scholarly journals A Note on Degenerate Bernstein and Degenerate Euler Polynomials

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.

scholarly journals Some Identities on Degenerate Bernstein and Degenerate Euler Polynomials

Mathematics ◽  
2019 ◽  
Vol 7 (1) ◽  
pp. 47 ◽  
Author(s):  
Taekyun Kim ◽  
Dae San Kim
Keyword(s):  
Differential Equations ◽  
Generating Functions ◽  
Bernstein Polynomials ◽  
Umbral Calculus ◽  
Euler Polynomials ◽  
Euler Numbers ◽  
Euler Numbers And Polynomials ◽  
Special Numbers And Polynomials ◽  
Degenerate Bernstein Polynomials ◽  
Combinatorial Methods

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.

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

Symmetry ◽  
2018 ◽  
Vol 10 (8) ◽  
pp. 311 ◽  
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.

scholarly journals Generalized -Euler Numbers and Polynomials

2012 ◽  
Vol 2012 ◽  
pp. 1-14
Author(s):  
H. Y. Lee ◽  
N. S. Jung ◽  
C. S. Ryoo
Keyword(s):  
Complex Plane ◽  
Euler Polynomials ◽  
Euler Numbers ◽  
Interesting Phenomenon ◽  
Euler Numbers And Polynomials

We generalize the Euler numbers and polynomials by the generalized -Euler numbers and polynomials . For the complement theorem, have interesting different properties from the Euler polynomials and we observe an interesting phenomenon of “scattering” of the zeros of the the generalized Euler polynomials in complex plane.

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

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.

scholarly journals Euler Numbers and Polynomials Associated with Zeta Functions

2008 ◽  
Vol 2008 ◽  
pp. 1-11 ◽  
Author(s):  
Taekyun Kim
Keyword(s):  
Zeta Function ◽  
Entire Functions ◽  
Zeta Functions ◽  
Euler Polynomials ◽  
Euler Numbers ◽  
Euler Numbers And Polynomials

Fors∈ℂ, the Euler zeta function and the Hurwitz-type Euler zeta function are defined byζE(s)=2∑n=1∞((−1)n/ns), andζE(s,x)=2∑n=0∞((−1)n/(n+x)s). Thus, we note that the Euler zeta functions are entire functions in whole complexs-plane, and these zeta functions have the values of the Euler numbers or the Euler polynomials at negative integers. That is,ζE(−k)=Ek∗, andζE(−k,x)=Ek∗(x). We give some interesting identities between the Euler numbers and the zeta functions. Finally, we will give the new values of the Euler zeta function at positive even integers.

scholarly journals Ordinary and degenerate Euler numbers and polynomials

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
Taekyun Kim ◽  
Dae San Kim ◽  
Han Young Kim ◽  
Jongkyum Kwon
Keyword(s):  
Stirling Numbers ◽  
Recursive Formula ◽  
Euler Polynomials ◽  
Euler Numbers ◽  
Integer Power ◽  
Power Sums ◽  
Power Sum ◽  
Euler Numbers And Polynomials

Abstract In this paper, we study some identities on Euler numbers and polynomials, and those on degenerate Euler numbers and polynomials which are derived from the fermionic p-adic integrals on $\mathbb{Z}_{p}$ Z p . Specifically, we obtain a recursive formula for alternating integer power sums and representations of alternating integer power sum polynomials in terms of Euler polynomials and Stirling numbers of the second kind, as well as various properties about Euler numbers and polynomials. In addition, we deduce representations of degenerate alternating integer power sum polynomials in terms of degenerate Euler polynomials and degenerate Stirling numbers of the second kind, as well as certain properties on degenerate Euler numbers and polynomials.

scholarly journals Generalized -Euler Numbers and Polynomials Associated with -Adic -Integral on

2012 ◽  
Vol 2012 ◽  
pp. 1-14
Author(s):  
H. Y. Lee ◽  
N. S. Jung ◽  
J. Y. Kang ◽  
C. S. Ryoo
Keyword(s):  
Complex Plane ◽  
Euler Polynomials ◽  
Euler Numbers ◽  
Interesting Phenomenon ◽  
Euler Numbers And Polynomials

We generalize the Euler numbers and polynomials by the generalized -Euler numbers and polynomials . We observe an interesting phenomenon of “scattering” of the zeros of the generalized -Euler polynomials in complex plane.

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