scholarly journals A Note on the -Euler Numbers and Polynomials with Weak Weight

2011 ◽  
Vol 2011 ◽  
pp. 1-14 ◽  
Author(s):  
H. Y. Lee ◽  
N. S. Jung ◽  
C. S. Ryoo
Keyword(s):  
Numerical Experiments ◽  
Regular Structure ◽  
Euler Polynomials ◽  
Euler Numbers ◽  
Euler Numbers And Polynomials ◽  
Complex Roots ◽  
New Type

We construct a new type of -Euler numbers and polynomials with weak weight : , , respectively. Some interesting results and relationships are obtained. Also, we observe the behavior of roots of the -Euler numbers and polynomials with weak weight . By means of numerical experiments, we demonstrate a remarkably regular structure of the complex roots of -Euler polynomials with weak weight .

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 A Numerical Investigation on the Structure of the Zeros of Euler Polynomials

2015 ◽  
Vol 2015 ◽  
pp. 1-9 ◽  
Author(s):  
C. S. Ryoo ◽  
J. Y. Kang
Keyword(s):  
Numerical Investigation ◽  
Numerical Experiments ◽  
Julia Set ◽  
Regular Structure ◽  
Newton Iteration ◽  
Euler Polynomials ◽  
Complex Roots ◽  
Iteration Function

Using numerical investigation, we observe the behavior of complex roots of the Euler polynomialsEn(x). By means of numerical experiments, we demonstrate a remarkably regular structure of the complex roots of the Euler polynomialsEn(x). Finally, we show the Julia set of Newton iteration functionR(x)=x-En(x)/En′(x).

scholarly journals Calculating Zeros of the -Genocchi Polynomials Associated with -Adic -Integral on

2012 ◽  
Vol 2012 ◽  
pp. 1-18
Author(s):  
C. S. Ryoo
Keyword(s):  
Numerical Investigation ◽  
Numerical Experiments ◽  
Regular Structure ◽  
Genocchi Polynomials ◽  
Complex Roots

In this paper we construct the new analogues of Genocchi the numbers and polynomials. We also observe the behavior of complex roots of the -Genocchi polynomials , using numerical investigation. By means of numerical experiments, we demonstrate a remarkably regular structure of the complex roots of the -Genocchi polynomials . Finally, we give a table for the solutions of the -Genocchi polynomials .

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 A Note on the -Extension of Bernoulli Numbers and Bernoulli Polynomials

2010 ◽  
Vol 2010 ◽  
pp. 1-11 ◽  
Author(s):  
C. S. Ryoo ◽  
T. Kim
Keyword(s):  
Numerical Experiments ◽  
Regular Structure ◽  
Bernoulli Polynomials ◽  
Bernoulli Numbers ◽  
Complex Roots

We observe the behavior of roots of the ()-extension of Bernoulli polynomials . By means of numerical experiments, we demonstrate a remarkably regular structure of the complex roots of theq-extension of Bernoulli polynomials . The main purpose of this paper is also to investigate the zeros of the ()-extension of Bernoulli polynomials . Furthermore, we give a table for the zeros of the ()-extension of Bernoulli polynomials .

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.

scholarly journals Hankel determinants of sequences related to Bernoulli and Euler polynomials

2021 ◽  
pp. 1-29
Author(s):  
Karl Dilcher ◽  
Lin Jiu
Keyword(s):  
Bernoulli Polynomials ◽  
Euler Polynomials ◽  
Hankel Determinant ◽  
Euler Numbers ◽  
Hankel Determinants ◽  
Euler Numbers And Polynomials ◽  
Dirichlet Characters ◽  
Special Cases ◽  
Generalized Bernoulli Polynomials ◽  
Derivatives Of

We evaluate the Hankel determinants of various sequences related to Bernoulli and Euler numbers and special values of the corresponding polynomials. Some of these results arise as special cases of Hankel determinants of certain sums and differences of Bernoulli and Euler polynomials, while others are consequences of a method that uses the derivatives of Bernoulli and Euler polynomials. We also obtain Hankel determinants for sequences of sums and differences of powers and for generalized Bernoulli polynomials belonging to certain Dirichlet characters with small conductors. Finally, we collect and organize Hankel determinant identities for numerous sequences, both new and known, containing Bernoulli and Euler numbers and polynomials.

scholarly journals Some Symmetry Identities for Carlitz’s Type Degenerate Twisted (p,q)-Euler Polynomials Related to Alternating Twisted (p,q)-Sums

Symmetry ◽  
2021 ◽  
Vol 13 (8) ◽  
pp. 1371
Author(s):  
Cheon-Seoung Ryoo
Keyword(s):  
Euler Polynomials ◽  
Euler Numbers ◽  
Explicit Formulas ◽  
Euler Numbers And Polynomials ◽  
New Form ◽  
Symmetric Properties

In this paper, we define a new form of Carlitz’s type degenerate twisted (p,q)-Euler numbers and polynomials by generalizing the degenerate Euler numbers and polynomials, Carlitz’s type degenerate q-Euler numbers and polynomials. Some interesting identities, explicit formulas, symmetric properties, a connection with Carlitz’s type degenerate twisted (p,q)-Euler numbers and polynomials are obtained. Finally, we investigate the zeros of the Carlitz’s type degenerate twisted (p,q)-Euler polynomials by using computer.

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