Let m∈N. For parameters a, b ∈Rand λ, α ∈C,
Ramírez et al. in [4] introduces three new classes of
the Apostol-Bernoulli polynomials B[m−1,α]
n(x;a, b;λ),
the degenerated generalized Apostol-Euler polynomi-
als E[m−1,α]
n(x;a, b;λ)and the degenerated general-
ized Apostol-Genocchi polynomials G[m−1,α]
n(x;a, b;λ)of
level mby means of the following generating functions,
defined in a suitable neighborhood of t= 0:
tmα[σ(λ;a, b;t)]α(1 + at)x
a=
∞
X
n=0
B[m−1,α]
n(x;a, b;λ)tn
n!,
2mα[ψ(λ;a, b;t)]α(1 + at)x
a=
∞
X
n=0
E[m−1,α]
n(x;a, b;λ)tn
n!,
(2t)mα[ψ(λ;a, b;t)]α(1+at)x
a=
∞
X
n=0
G[m−1,α]
n(x;a, b;λ)tn
n!,
where
σ(λ;a, b;t) = λ(1 + at)1
a−
m−1
X
l=0
(tlog b)l
l!!−1
and,
ψ(λ;a, b;t) = λ(1 + at)1
a+
m−1
X
l=0
(tlog b)l
l!!−1
.
The following proposition summarizes some ele-
mentary properties of the degenerated generalized the
Apostol-Bernoulli polynomials, the degenerated gener-
alized Apostol-Euler polynomials and the degenerated
generalized Apostol-Genocchi polynomials, in the vari-
able x, (cf. [4]).
Proposition I..1 For a m∈Nfixed, let
{B[m−1,α]
n(x;a, b;λ)}n≥0,{E[m−1,α]
n(x;a, b;λ)}n≥0
and {G[m−1,α]
n(x;a, b;λ)}n≥0be the sequence of degener-
ated generalized Apostol-type polynomials in the variable
x,a, b ∈R+, order α∈Cand level m. Then the
followings identities (Addition theorem of the argument)
hold.
B[m−1,α+β]
n(x+y;a, b;λ) =
n
X
k=0 n
kB[m−1,α]
k(x;a, b;λ)B[m−1,β]
n−k(y;a, b;λ),
B[m−1,α]
n(x+y;a, b;λ) =
n
X
k=0n
kB[m−1,α]
k(y;a, b;λ)(x|a)n−k,(1)
E[m−1,α+β]
n(x+y;a, b;λ) =
n
X
k=0 n
kE[m−1,α]
k(x;a, b;λ)E[m−1,β]
n−k(y;a, b;λ),
E[m−1,α]
n(x+y;a, b;λ) =
n
X
k=0n
kE[m−1,α]
k(y;a, b;λ)(x|a)n−k,
G[m−1,α+β]
n(x+y;a, b;λ) =
n
X
k=0 n
kG[m−1,α]
k(x;a, b;λ)G[m−1,β]
n−k(y;a, b;λ),
G[m−1,α]
n(x+y;a, b;λ) =
n
X
k=0n
kG[m−1,α]
k(y;a, b;λ)(x|a)n−k.
New results for degenerated generalized Apostol–Bernoulli,
Apostol–Euler and Apostol–Genocchi polynomials
1Universidad de la Costa, Barranquilla, COLOMBIA
2Universitá Telematica Internazionale Uninettuno, Rome, ITALY
Abstract- The main objective of this work is to deduce some interesting algebraic relationships that connect the
degenerated generalized Apostol–Bernoulli, Apostol–Euler and Apostol– Genocchi polynomials and other
families of polynomials such as the generalized Bernoulli polynomials of level m and the Genocchi polynomials.
Futher, find new recurrence formulas for these three families of polynomials to study.
Keywords- Apostol–type polynomials; degenerate Apostol-type polynomials.
Received: July 2, 2021. Revised: June 2, 2022. Accepted: July 12, 2022. Published: August 5, 2022.
1. Introduction
1WILLIAM RAMÍREZ, 2CLEMENTE CESARANO, 1STIVEN DÍAZ
WSEAS TRANSACTIONS on MATHEMATICS
DOI: 10.37394/23206.2022.21.69
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On the subject of the Appell–type polynomials and
their various extensions, a remarkably large number of
investigations have appeared in the literature, see for
example (see, [1, 3, 7, 10]).
