j+1

}
.

where Ĝ(k,α)
m (x, y;λ, ρ, u, a, b) = 0 for m = 0, 1, . . . , α−1, An(u) is the Eulerian polynomial

defined in (30) and Bm1,m2,...,mk−1
(i, ρ− 1) satisfies (29).

Remark 2.4. It can easily be seen that, when α = 1, the explicit formula in Theorem 2.3
reduces to that in Theorem 2.2.

3. Relation With Some Genocchi-type Polynomials

By giving special values to the parameters involved, Ĝ(k,α)
n (x, y;λ, ρ, u, a, b) reduces to

some interesting Genocchi-type polynomials.

(i) Using (23), when k = 1, (27) yields

∞∑
n=0

Ĝ(α)
n (x, y;λ, ρ, u, a, b)

tn

n!
=

(
(1− u)t ln ab

λbt − ua−t

)α

exρ(t)e
y
ρ(t

2), (31)

where the polynomials Ĝ(α)
n (x, y;λ, ρ, u, a, b) = Ĝ(1,α)

n (x;λ, ρ, u, a, b) are called the
degenerate Hermite-based Apostol-Frobenius-type Genocchi polynomials of higher
order with parameters a, b and c. When α = 1, (31) yields

∞∑
n=0

Ĝ(1)
n (x, y;λ, ρ, u, a, b)

tn

n!
=

(1− u)t ln ab

λbt − ua−t
exρ(t)e

y
ρ(t

2), (32)

where the polynomials Ĝ(1)
n (x, y;λ, ρ, u, a, b) are called the degenerate Hermite-based

Apostol-Frobenius-type Genocchi polynomials with parameters a and b.

(ii) When x = y = 0, equation (27) reduces to

∞∑
n=0

Ĝ(k,α)
n (λ, ρ, u, a, b)

tn

n!
=

(
Eik,ρ(logρ(1 + (1− u)t ln ab))

λbt − ua−t

)α

, (33)

the type 2 degenerate Apostol-Frobenius-type poly-Genocchi numbers with param-
eters a and b.



R. B. Corcino, C. B. Corcino / Eur. J. Pure Appl. Math, 18 (3) (2025), 6167 13 of 20

(iii) When a = 1, b = e, (27) will reduce to

∞∑
n=0

Ĝ(k,α)
n (x, y;λ, ρ, u)

tn

n!
=

(
Eik,ρ(logρ(1 + (1− u)t))

λet − u

)α

exρ(t)e
y
ρ(t

2), (34)

and call Ĝ(k,α)
n (x, y;λ, ρ, u), the type 2 degenerate Apostol-Frobenius-type poly-

Genocchi polynomials of higher order. When x = y = 0, we get

∞∑
n=0

Ĝ(k,α)
n (λ, ρ, u)

tn

n!
=

(
Eik,ρ(logρ(1 + (1− u)t)

λet − u

)α

, (35)

the type 2 degenerate Apostol-Frobenius-type Genocchi numbers of higher order.

(iv) When ρ → 0, equation (27) reduces to

∞∑
n=0

Ĝ(k,α)
n (x, y;λ, 0, u, a, b)

tn

n!
=

(
Eik,0(log0(1 + (1− u)t ln ab))

λbt − ua−t

)α

ex0(t)e
y
0(t

2)

∞∑
n=0

Ĝ(k,α)
n (x, y;λ, u, a, b)

tn

n!
=

(
Eik(log(1 + (1− u)t ln ab))

λbt − ua−t

)α

ext+yt2 , (36)

where the polynomials G(k,α)
n (x, y;λ, u, a, b) are the type 2 Hermite-based Apostol-

Frobenius-type poly-Genocchi polynomials of higher order with parameters a, b and
c with c = e in [29].

(v) When λ = 1, (34) gives

∞∑
n=0

Ĝ(k,α)
n (x, y;u, 1, e)

tn

n!
=

(
Eik,ρ(logρ(1 + (1− u)t))

et − u

)α

ext+yt2 . (37)

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

2 Hermite-based poly-Genocchi polynomials. We may use Ĝ(k,α)
n (x;u) to denote

Ĝ(k,α)
n (x;u, 1, e).

