Some Identities on Degenerate Bernstein and Degenerate Euler Polynomials

In recent years, intensive studies on degenerate versions of various special numbers and polynomials have been done by means of generating functions, combinatorial methods, umbral calculus, p-adic analysis and differential equations. The degenerate Bernstein polynomials and operators were recently introduced as degenerate versions of the classical Bernstein polynomials and operators. Herein, we firstly derive some of their basic properties. Secondly, we explore some properties of the degenerate Euler numbers and polynomials and also their relations with the degenerate Bernstein polynomials.

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Degenerate Bell polynomials associated with umbral calculus

Journal of Inequalities and Applications ◽

10.1186/s13660-020-02494-7 ◽

2020 ◽

Vol 2020 (1) ◽

Cited By ~ 1

Author(s):

Taekyun Kim ◽

Dae San Kim ◽

Han-Young Kim ◽

Hyunseok Lee ◽

Lee-Chae Jang

Keyword(s):

Generating Functions ◽

Recurrence Relations ◽

Bernoulli Polynomials ◽

Umbral Calculus ◽

Bell Polynomials ◽

Euler Polynomials ◽

Euler Numbers ◽

Type Formula ◽

Euler Numbers And Polynomials ◽

Special Numbers And Polynomials

Abstract Carlitz initiated a study of degenerate Bernoulli and Euler numbers and polynomials which is the pioneering work on degenerate versions of special numbers and polynomials. In recent years, studying degenerate versions regained lively interest of some mathematicians. The purpose of this paper is to study degenerate Bell polynomials by using umbral calculus and generating functions. We derive several properties of the degenerate Bell polynomials including recurrence relations, Dobinski-type formula, and derivatives. In addition, we represent various known families of polynomials such as Euler polynomials, modified degenerate poly-Bernoulli polynomials, degenerate Bernoulli polynomials of the second kind, and falling factorials in terms of degenerate Bell polynomials and vice versa.

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A Note on Degenerate Bernstein and Degenerate Euler Polynomials

10.20944/preprints201807.0182.v1 ◽

2018 ◽

Author(s):

Taekyun Kim ◽

Dae San Kim

Keyword(s):

Bernstein Polynomials ◽

Euler Polynomials ◽

Euler Numbers ◽

Euler Numbers And Polynomials ◽

Degenerate Bernstein Polynomials

In this paper, we investigate the recently introduced degenerate Bernstein polynomials and operators and derive some of their properties. Also, we give some properties of the degenerate Euler numbers and polynomials and their connection with the degenerate Euler polynomials.

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New classes of recurrence relations involving hyperbolic functions, special numbers and polynomials

Applicable Analysis and Discrete Mathematics ◽

10.2298/aadm201020015s ◽

2021 ◽

pp. 15-15

Author(s):

Yilmaz Simsek

Keyword(s):

Generating Functions ◽

Recurrence Relations ◽

Stirling Numbers ◽

Umbral Calculus ◽

Discrete Mathematics ◽

New Class ◽

Integral Formulas ◽

Euler Numbers And Polynomials ◽

Derivative Formulas ◽

Special Numbers And Polynomials

By using the calculus of finite differences methods and the umbral calculus, we construct recurrence relations for a new class of special numbers. Using this recurrence relation, we define generating functions for this class of special numbers and also new classes of special polynomials. We investigate some properties of these generating functions. By using these generating functions with their functional equations, we obtain many new and interesting identities and relations related to these classes of special numbers and polynomials, the Bernoulli numbers and polynomials, the Euler numbers and polynomials, the Stirling numbers. Finally, some derivative formulas and integral formulas for these classes of special numbers and polynomials are given. In general, this article includes results that have the potential to be used in areas such as discrete mathematics, combinatorics analysis and their applications.

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A Note on a New Type of Degenerate poly-Euler Numbers and Polynomials

10.20944/preprints202007.0202.v1 ◽

2020 ◽

Author(s):

Waseem Khan

Keyword(s):

Arithmetic Function ◽

Bernoulli Numbers ◽

Euler Polynomials ◽

Euler Numbers ◽

Polylogarithm Function ◽

Euler Numbers And Polynomials ◽

Special Numbers And Polynomials ◽

Euler Polynomials And Numbers ◽

New Type ◽

Logarithm Function

Kim-Kim [12] introduced the new type of degenerate Bernoulli numbers and polynomials arising from the degenerate logarithm function. In this paper, we introduce a new type of degenerate poly-Euler polynomials and numbers, are called degenerate poly-Euler polynomials and numbers, by using the degenerate polylogarithm function and derive several properties on the degenerate poly-Euler polynomials and numbers. In the last section, we also consider the degenerate unipoly-Euler polynomials attached to an arithmetic function, by using the degenerate polylogarithm function and investigate some identities of those polynomials. In particular, we give some new explicit expressions and identities of degenerate unipoly polynomials related to special numbers and polynomials.

