scholarly journals On the type 2 poly-Bernoulli polynomials associated with umbral calculus

2021 ◽  
Vol 19 (1) ◽  
pp. 878-887
Author(s):  
Taekyun Kim ◽  
Dae San Kim ◽  
Dmitry V. Dolgy ◽  
Jin-Woo Park
Keyword(s):  
Bernoulli Polynomials ◽  
Higher Order ◽  
Umbral Calculus ◽  
Euler Polynomials ◽  
Special Polynomials

Abstract Type 2 poly-Bernoulli polynomials were introduced recently with the help of modified polyexponential functions. In this paper, we investigate several properties and identities associated with those polynomials arising from umbral calculus techniques. In particular, we express the type 2 poly-Bernoulli polynomials in terms of several special polynomials, like higher-order Cauchy polynomials, higher-order Euler polynomials, and higher-order Frobenius-Euler polynomials.

scholarly journals Type 2 Degenerate Poly-Euler Polynomials

Symmetry ◽  
2020 ◽  
Vol 12 (6) ◽  
pp. 1011 ◽  
Author(s):  
Dae Sik Lee ◽  
Hye Kyung Kim ◽  
Lee-Chae Jang
Keyword(s):  
Final Section ◽  
Bernoulli Polynomials ◽  
Stirling Numbers ◽  
Euler Polynomials ◽  
Special Polynomials ◽  
Special Numbers And Polynomials ◽  
Euler Polynomials And Numbers ◽  
New Type ◽  
Explicit Expressions

In recent years, many mathematicians have studied the degenerate versions of many special polynomials and numbers. The polyexponential functions were introduced by Hardy and rediscovered by Kim, as inverses to the polylogarithms functions. The paper is divided two parts. First, we introduce a new type of the type 2 poly-Euler polynomials and numbers constructed from the modified polyexponential function, the so-called type 2 poly-Euler polynomials and numbers. We show various expressions and identities for these polynomials and numbers. Some of them involving the (poly) Euler polynomials and another special numbers and polynomials such as (poly) Bernoulli polynomials, the Stirling numbers of the first kind, the Stirling numbers of the second kind, etc. In final section, we introduce a new type of the type 2 degenerate poly-Euler polynomials and the numbers defined in the previous section. We give explicit expressions and identities involving those polynomials in a similar direction to the previous section.

scholarly journals Some Identities on Type 2 Degenerate Bernoulli Polynomials of the Second Kind

Symmetry ◽  
2020 ◽  
Vol 12 (4) ◽  
pp. 510 ◽  
Author(s):  
Taekyun Kim ◽  
Lee-Chae Jang ◽  
Dae San Kim ◽  
Han Young Kim
Keyword(s):  
Bernoulli Polynomials ◽  
Stirling Numbers ◽  
Order Type ◽  
Higher Order ◽  
Special Polynomials

In recent years, many mathematicians studied various degenerate versions of some special polynomials for which quite a few interesting results were discovered. In this paper, we introduce the type 2 degenerate Bernoulli polynomials of the second kind and their higher-order analogues, and study some identities and expressions for these polynomials. Specifically, we obtain a relation between the type 2 degenerate Bernoulli polynomials of the second and the degenerate Bernoulli polynomials of the second, an identity involving higher-order analogues of those polynomials and the degenerate Stirling numbers of the second kind, and an expression of higher-order analogues of those polynomials in terms of the higher-order type 2 degenerate Bernoulli polynomials and the degenerate Stirling numbers of the first kind.

scholarly journals Some recurrence formulas for the q-Bernoulli and q-Euler polynomials

Filomat ◽  
2020 ◽  
Vol 34 (2) ◽  
pp. 663-669
Author(s):  
Paçin Dere
Keyword(s):  
Recurrence Relations ◽  
Bernoulli Polynomials ◽  
Umbral Calculus ◽  
Euler Polynomials ◽  
Appell Polynomials ◽  
Derivative Operator ◽  
Important Place ◽  
Quantum Calculus ◽  
Recurrence Formulas ◽  
Special Polynomials