(vi) When k = 1, a = 1 and b = e, (36) gives

∞∑
n=0

Ĝ(1,α)
n (x, y;λ, u)

tn

n!
=

(
(1− u)t

λet − u

)α

ext+yt2 , (38)

and when λ = 1, (38) gives

∞∑
n=0

Ĝ(1,α)
n (x, y; 1, u)

tn

n!
=

(
(1− u)t

et − u

)α

ext+yt2 ,

where Ĝ(1,α)
n (x, y;λ, u) = Ĝ(α)

n (x, y;λ, u) and Ĝ(1,α)
n (x; 1, u) = Ĝ(α)

n (x;u) are called
the degenerate Hermite-based Apostol-Frobenius-type Genocchi polynomials and



R. B. Corcino, C. B. Corcino / Eur. J. Pure Appl. Math, 18 (3) (2025), 6167 14 of 20

Hermite-based Frobenius-Genocchi polynomials of higher order in (5) and (3), re-
spectively. Furthermore, when α = 1, we have

∞∑
n=0

Ĝn(x;λ, u)
tn

n!
=

(1− u)t

λet − u
ext+yt2 , (39)

and
∞∑
n=0

Ĝn(x, y;u)
tn

n!
=

(1− u)t

et − u
ext+yt2 ,

where Ĝn(x;λ, u) and Ĝn(x;u) are called the Hermite-based Apostol-Frobenius-type
Genocchi polynomials and Hermite-based Frobenius-Genocchi polynomials

Now, let us consider some some relations of Ĝ(k,α)
n (x, y;λ, ρ, u, a, b) with other Genocchi-

type polynomials. First is to establish a kind of addition formula for Ĝ(k,α)
n (x, y;λ, ρ, u, a, b)

expressing them as polynomials in x.

Theorem 3.1. The type 2 degenerate Hermite-based Apostol-Frobenius-type poly-Genocchi
polynomials of higher order with parameters a and b satisfy the relation

Ĝ(k,α)
n (x, y;λ, ρ, u, a, b)

=
n∑

m=0

m∑
j=0

(
n

m

)
m! ρm− j

2 Ĝ(k,α)
n−m(λ, ρ, u, a, b)

( x
ρ

m− j

)(y
ρ
j
2

)
. (40)

Moreover, the expression of Ĝ(k,α)
n (x, y;λ, ρ, u, a, b) as polynomial in x and y is given by

Ĝ(k,α)
n (x, y;λ, ρ, u, a, b) =

n∑
i=0

i∑
l=0

Ĝ(k,α)
n,i,l,w̃(λ, ρ, u, a, b)x

i−lyl, (41)

where

Ĝ(k,α)
n,i,l,w̃(λ, ρ, u, a, b) =

n∑
m=i

m∑
j=0

j is even

(
n

m

)
m!

(m− j)!(j/2)!

× Ĝ(k,α)
n−m(λ, ρ, u, a, b)w̃ρ(m− j, i− l)w̃ρ(j/2, l).

Proof. Using (33), we can write (27) as follows:

∞∑
n=0

Ĝ(k,α)
n (x;λ, ρ, u, a, b)

tn

n!

=

(
Eik,ρ(logρ(1 + (1− u)t ln ab))

λbt − ua−t

)α

exρ(t)e
y
ρ(t

2)



R. B. Corcino, C. B. Corcino / Eur. J. Pure Appl. Math, 18 (3) (2025), 6167 15 of 20

=

( ∞∑
n=0

Ĝ(k,α)
n (λ, ρ, u, a, b)

tn

n!

) ∞∑
n=0


n∑

j=0

n! ρn−
j
2

( x
ρ

n− j

)(y
ρ
j
2

) tn

n!


=

∞∑
n=0

 n∑
m=0

(
n

m

)
Ĝ(k,α)
n−m(λ, ρ, u, a, b)


m∑
j=0

m! ρm− j
2

( x
ρ

m− j

)(y
ρ
j
2

)
 tn

n!

Comparing the coefficients of tn

n! yields (46). To prove (41), we first recall that the r-
Whitney numbers of the first kind, denoted by wm,r(n, k) were defined by Mező [54] by
means of the following horizontal generating function:

mn(x)n =

n∑
j=0

wm,r(n, j)(mx+ r)j , (42)

where (x)n = x(x− 1)(x− 2) . . . (x−n+1). Replacing x with x/m and letting r = 0 yield

(x)n,m =
n∑

j=0

w̃m(n, j)xj ,

where w̃m(n, j) = wm,0(n, j), a certain of generalization of Stirling numbers of the first
kind, i.e. S1(n, j) = w̃0(n, j). Using (42), equation (46) can further be written as polyno-
mial in x

Ĝ(k,α)
n (x, y;λ, ρ, u, a, b)

=
n∑

m=0

m∑
j=0

j is even

(
n

m

)
m!

