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 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 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 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 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 NOTE ON q-DEDEKIND-TYPE SUMS RELATED TO q-EULER POLYNOMIALS

2011 ◽  
Vol 54 (1) ◽  
pp. 121-125 ◽  
Author(s):  
TAEKYUN KIM
Keyword(s):  
Continuous Function ◽  
Zeta Function ◽  
Higher Order ◽  
Euler Polynomials ◽  
Euler Numbers ◽  
Euler Numbers And Polynomials ◽  
Euler Polynomials And Numbers

AbstractRecently, q-Dedekind-type sums related to q-zeta function and basic L-series are studied by Simsek in [13] (Y. Simsek, q-Dedekind type sums related to q-zeta function and basic L-series, J. Math. Anal. Appl. 318 (2006), 333–351) and Dedekind-type sums related to Euler numbers and polynomials are introduced in the previous paper [11] (T. Kim, Note on Dedekind type DC sums, Adv. Stud. Contem. Math. 18 (2009), 249–260). It is the purpose of this paper to construct a p-adic continuous function for an odd prime to contain a p-adic q-analogue of the higher order Dedekind the type sums related to q-Euler polynomials and numbers by using an invariant p-adic q-integrals.

scholarly journals On theq-Extension of Apostol-Euler Numbers and Polynomials

2008 ◽  
Vol 2008 ◽  
pp. 1-10 ◽  
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.

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.

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