scholarly journals Dynamics of the Zeros of Analytic Continued(h,q)-Euler Polynomials

2014 ◽  
Vol 2014 ◽  
pp. 1-9
Author(s):  
C. S. Ryoo
Keyword(s):  
Euler Polynomials ◽  
Bernoulli Equation ◽  
Euler Numbers ◽  
Interesting Phenomenon

In this paper, we study that the(h,q)-Euler numbersEn,q(h)and(h,q)-Euler polynomialsEn,q(h)(x)are analytic continued toEq(h)(s)andEq(h)(s,w). We investigate the new concept of dynamics of the zeros of analytic continued polynomials related to solution of Bernoulli equation. Finally, we observe an interesting phenomenon of “scattering” of the zeros ofEq(h)(s,w).

scholarly journals Zeros of Analytic Continuedq-Euler Polynomials andq-Euler Zeta Function

2014 ◽  
Vol 2014 ◽  
pp. 1-7 ◽  
Author(s):  
C. S. Ryoo
Keyword(s):  
Zeta Function ◽  
Euler Polynomials ◽  
Euler Numbers ◽  
Interesting Phenomenon

We study that theq-Euler numbersEn,qandq-Euler polynomialsEn,q(x)are analytic continued toEq(s)andEq(s,w). We investigate the new concept of dynamics of the zeros of analytic continued polynomials. Finally, we observe an interesting phenomenon of ‘‘scattering’’ of the zeros ofEq(s,w).

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 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.

scholarly journals Analytic Continuation of Euler Polynomials and the Euler Zeta Function

2014 ◽  
Vol 2014 ◽  
pp. 1-6 ◽  
Author(s):  
C. S. Ryoo
Keyword(s):  
Zeta Function ◽  
Analytic Continuation ◽  
Euler Polynomials ◽  
Euler Numbers ◽  
Interesting Phenomenon

We study that the Euler numbersEnand Euler polynomialsEnzare analytically continued toEsandE(s,w). We investigate the new concept of dynamics of the zeros of analytica continued polynomials. Finally, we observe an interesting phenomenon of “scattering” of the zeros ofE(s,w).

scholarly journals Some Identities Involving the Fubini Polynomials and Euler Polynomials

Mathematics ◽  
2018 ◽  
Vol 6 (12) ◽  
pp. 300 ◽  
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.

scholarly journals On the generalized q-poly-Euler polynomials of the second kind

Filomat ◽  
2020 ◽  
Vol 34 (2) ◽  
pp. 475-482
Author(s):  
Veli Kurt
Keyword(s):  
Bernoulli Polynomials ◽  
Stirling Numbers ◽  
Euler Polynomials ◽  
Euler Numbers ◽  
Basic Properties

In this work, we define the generalized q-poly-Euler numbers of the second kind of order ? and the generalized q-poly-Euler polynomials of the second kind of order ?. We investigate some basic properties for these polynomials and numbers. In addition, we obtain many identities, relations including the Roger-Sz?go polynomials, the Al-Salam Carlitz polynomials, q-analogue Stirling numbers of the second kind and two variable Bernoulli polynomials.

scholarly journals Hermite Polynomials and their Applications Associated with Bernoulli and Euler Numbers

2012 ◽  
Vol 2012 ◽  
pp. 1-13 ◽  
Author(s):  
Dae San Kim ◽  
Taekyun Kim ◽  
Seog-Hoon Rim ◽  
Sang Hun Lee
Keyword(s):  
Vector Space ◽  
Hermite Polynomials ◽  
Euler Polynomials ◽  
Dimensional Vector ◽  
Euler Numbers ◽  
Dimensional Vector Space ◽  
Good Basis ◽  
Arithmetic Properties

We derive some interesting identities and arithmetic properties of Bernoulli and Euler polynomials from the orthogonality of Hermite polynomials. LetPn={p(x)∈ℚ[x]∣deg p(x)≤n}be the(n+1)-dimensional vector space overℚ. Then we show that{H0(x),H1(x),…,Hn(x)}is a good basis for the spacePnfor our purpose of arithmetical and combinatorial applications.

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.

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