(m− j)!(j/2)!
Ĝ(k,α)
n−m(λ, ρ, u, a, b)(x)m−j,ρ(y)j/2,ρ (43)

(x)m−j,ρ(y)j/2,ρ =

( ∞∑
i=0

w̃ρ(m− j, i)xi

)( ∞∑
i=0

w̃ρ(j/2, i)y
i

)
(44)

=

∞∑
i=0

i∑
l=0

w̃ρ(m− j, i− l)w̃ρ(j/2, l)x
i−lyl. (45)

Ĝ(k,α)
n (x, y;λ, ρ, u, a, b)

=

n∑
m=0

m∑
j=0

j is even

(
n

m

)
m!

(m− j)!(j/2)!
Ĝ(k,α)
n−m(λ, ρ, u, a, b)

×
m∑
i=0

i∑
l=0

w̃ρ(m− j, i− l)w̃ρ(j/2, l)x
i−lyl



R. B. Corcino, C. B. Corcino / Eur. J. Pure Appl. Math, 18 (3) (2025), 6167 16 of 20

=
n∑

m=0

m∑
i=0

i∑
l=0

m∑
j=0

j is even

(
n

m

)
m!

(m− j)!(j/2)!
Ĝ(k,α)
n−m(λ, ρ, u, a, b)

× w̃ρ(m− j, i− l)w̃ρ(j/2, l)x
i−lyl

=
n∑

i=0

i∑
l=0

n∑
m=i

m∑
j=0

j is even

(
n

m

)
m!

(m− j)!(j/2)!
Ĝ(k,α)
n−m(λ, ρ, u, a, b)

× w̃ρ(m− j, i− l)w̃ρ(j/2, l)x
i−lyl

Ĝ(k,α)
n (x, y;λ, ρ, u, a, b) =

n∑
i=0

i∑
l=0

Ĝ(k,α)
n,i,l,w̃(λ, ρ, u, a, b)x

i−lyl

with coefficients

Ĝ(k,α)
n,i,l,w̃(λ, ρ, u, a, b) =

n∑
m=i

m∑
j=0

j is even

(
n

m

)
m!

(m− j)!(j/2)!

× Ĝ(k,α)
n−m(λ, ρ, u, a, b)w̃ρ(m− j, i− l)w̃ρ(j/2, l)

the convolution of Ĝ(k,α)
n (λ, ρ, u, a, b) and w̃ρ(n, k).

The next result is a kind of addition formula for Ĝ(k,α)
n (x;λ, ρ, u, a, b).

Theorem 3.2. The type 2 degenerate Hermite-based Apostol-Frobenius-type poly-Genocchi
polynomials of higher order with parameters a and b satisfy the relation

Ĝ(k,α)
n (x1 + x2, y1 + y2;λ, ρ, u, a, b)

=

n∑
q=0

n−q∑
j=0

j is even

(n− q)!Ĝ(k,α)
q (x1, y1;λ, ρ, u, a, b)

(n− q − j)!(j/2)!
(x2)n−q,ρ(y2)j/2,ρ. (46)

Proof. We can write (27) as follows:

∞∑
n=0

Ĝ(k,α)
n (x1 + x2, y1 + y2;λ, ρ, u, a, b)

tn

n!

=

(
Eik,ρ(logρ(1 + (1− u)t ln ab))

λbt − ua−t

)α

ex1+x2
ρ (t)ey1+y2

ρ (t2)

=

(
Eik,ρ(logρ(1 + (1− u)t ln ab))

λbt − ua−t

)α

ex1
ρ (t)ey1ρ (t2)ex2

ρ (t)ey2ρ (t2)

=

( ∞∑
n=0

Ĝ(k,α)
n (x1, y1;λ, ρ, u, a, b)

tn

n!

) ∞∑
n=0

 n∑
j=0

n! ρn−
j
2

( x2
ρ

n− j

)(y2
ρ
j
2

) tn

n!





R. B. Corcino, C. B. Corcino / Eur. J. Pure Appl. Math, 18 (3) (2025), 6167 17 of 20

=
∞∑
n=0

 n∑
q=0

n−q∑
j=0

(n− q)! ρn−q− j
2

( x2
ρ

n− q − j

)(y2
ρ
j
2

)
Ĝ(k,α)
q (x1, y1;λ, ρ, u, a, b)

 tn

n!

Comparing the coefficients of tn

n! completes the proof of the theorem.

Acknowledgements

The authors are grateful to Cebu Normal University (CNU) for funding this research
project through its Research Institute for Computational Mathematics and Physics (RICMP).

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