Generalized -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.

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

International Journal of Mathematics and Mathematical Sciences ◽

10.1155/2012/817157 ◽

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 ◽

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.

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

Journal of Applied Mathematics ◽

10.1155/2014/806239 ◽

2014 ◽

Vol 2014 ◽

pp. 1-7 ◽

Cited By ~ 1

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).

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 ◽

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.

Euler Numbers and Polynomials Associated with Zeta Functions

Abstract and Applied Analysis ◽

10.1155/2008/581582 ◽

2008 ◽

Vol 2008 ◽

pp. 1-11 ◽

Cited By ~ 32

Author(s):

Taekyun Kim

Keyword(s):

Zeta Function ◽

Entire Functions ◽

Zeta Functions ◽

Euler Polynomials ◽

Euler Numbers ◽

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.

Ordinary and degenerate Euler numbers and polynomials

Journal of Inequalities and Applications ◽

10.1186/s13660-019-2221-5 ◽

2019 ◽

Vol 2019 (1) ◽

Cited By ~ 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 ◽

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.

Hankel determinants of sequences related to Bernoulli and Euler polynomials

International Journal of Number Theory ◽

10.1142/s179304212250021x ◽

2021 ◽

pp. 1-29

Author(s):

Karl Dilcher ◽

Lin Jiu

Keyword(s):

Bernoulli Polynomials ◽

Euler Polynomials ◽

Hankel Determinant ◽

Euler Numbers ◽

Hankel Determinants ◽

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.

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

Symmetry ◽

10.3390/sym13081371 ◽

2021 ◽

Vol 13 (8) ◽

pp. 1371

Author(s):

Cheon-Seoung Ryoo

Keyword(s):

Euler Polynomials ◽

Euler Numbers ◽

Explicit Formulas ◽

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.

Some Identities on Degenerate Bernstein and Degenerate Euler Polynomials

Mathematics ◽

10.3390/math7010047 ◽

2019 ◽

Vol 7 (1) ◽

pp. 47 ◽

Cited By ~ 4

Author(s):

Taekyun Kim ◽

Dae San Kim

Keyword(s):

Differential Equations ◽

Generating Functions ◽

Bernstein Polynomials ◽

Umbral Calculus ◽

Euler Polynomials ◽

Euler Numbers ◽

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.

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 ◽

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.

Some Symmetric Identities for Degenerate Carlitz-type (p, q)-Euler Numbers and Polynomials

Symmetry ◽

10.3390/sym11060830 ◽

2019 ◽

Vol 11 (6) ◽

pp. 830 ◽

Cited By ~ 2

Author(s):

Kyung-Won Hwang ◽

Cheon Seoung Ryoo

Keyword(s):

Euler Polynomials ◽

Euler Numbers ◽

Symmetric Identities

In this paper we define the degenerate Carlitz-type ( p , q ) -Euler polynomials by generalizing the degenerate Euler numbers and polynomials, degenerate Carlitz-type q-Euler numbers and polynomials. We also give some theorems and exact formulas, which have a connection to degenerate Carlitz-type ( p , q ) -Euler numbers and polynomials.