On the other hand, the first–kind Stirling number
s(n, k)is the number of ways in which nobjects can
be divided among knon–empty cycles and the second-
kind Stirling numbers S(n, k)count the number of ways
to partition a set of nelements into exactly knonempty
subsets. The generating functions are given, respectively,
by (see [8]):
1
k![ln(1 + t)]k=
∞
X
n=k
s(n, k)tn
n!
and,
1
k!(et−1)k=
∞
X
n=k
S(n, k)tn
n!.
The generalized falling factorial (x|a)nwith increment a
is defined by (see [9, Definition 2.3]):
(x|a)n=
n−1
Y
k=0
(x−ak),
for positive integer n, with the convention (x|a)0= 1, it
follows that
(x|a)n=
n
X
k=0
s(n, k)an−kxk.(2)
Proposition I..2 For m∈N. Let {B[m−1]
n(x)}n≥0and
{Gn(x)}n≥0be the sequences of generalized Bernoulli
polynomials of level mand Genocchi polynomials, respec-
tively. Then, the following identities are satisfied.
1) [6, Equation (2.6)].
xn=
n
X
k=0 n
kk!
(k+m)!B[m−1]
n−k(x),(3)
2) [5, Remark 7].
xn=1
2(n+ 1) "n+1
X
k=0 n+ 1
kGk(x) + Gn+1(x)#.
(4)
From the Proposition I..2 it is possible to deduce
some interesting algebraic relations connecting the de-
generated generalized Apostol–Bernoulli, Apostol–Euler
and Apostol–Genocchi polynomials and other families of
polynomials such as generalized Bernoulli polynomials of
level m, Genocchi polynomials and Apostol-Euler poly-
nomials.
Theorem II..1 For m∈N, degenerated general-
ized Apostol–Bernoulli polynomials B[m−1,α]
n(x;a, b;λ),
are related with the generalized Bernoulli polynomials
B[m−1]
n(x)of level m, by means of the following identity.
B[m−1,α]
n(x+y;a, b;λ) =
n
X
k=0
n−k
X
j=0
ν
X
r=0 n
kν
rr!an−k−j
(r+m)!
×B[m−1,α]
k(y;a, b;λ)s(n−k, j)B[m−1]
ν−r(x).
Proof 1 By substituting (3) and (2) into the right-hand
side of (1), we have
B[m−1,α]
n(x+y;a, b;λ)
=
n
X
k=0 n
kB[m−1,α]
k(y;a, b;λ)(x|a)n−k
=
n
X
k=0 n
kB[m−1,α]
k(y;a, b;λ)
n−k
X
j=0
s(n−k, j)an−k−jxj.
=
n
X
k=0 n
kB[m−1,α]
k(y;a, b;λ)
n−k
X
j=0
s(n−k, j)an−k−j
×
ν
X
r=0 ν
rr!
(r+m)!B[m−1]
ν−r(x)
=
n
X
k=0
n−k
X
j=0
ν
X
r=0 n
kν
rr!an−k−j
(r+m)!
×B[m−1,α]
k(y;a, b;λ)s(n−k, j)B[m−1]
ν−r(x).
Therefore, Theorem II..1 holds.
The proofs of Theorem II..2 and Theorem II..3, it is
analogously to Theorem II..1.
Theorem II..2 For m∈N, degenerated general-
ized Apostol–Euler polynomials E[m−1,α]
n(x;a, b;λ), are
related with the generalized Bernoulli polynomials
B[m−1]
n(x)of level m, by means of the following identity.
E[m−1,α]
n(x+y;a, b;λ)
=
n
X
k=0
n−k
X
j=0
ν
X
r=0 n
kν
rr!an−k−j
(r+m)!
×E[m−1,α]
k(y;a, b;λ)s(n−k, j)B[m−1]
ν−r(x).
Theorem II..3 For m∈N, degenerated general-
ized Apostol–Genocchi polynomials G[m−1,α]
n(x;a, b;λ),
are related with the generalized Bernoulli polynomials
B[m−1]
n(x)of level m, by means of the following identity.
G[m−1,α]
n(x+y;a, b;λ)
=
n
X
k=0
n−k
X
j=0
ν
X
r=0 n
kν
rr!an−k−j
(r+m)!
×G[m−1,α]
k(y;a, b;λ)s(n−k, j)B[m−1]
ν−r(x).
Theorem II..4 For m∈N, degenerated generalized
Apostol–Bernoulli polynomials B[m−1,α]
n(x;a, b;λ), are
2. Some connection formulas for
degenerated generalized
Apostol–Bernoulli,$SRVWRO±(XOHU
DQG$SRVWRO±*HQRFFKLSRO\QRPLDOV
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related with the Genocchi polynomials Gn(x), by means
of the following identity.
