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 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 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 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 Generalized Tepper’s Identity and Its Application

Mathematics ◽  
2020 ◽  
Vol 8 (2) ◽  
pp. 243
Author(s):  
Dmitry Kruchinin ◽  
Vladimir Kruchinin ◽  
Yilmaz Simsek
Keyword(s):  
Number Theory ◽  
Generating Functions ◽  
Stirling Numbers ◽  
Bernoulli Numbers ◽  
Combinatorial Analysis ◽  
Euler Numbers ◽  
Bernoulli Numbers And Polynomials ◽  
Euler Numbers And Polynomials

The aim of this paper is to study the Tepper identity, which is very important in number theory and combinatorial analysis. Using generating functions and compositions of generating functions, we derive many identities and relations associated with the Bernoulli numbers and polynomials, the Euler numbers and polynomials, and the Stirling numbers. Moreover, we give applications related to the Tepper identity and these numbers and polynomials.

scholarly journals On Multiple Interpolation Functions of the Nörlund-Typeq-Euler Polynomials

2009 ◽  
Vol 2009 ◽  
pp. 1-14 ◽  
Author(s):  
Mehmet Acikgoz ◽  
Yilmaz Simsek
Keyword(s):  
Zeta Function ◽  
Generating Functions ◽  
Euler Polynomials ◽  
Multiple Interpolation ◽  
Definition Of ◽  
Interpolation Functions

In (2006) and (2009), Kim defined new generating functions of the Genocchi, Nörlund-typeq-Euler polynomials and their interpolation functions. In this paper, we give another definition of the multiple Hurwitz typeq-zeta function. This function interpolates Nörlund-typeq-Euler polynomials at negative integers. We also give some identities related to these polynomials and functions. Furthermore, we give some remarks about approximations of Bernoulli and Euler polynomials.

scholarly journals On the Higher-Orderq-Euler Numbers and Polynomials with Weightα

2011 ◽  
Vol 2011 ◽  
pp. 1-12 ◽  
Author(s):  
K.-W. Hwang ◽  
D. V. Dolgy ◽  
T. Kim ◽  
S. H. Lee
Keyword(s):  
Higher Order ◽  
Euler Numbers ◽  
Euler Numbers And Polynomials

The main purpose of this paper is to present a systemic study of some families of higher-orderq-Euler numbers and polynomials with weightα. In particular, by using the fermionicp-adicq-integral onℤp, we give a new concept ofq-Euler numbers and polynomials with weightα.

scholarly journals Identities and relations for Fubini type numbers and polynomials via generating functions and p-adic integral approach

2019 ◽  
Vol 106 (120) ◽  
pp. 113-123
Author(s):  
Neslihan Kilar ◽  
Yilmaz Simsek
Keyword(s):  
Functional Equation ◽  
Generating Functions ◽  
Stirling Numbers ◽  
Bernoulli Numbers ◽  
Combinatorial Analysis ◽  
Integral Approach ◽  
Euler Numbers ◽  
Bernoulli Numbers And Polynomials ◽  
Array Polynomials ◽  
Euler Numbers And Polynomials

The Fubini type polynomials have many application not only especially in combinatorial analysis, but also other branches of mathematics, in engineering and related areas. Therefore, by using the p-adic integrals method and functional equation of the generating functions for Fubini type polynomials and numbers, we derive various different new identities, relations and formulas including well-known numbers and polynomials such as the Bernoulli numbers and polynomials, the Euler numbers and polynomials, the Stirling numbers of the second kind, the ?-array polynomials and the Lah numbers.

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