The recurrence relations have a very important place for the special polynomials such as q-Appell polynomials. In this paper, we give some recurrence formulas that allow us a better understanding of q-Appell polynomials. We investigate the q-Bernoulli polynomials and q-Euler polynomials, which are q-Appell polynomials, and we obtain their recurrence formulas by using the methods of the q-umbral calculus and the quantum calculus. Our methods include some operators which are quite handy for obtaining relations for the q-Appell polynomials. Especially, some applications of q-derivative operator are used in this work.

scholarly journals Two-Variable Type 2 Poly-Fubini Polynomials

Mathematics ◽  
2021 ◽  
Vol 9 (3) ◽  
pp. 281
Author(s):  
Ghulam Muhiuddin ◽  
Waseem Ahmad Khan ◽  
Ugur Duran
Keyword(s):  
Integral Representation ◽  
Bernoulli Polynomials ◽  
Stirling Numbers ◽  
Higher Order ◽  
Variable Type ◽  
Summation Formulas

In the present work, a new extension of the two-variable Fubini polynomials is introduced by means of the polyexponential function, which is called the two-variable type 2 poly-Fubini polynomials. Then, some useful relations including the Stirling numbers of the second and the first kinds, the usual Fubini polynomials, and the higher-order Bernoulli polynomials are derived. Also, some summation formulas and an integral representation for type 2 poly-Fubini polynomials are investigated. Moreover, two-variable unipoly-Fubini polynomials are introduced utilizing the unipoly function, and diverse properties involving integral and derivative properties are attained. Furthermore, some relationships covering the two-variable unipoly-Fubini polynomials, the Stirling numbers of the second and the first kinds, and the Daehee polynomials are acquired.

scholarly journals Degenerate Catalan-Daehee numbers and polynomials of order $ r $ arising from degenerate umbral calculus

2021 ◽  
Vol 7 (3) ◽  
pp. 3845-3865
Author(s):  
Hye Kyung Kim ◽  
◽  
Dmitry V. Dolgy ◽  
Keyword(s):  
Bernoulli Polynomials ◽  
Umbral Calculus ◽  
Bell Polynomials ◽  
Psychological Burden ◽  
Special Polynomials ◽  
Sheffer Sequences ◽  
Special Numbers And Polynomials ◽  
Bernoulli Polynomials And Numbers ◽  
Euler Polynomials And Numbers ◽  
Degenerate Version

<abstract><p>Many mathematicians have studied degenerate versions of some special polynomials and numbers that can take into account the surrounding environment or a person's psychological burden in recent years, and they've discovered some interesting results. Furthermore, one of the most important approaches for finding the combinatorial identities for the degenerate version of special numbers and polynomials is the umbral calculus. The Catalan numbers and the Daehee numbers play important role in connecting relationship between special numbers.</p> <p>In this paper, we first define the degenerate Catalan-Daehee numbers and polynomials and aim to study the relation between well-known special polynomials and degenerate Catalan-Daehee polynomials of order $ r $ as one of the generalizations of the degenerate Catalan-Daehee polynomials by using the degenerate Sheffer sequences. Some of them include the degenerate and other special polynomials and numbers such as the degenerate falling factorials, the degenerate Bernoulli polynomials and numbers of order $ r $, the degenerate Euler polynomials and numbers of order $ r $, the degenerate Daehee polynomials of order $ r $, the degenerate Bell polynomials, and so on.</p></abstract>

scholarly journals A Note on Type 2 w-Daehee Polynomials

Mathematics ◽  
2019 ◽  
Vol 7 (8) ◽  
pp. 697
Author(s):  
Minyoung Ma ◽  
Dongkyu Lim
Keyword(s):  
Order Type ◽  
Higher Order ◽  
Special Polynomials ◽  
Invariant Integral ◽  
Symmetric Identities

In the paper, by virtue of the p-adic invariant integral on Z p , the authors consider a type 2 w-Daehee polynomials and present some properties and identities of these polynomials related with well-known special polynomials. In addition, we present some symmetric identities involving the higher order type 2 w-Daehee polynomials. These identities extend and generalize some known results.

scholarly journals Some Identities of the Degenerate Higher Order Derangement Polynomials and Numbers