B[m−1,α]
n(x+y;a, b;λ)
=
n
X
k=0
n−k
X
j=0 n
kan−k−j
2(ν+ 1)
ν+1
X
r=0
Gr(x)
×B[m−1,α]
k(y;a, b;λ)s(n−k, j)ν+ 1
r
+
n
X
k=0
n−k
X
j=0 n
ks(n−k, j)Gν+1(x).
Proof 2 By substituting (4) and (2) into the right-hand
side of (1), we obtain
B[m−1,α]
n(x+y;a, b;λ)
=
n
X
k=0 n
kB[m−1,α]
k(y;a, b;λ)(x|a)n−k
=
n
X
k=0 n
kB[m−1,α]
k(y;a, b;λ)
n−k
X
j=0
s(n−k, j)an−k−jxj
=
n
X
k=0 n
kB[m−1,α]
k(y;a, b;λ)
n−k
X
j=0
s(n−k, j)an−k−j
×"1
2(ν+ 1)
ν+1
X
r=0 ν+ 1
kGr(x) + 1
2(ν+ 1)Gν+1(x)
=
n
X
k=0
n−k
X
j=0 n
kan−k−j
2(ν+ 1)
ν+1
X
r=0 ν+ 1
ks(n−k, j)
×B[m−1,α]
k(y;a, b;λ)Gr(x)
+
n
X
k=0
n−k
X
j=0 n
kan−k−j
2(ν+ 1)Gν+1(x)
×B[m−1,α]
k(y;a, b;λ)s(n−k, j).
Therefore, Theorem II..4 holds.
The proofs of Theorem II..5 and Theorem II..6, it is
analogously to Theorem II..4.
Theorem II..5 For m∈N, degenerated generalized
Apostol–Euler polynomials E[m−1,α]
n(x;a, b;λ), are re-
lated with the Genocchi polynomials Gn(x), by means of
the following identity.
E[m−1,α]
n(x+y;a, b;λ) =
n
X
k=0
n−k
X
j=0 n
kan−k−j
2(ν+ 1)
ν+1
X
r=0
Gr(x)×E[m−1,α]
k(y;a, b;λ)s(n−k, j)ν+ 1
r
+
n
X
k=0
n−k
X
j=0 n
kan−k−j
2(ν+ 1)E[m−1,α]
k(y;a, b;λ)
×s(n−k, j)Gν+1(x).
Theorem II..6 For m∈N, degenerated generalized
Apostol–Genocchi polynomials G[m−1,α]
n(x;a, b;λ), are
related with the Genocchi polynomials Gn(x), by means
of the following identity.
G[m−1,α]
n(x+y;a, b;λ)
=
n
X
k=0
n−k
X
j=0 n
kan−k−j
2(ν+ 1)
ν+1
X
r=0
Gr(x)
×G[m−1,α]
k(y;a, b;λ)s(n−k, j)ν+ 1
r
+
n
X
k=0
n−k
X
j=0 n
kan−k−j
2(ν+ 1)G[m−1,α]
k(y;a, b;λ)
×s(n−k, j)Gν+1(x).
Theorem II..7 For m∈N, degenerated generalized
Apostol–Bernoulli polynomials B[m−1,α]
n(x;a, b;λ), they
satisfy the following relation.
B[m−1,α]
n(ax +x;a, b;λ) =
n−1
X
k=0 x
k+ 1n−1
kak+1nk!
×B[m−1,α]
n−1−k(x;a, b;λ) + B[m−1,α]
n(x;a, b;λ).
Proof 3 By the generating function of degen-
erated generalized Apostol–Bernoulli polynomi-
als B[m−1,α]
n(x;a, b;λ)and considering ϕn=
B[m−1,α]
n(ax +x;a, b;λ)and ψn=B[m−1,α]
n(x;a, b;λ),
we have
∞
X
n=0
[ϕn−ψn]tn
n!=tmα[σ(λ;a, b;t)]α(1 + at)ax+x
a
−tmα[σ(λ;a, b;t)]α(1 + at)x
a
=tmα[σ(λ;a, b;t)]α(1 + at)x
a[(1 + at)x−1]
=
∞
X
n=0
B[m−1,α]
n(x;a, b;λ)tn
n!
∞
X
n=0 x
n+ 1ak+1zn+1
=
∞
X
n=0
n
X
k=0 x
k+ 1n
kk!ak+1B[m−1,α]
n(x;a, b;λ)tn+1
n!
