scholarly journals Some Properties for Multiple Twisted (p, q)-L-Function and Carlitz’s Type Higher-Order Twisted (p, q)-Euler Polynomials

Mathematics ◽  
2019 ◽  
Vol 7 (12) ◽  
pp. 1205 ◽  
Author(s):  
Kyung-Won Hwang ◽  
Cheon Seoung Ryoo
Keyword(s):  
Generating Functions ◽  
Higher Order ◽  
Integral Representations ◽  
Euler Polynomials ◽  
Euler Numbers ◽  
Mellin Transformation ◽  
Euler Numbers And Polynomials ◽  
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.

scholarly journals Symmetric Properties for Dirichlet-Type Multiple (p,q)-L-Function

Symmetry ◽  
2021 ◽  
Vol 13 (1) ◽  
pp. 95
Author(s):  
Kyung-Won Hwang ◽  
Ravi P. Agarwal ◽  
Cheon Seoung Ryoo
Keyword(s):  
Higher Order ◽  
Euler Polynomials ◽  
Euler Numbers ◽  
Dirichlet Type ◽  
Euler Numbers And Polynomials ◽  
Symmetric Identities ◽  
Symmetric Properties

The main aim of this article is to investigate some interesting symmetric identities for the Dirichlet-type multiple (p,q)-L function. We use this function to examine the symmetry of the generalized higher-order (p,q)-Euler polynomials related to χ. First, the generalized higher-order (p,q)-Euler numbers and polynomials related to χ are defined. We also give a few new symmetric properties for the Dirichlet-type multiple (p,q)-L-function and generalized higher-order (p,q)-Euler polynomials related to χ.

scholarly journals Some Symmetric Identities for the Multiple (p, q)-Hurwitz-Euler eta Function

Symmetry ◽  
2019 ◽  
Vol 11 (5) ◽  
pp. 645 ◽  
Author(s):  
Kyung-Won Hwang ◽  
Cheon Seoung Ryoo
Keyword(s):  
Higher Order ◽  
Complex Field ◽  
Euler Numbers ◽  
Euler Numbers And Polynomials ◽  
Symmetric Identities ◽  
Symmetric Properties

The main purpose of this paper is to find some interesting symmetric identities for the ( p , q ) -Hurwitz-Euler eta function in a complex field. Firstly, we define the multiple ( p , q ) -Hurwitz-Euler eta function by generalizing the Carlitz’s form ( p , q ) -Euler numbers and polynomials. We find some formulas and properties involved in Carlitz’s form ( p , q ) -Euler numbers and polynomials with higher order. We find new symmetric identities for multiple ( p , q ) -Hurwitz-Euler eta functions. We also obtain symmetric identities for Carlitz’s form ( p , q ) -Euler numbers and polynomials with higher order by using symmetry about multiple ( p , q ) -Hurwitz-Euler eta functions. Finally, we study the distribution and symmetric properties of the zero of Carlitz’s form ( p , q ) -Euler numbers and polynomials with higher order.

scholarly journals Multivariate Interpolation Functions of Higher-Orderq-Euler Numbers and Their Applications

2008 ◽  
Vol 2008 ◽  
pp. 1-16 ◽  
Author(s):  
Hacer Ozden ◽  
Ismail Naci Cangul ◽  
Yilmaz Simsek
Keyword(s):  
Zeta Function ◽  
Generating Functions ◽  
Higher Order ◽  
Multivariate Interpolation ◽  
Euler Numbers ◽  
Euler Numbers And Polynomials ◽  
Sums Of Products ◽  
Interpolation Functions

The aim of this paper, firstly, is to construct generating functions ofq-Euler numbers and polynomials of higher order by applying the fermionicp-adicq-Volkenborn integral, secondly, to define multivariateq-Euler zeta function (Barnes-type Hurwitzq-Euler zeta function) andl-function which interpolate these numbers and polynomials at negative integers, respectively. We give relation between Barnes-type Hurwitzq-Euler zeta function and multivariateq-Eulerl-function. Moreover, complete sums of products of these numbers and polynomials are found. We give some applications related to these numbers and functions as well.

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 Symmetric Identities for Carlitz-Type Higher-Order Degenerate (p,q)-Euler Numbers and Polynomials

Symmetry ◽  
2019 ◽  
Vol 11 (12) ◽  
pp. 1432
Author(s):  
Kyung-Won Hwang ◽  
Cheon Seoung Ryoo
Keyword(s):  
Higher Order ◽  
Euler Numbers ◽  
Euler Numbers And Polynomials ◽  
Order Degenerate ◽  
Symmetric Identities

The main goal of this paper is to investigate some interesting symmetric identities for Carlitz-type higher-order degenerate ( p , q ) -Euler numbers, and polynomials. At first, the Carlitz-type higher-order degenerate ( p , q ) -Euler numbers and polynomials are defined. We give few new symmetric identities for Carlitz-type higher-order degenerate ( p , q ) -Euler numbers and polynomials.

scholarly journals Identities of Symmetry for Type 2 Bernoulli and Euler Polynomials

Symmetry ◽  
2019 ◽  
Vol 11 (5) ◽  
pp. 613 ◽  
Author(s):  
Dae San Kim ◽  
Han Young Kim ◽  
Dojin Kim ◽  
Taekyun Kim
Keyword(s):  
Prime Number ◽  
Random Variables ◽  
Bernoulli Polynomials ◽  
Euler Polynomials ◽  
Euler Numbers ◽  
Power Sums ◽  
Symmetric Identities ◽  
Positive Integers

The main purpose of this paper is to give several identities of symmetry for type 2 Bernoulli and Euler polynomials by considering certain quotients of bosonic p-adic and fermionic p-adic integrals on Z p , where p is an odd prime number. Indeed, they are symmetric identities involving type 2 Bernoulli polynomials and power sums of consecutive odd positive integers, and the ones involving type 2 Euler polynomials and alternating power sums of odd positive integers. Furthermore, we consider two random variables created from random variables having Laplace distributions and show their moments are given in terms of the type 2 Bernoulli and Euler numbers.

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

Symmetry ◽  
2019 ◽  
Vol 11 (6) ◽  
pp. 830 ◽  
Author(s):  
Kyung-Won Hwang ◽  
Cheon Seoung Ryoo
Keyword(s):  
Euler Polynomials ◽  
Euler Numbers ◽  
Euler Numbers And Polynomials ◽  
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.

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.

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