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 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 On the Symmetric Properties of Higher-Order Twisted -Euler Numbers and Polynomials

2010 ◽  
Vol 2010 (1) ◽  
pp. 765259 ◽  
Author(s):  
Eun-Jung Moon ◽  
Seog-Hoon Rim ◽  
Jeong-Hee Jin ◽  
Sun-Jung Lee
Keyword(s):  
Higher Order ◽  
Euler Numbers ◽  
Euler Numbers And Polynomials ◽  
Symmetric Properties

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.

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 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 On the Symmetric Properties of Higher-Order Twisted q-Euler Numbers and Polynomials

2010 ◽  
Vol 2010 ◽  
pp. 1-9
Author(s):  
Eun-Jung Moon ◽  
Seog-Hoon Rim ◽  
Jeong-Hee Jin ◽  
Sun-Jung Lee
Keyword(s):  
Higher Order ◽  
Euler Numbers ◽  
Euler Numbers And Polynomials ◽  
Symmetric Properties

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 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.

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