=
∞
X
n=0
n−1
X
k=0 x
k+ 1n−1
kk!nak+1B[m−1,α]
n(x;a, b;λ)tn
n!.
Comparing the coefficients of tn
n!in both sides of the
equation, the result is
B[m−1,α]
n(ax +x;a, b;λ) =
n−1
X
k=0 x
k+ 1n−1
kak+1nk!
×B[m−1,α]
n−1−k(x;a, b;λ) + B[m−1,α]
n(x;a, b;λ).
Theorem II..8 For m∈N, degenerated generalized
Apostol–Euler polynomials E[m−1,α]
n(x;a, b;λ), they sat-
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isfy the following relation.
E[m−1,α]
n(ax +x;a, b;λ)
=
n−1
X
k=0 x
k+ 1n−1
kak+1nk!
×E[m−1,α]
n−1−k(x;a, b;λ) + E[m−1,α]
n(x;a, b;λ).
Proof 4 By the generating function of degenerated gen-
eralized Apostol–Euler polynomials E[m−1,α]
n(x;a, b;λ)
and considering ϕn=E[m−1,α]
n(ax +x;a, b;λ)and ψn=
E[m−1,α]
n(x;a, b;λ),we have
∞
X
n=0
[ϕn−ψn]tn
n!= 2mα[ψ(λ;a, b;t)]α(1 + at)ax+x
a
−2mα[ψ(λ;a, b;t)]α(1 + at)x
a
= 2mα[ψ(λ;a, b;t)]α(1 + at)x
a[(1 + at)x−1]
=
∞
X
n=0
E[m−1,α]
n(x;a, b;λ)tn
n!
∞
X
n=0 x
n+ 1ak+1zn+1
=
∞
X
n=0
n
X
k=0 x
k+ 1n
kk!ak+1
×E[m−1,α]
n(x;a, b;λ)tn+1
n!
=
∞
X
n=0
n−1
X
k=0 x
k+ 1n−1
kk!nak+1
×E[m−1,α]
n(x;a, b;λ)tn
n!.
Comparing the coefficients of tn
n!in both sides of the
equation, the result is
E[m−1,α]
n(ax +x;a, b;λ) =
n−1
X
k=0 x
k+ 1n−1
kak+1nk!
×E[m−1,α]
n−1−k(x;a, b;λ) + E[m−1,α]
n(x;a, b;λ).
Theorem II..9 For m∈N, degenerated generalized
Apostol–Genocchi polynomials G[m−1,α]
n(x;a, b;λ), they
satisfy the following relation.
G[m−1,α]
n(ax +x;a, b;λ) =
n−1
X
k=0 x
k+ 1n−1
kak+1nk!
×G[m−1,α]
n−1−k(x;a, b;λ) + G[m−1,α]
n(x;a, b;λ).
Proof 5 By the generating function of degen-
erated generalized Apostol–Genocchi polynomi-
als G[m−1,α]
n(x;a, b;λ)and considering ϕn=
G[m−1,α]
n(ax +x;a, b;λ)and ψn=G[m−1,α]
n(x;a, b;λ),
we have
∞
X
n=0
[ϕn−ψn]tn
n!= (2t)mα[σ(λ;a, b;t)]α(1 + at)ax+x
a
−(2t)mα[ψ(λ;a, b;t)]α(1 + at)x
a
= (2t)mα[ψ(λ;a, b;t)]α(1 + at)x
a[(1 + at)x−1]
=
∞
X
n=0
G[m−1,α]
n(x;a, b;λ)tn
n!
∞
X
n=0 x
n+ 1ak+1zn+1
=
∞
X
n=0
n
X
k=0 x
k+ 1n
kk!ak+1G[m−1,α]
n(x;a, b;λ)tn+1
n!
=
∞
X
n=0
n−1
X
k=0 x
k+ 1n−1
kk!nak+1G[m−1,α]
n(x;a, b;λ)tn
n!.
Comparing the coefficients of tn
n!in both sides of the
equation, the result is
G[m−1,α]
n(ax +x;a, b;λ) =
n−1
X
k=0 x
k+ 1n−1
kak+1nk!
×G[m−1,α]
n−1−k(x;a, b;λ) + G[m−1,α]
n(x;a, b;λ).
In this work, new properties of the degenerated gen-
eralized Apostol–Bernoulli, Apostol–Euler and Apostol–
Genocchi polynomials are studied, using various generat-
ing function methods. The generalization of these results
can lead to other interesting results, which can be useful
for fractional calculus theory.