Symmetry ◽  
2021 ◽  
Vol 13 (2) ◽  
pp. 176
Author(s):  
Hye Kyung Kim
Keyword(s):  
Higher Order ◽  
Umbral Calculus ◽  
Inversion Formulas ◽  
Special Polynomials ◽  
Sheffer Sequence ◽  
Sheffer Sequences

Recently, Kim-Kim (J. Math. Anal. Appl. (2021), Vol. 493(1), 124521) introduced the λ-Sheffer sequence and the degenerate Sheffer sequence. In addition, Kim et al. (arXiv:2011.08535v1 17 November 2020) studied the degenerate derangement polynomials and numbers, and investigated some properties of those polynomials without using degenerate umbral calculus. In this paper, the y the degenerate derangement polynomials of order s (s∈N) and give a combinatorial meaning about higher order derangement numbers. In addition, the author gives some interesting identities related to the degenerate derangement polynomials of order s and special polynomials and numbers by using degenerate Sheffer sequences, and at the same time derive the inversion formulas of these identities.

scholarly journals Some Relations of Two Type 2 Polynomials and Discrete Harmonic Numbers and Polynomials

Symmetry ◽  
2020 ◽  
Vol 12 (6) ◽  
pp. 905 ◽  
Author(s):  
Taekyun Kim ◽  
Dae San Kim
Keyword(s):  
Zeta Functions ◽  
Stirling Numbers ◽  
Higher Order ◽  
High Order ◽  
Euler Polynomials ◽  
Harmonic Numbers ◽  
Order Degenerate ◽  
Riemann Zeta ◽  
Changhee Polynomials

Harmonic numbers appear, for example, in many expressions involving Riemann zeta functions. Here, among other things, we introduce and study discrete versions of those numbers, namely the discrete harmonic numbers. The aim of this paper is twofold. The first is to find several relations between the Type 2 higher-order degenerate Euler polynomials and the Type 2 high-order Changhee polynomials in connection with the degenerate Stirling numbers of both kinds and Jindalrae–Stirling numbers of both kinds. The second is to define the discrete harmonic numbers and some related polynomials and numbers, and to derive their explicit expressions and an identity.

scholarly journals Symmetric identity for polynomial sequences satisfying A′ n +1(x) = (n + 1)An (x)

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Farid Bencherif ◽  
Rachid Boumahdi ◽  
Tarek Garici
Keyword(s):  
Bernoulli Polynomials ◽  
Dirichlet Character ◽  
Umbral Calculus ◽  
Euler Polynomials ◽  
Polynomial Sequences ◽  
Symmetric Identity ◽  
Primitive Dirichlet Character ◽  
Generalized Bernoulli Polynomials

Abstract Using umbral calculus, we establish a symmetric identity for any sequence of polynomials satisfying A′ n +1(x) = (n + 1)An (x) with A 0(x) a constant polynomial. This identity allows us to obtain in a simple way some known relations involving Apostol-Bernoulli polynomials, Apostol-Euler polynomials and generalized Bernoulli polynomials attached to a primitive Dirichlet character.

scholarly journals Representations of degenerate poly-Bernoulli polynomials

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Taekyun Kim ◽  
Dae San Kim ◽  
Jongkyum Kwon ◽  
Hyunseok Lee
Keyword(s):  
Bernoulli Polynomials ◽  
Higher Order ◽  
Umbral Calculus ◽  
Order Degenerate

AbstractAs is well known, poly-Bernoulli polynomials are defined in terms of polylogarithm functions. Recently, as degenerate versions of such functions and polynomials, degenerate polylogarithm functions were introduced and degenerate poly-Bernoulli polynomials were defined by means of the degenerate polylogarithm functions, and some of their properties were investigated. The aim of this paper is to further study some properties of the degenerate poly-Bernoulli polynomials by using three formulas coming from the recently developed ‘λ-umbral calculus’. In more detail, among other things, we represent the degenerate poly-Bernoulli polynomials by higher-order degenerate Bernoulli polynomials and by higher-order degenerate derangement polynomials.

Sign in / Sign up

Export Citation Format

Share Document