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Some Properties for Multiple Twisted (p, q)-L-Function and Carlitz’s Type Higher-Order Twisted (p, q)-Euler Polynomials

Mathematics ◽

10.3390/math7121205 ◽

2019 ◽

Vol 7 (12) ◽

pp. 1205 ◽

Cited By ~ 2

Author(s):

Kyung-Won Hwang ◽

Cheon Seoung Ryoo

Keyword(s):

Generating Functions ◽

Higher Order ◽

Integral Representations ◽

Euler Polynomials ◽

Euler Numbers ◽

Mellin Transformation ◽

Euler Numbers And Polynomials ◽

Symmetric Identities ◽

Positive Integers ◽

Symmetric Property

The main goal of this paper is to study some interesting identities for the multiple twisted ( p , q ) -L-function in a complex field. First, we construct new generating functions of the new Carlitz-type higher order twisted ( p , q ) -Euler numbers and polynomials. By applying the Mellin transformation to these generating functions, we obtain integral representations of the multiple twisted ( p , q ) -Euler zeta function and multiple twisted ( p , q ) -L-function, which interpolate the Carlitz-type higher order twisted ( p , q ) -Euler numbers and Carlitz-type higher order twisted ( p , q ) -Euler polynomials at non-positive integers, respectively. Second, we get some explicit formulas and properties, which are related to Carlitz-type higher order twisted ( p , q ) -Euler numbers and polynomials. Third, we give some new symmetric identities for the multiple twisted ( p , q ) -L-function. Furthermore, we also obtain symmetric identities for Carlitz-type higher order twisted ( p , q ) -Euler numbers and polynomials by using the symmetric property for the multiple twisted ( p , q ) -L-function.

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Viewing Some Ordinary Differential Equations from the Angle of Derivative Polynomials

10.20944/preprints201610.0043.v1 ◽

2016 ◽

Cited By ~ 7

Author(s):

Feng Qi ◽

Bai-Ni Guo

Keyword(s):

Differential Equations ◽

Ordinary Differential Equations ◽

Stirling Numbers ◽

Bernoulli Numbers ◽

Euler Polynomials ◽

Euler Numbers ◽

Generalized Derivative ◽

Bernoulli Numbers And Polynomials ◽

Euler Numbers And Polynomials

In the paper, the authors view some ordinary differential equations and their solutions from the angle of (the generalized) derivative polynomials and simplify some known identities for the Bernoulli numbers and polynomials, the Frobenius-Euler polynomials, the Euler numbers and polynomials, in terms of the Stirling numbers of the first and second kinds.

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On theq-Extension of Apostol-Euler Numbers and Polynomials

Abstract and Applied Analysis ◽

10.1155/2008/296159 ◽

2008 ◽

Vol 2008 ◽

pp. 1-10 ◽

Cited By ~ 5

Author(s):

Young-Hee Kim ◽

Wonjoo Kim ◽

Lee-Chae Jang

Keyword(s):

Zeta Function ◽

Generating Functions ◽

Hurwitz Zeta Function ◽

Euler Polynomials ◽

Euler Numbers ◽

Euler Numbers And Polynomials

Recently, Choi et al. (2008) have studied theq-extensions of the Apostol-Bernoulli and the Apostol-Euler polynomials of ordernand multiple Hurwitz zeta function. In this paper, we define Apostol's typeq-Euler numbersEn,q,ξandq-Euler polynomialsEn,q,ξ(x). We obtain the generating functions ofEn,q,ξandEn,q,ξ(x), respectively. We also have the distribution relation for Apostol's typeq-Euler polynomials. Finally, we obtainq-zeta function associated with Apostol's typeq-Euler numbers and Hurwitz's typeq-zeta function associated with Apostol's typeq-Euler polynomials for negative integers.

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Some Identities Involving the Fubini Polynomials and Euler Polynomials

Mathematics ◽

10.3390/math6120300 ◽

2018 ◽

Vol 6 (12) ◽

pp. 300 ◽

Cited By ~ 4

Author(s):

Guohui Chen ◽

Li Chen

Keyword(s):

Power Series ◽

Second Order ◽

Euler Polynomials ◽

Euler Numbers ◽

Non Linear ◽

Combinatorial Methods

In this paper, we first introduce a new second-order non-linear recursive polynomials U h , i ( x ) , and then use these recursive polynomials, the properties of the power series and the combinatorial methods to prove some identities involving the Fubini polynomials, Euler polynomials and Euler numbers.

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On p-Adic Fermionic Integrals of q-Bernstein Polynomials Associated with q-Euler Numbers and Polynomials †

Symmetry ◽

10.3390/sym10080311 ◽

2018 ◽

Vol 10 (8) ◽

pp. 311 ◽

Cited By ~ 2

Author(s):

Lee-Chae Jang ◽

Taekyun Kim ◽

Dae Kim ◽

Dmitry Dolgy

Keyword(s):

Bernstein Polynomials ◽

Euler Numbers ◽

Euler Numbers And Polynomials

We study a q-analogue of Euler numbers and polynomials naturally arising from the p-adic fermionic integrals on Zp and investigate some properties for these numbers and polynomials. Then we will consider p-adic fermionic integrals on Zp of the two variable q-Bernstein polynomials, recently introduced by Kim, and demonstrate that they can be written in terms of the q-analogues of Euler numbers. Further, from such p-adic integrals we will derive some identities for the q-analogues of Euler numbers.

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A note on characteristic function for Bernstein polynomials involving special numbers and polynomials

Filomat ◽

10.2298/fil2002543s ◽

2020 ◽

Vol 34 (2) ◽

pp. 543-549

Author(s):

Buket Simsek

Keyword(s):

Characteristic Function ◽

Binomial Distribution ◽

Bernstein Polynomials ◽

Random Variable ◽

Stirling Numbers ◽

Bernoulli Numbers ◽

Discrete Random Variable ◽

Euler Numbers ◽

Negative Order ◽

Special Numbers And Polynomials

The aim of this present paper is to establish and study generating function associated with a characteristic function for the Bernstein polynomials. By this function, we derive many identities, relations and formulas relevant to moments of discrete random variable for the Bernstein polynomials (binomial distribution), Bernoulli numbers of negative order, Euler numbers of negative order and the Stirling numbers.

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