This research was funded by the International
Telematic University Uninettuno (Italia) and the
Universidad de la Costa (Colombia) for all the
support provided supported by the project whit
order code SAP E11P1070121C.
[1] Bedoya, D., Ortega, M., Ramírez, W., Urieles, A.,
“New biparametric families of Apostol-Frobenius-
Euler polynomials of level m”, Mat. Stud., Vol. 55,
pp. 10–23, 2021.
[2] Comtet, L., “Advanced Combinatorics: The Art of
Finite and Infinite Expansions”, Reidel, Dordrecht
and Boston (1974). (Traslated from French by Nien-
huys, J.W.)
[3] Cesarano, C., Ramírez, W., Khan, S, “A new class
of degenerate Apostol–type Hermite polynomials and
applications”, Dolomites Res. Notes Approx, Vol. 15,
pp. 1–10, 2022.
[4] Cesarano, C and Ramírez, W, “Some new classes of
degenerated generalized Apostol–Bernoulli, Apostol–
Euler and Apostol–Genocchi polynomials”, (Ac-
cepted). Carpathian Math. Publ, Vol. 14 no. 2, 2022.
[5] Luo, Q.-M., “Extensions of the Genocchi polynomials
and its Fourier expansions and integral representa-
tions”, Osaka J. Math, Vol. 48, pp. 291-309, 2011.
3. Conclusion
Acknowledgment
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DOI: 10.37394/23206.2022.21.69
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Volume 21, 2022
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William Ramírez is Professor and researcher in Fac-
ulty of Natural and Exact Sciences at Universidad de
la Costa (Colombia), and previously was member of De-
partment of Pure Mathematics at Universidad del Atlán-
tico (Colombia). He received the MSc degree in Mathe-
matical Sciences at Universidad del Atlántico. Also, he is
a junior researcher recognized by Colciencias (Colombia).
His main research interests are: mathematical analysis,
number theory and special functions.
Clemente Cesarano is associate professor of Numeri-
cal Analysis at the Section of Mathematics -Uninettuno
University, Rome Italy; he is the coordinator of the doc-
toral college in Technological Innovation Engineering,
coordinator of the Section of Mathematics, vice-dean
of the Faculty of Engineering, president of the Degree
Course in Management Engineering, director of the Mas-
ter in Project Management Techniques, and coordinator
of the Master in Applied and Industrial Mathematics.
He is also a member of the Research Project “Modeling
and Simulation of the Fractionary and Medical Center”,
Complutense University of Madrid (Spain) and head of
the national group from 2015, member of the Research
Project (Serbian Ministry of Education and Science)
“Approximation of Integral and Differential Operators
and Applications”, University of Belgrade (Serbia) and
coordinator of the national group from 2011-), a member
of the Doctoral College in Mathematics at the Depart-
ment of Mathematics of the University of Mazandaran
(Iran), expert (Reprise) at the Ministry of Education,
University and Research, for the ERC sectors: Analy-
sis, Operator algebras and functional analysis, Numeri-
cal analysis. Clemente Cesarano is Honorary Fellows of
the Australian Institute of High Energetic Materials, af-
filiated with the National Institute of High Mathematics
(INdAM), is affiliated with the International Research
Center for the “Mathematics Mechanics of Complex Sys-
tems” (MEMOCS) - University of L’Aquila, associate of
the CNR at the Institute of Complex Systems (ISC), af-
filiated with the “Research ITalian network on Approxi-
mation (RITA)” network as the head of the Uninettuno
office, UMI member, SIMAI member.
Stiven Díaz is professor and researcher in Faculty of
Natural and Exact Sciences at Universidad de la Costa
(Colombia). He received the doctor in Natural Sciences
and magister in Mathematical Sciences at Universidad
del Norte. His main research interests are mathemat-
ical analysis, numerical analysis, special functions and
difference equations.
Contribution of individual authors to the creation
of a scientific article (ghostwriting policy).
William Ramírez, Clemente Cesarano and Stiven
Díaz developed the theory and performed the compu-
tations. All authors discussed the results, read and ap-
proved the final manuscript.
Creative Commons Attribution License 4.0 (At-
tribution 4.0 International , CC BY 4.0)
This article is published under the terms of the Creative
Commons Attribution License 4.0
https://creativecommons.org/licenses/by/4.0/deed.en_US
WSEAS TRANSACTIONS on MATHEMATICS
DOI: 10.37394/23206.2022.21.69
William Ramírez, Clemente Cesarano, Stiven Díaz
E-ISSN: 2224-2880
608
Volume 21, 2022