scholarly journals A note on type 2 q-Bernoulli and type 2 q-Euler polynomials

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
Dae San Kim ◽  
Taekyun Kim ◽  
Han Young Kim ◽  
Jongkyum Kwon
Keyword(s):  
Euler Polynomials

scholarly journals A Note on Type 2 Degenerate Poly-Frobenius-Euler Polynomials

2020 ◽  
Author(s):  
Waseem Khan
Keyword(s):  
Euler Polynomials ◽  
Genocchi Polynomials ◽  
Explicit Expressions

In this paper, we construct the degenerate poly-Frobenius-Genocchi polynomials, called the type 2 degenerate poly-Frobenius-Euler polynomials, by means of polyexponential function. We derive explicit expressions and some identities of those polynomials. In the last section, we introduce type 2 degenerate unipoly-Frobenius-Genocchi polynomials by means of unipoly function and derive explicit multifarious properties.

scholarly journals Some identities and reciprocity relationsof unipoly-Dedekind type DC sums

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Hye Kyung Kim ◽  
Dae Sik Lee
Keyword(s):  
The Other ◽  
Reciprocity Relation ◽  
Dedekind Sums ◽  
Euler Polynomials ◽  
Euler Function ◽  
Polylogarithm Function

AbstractDedekind type DC sums and their generalizations are defined in terms of Euler functions and their generalization. Recently, Ma et al. (Adv. Differ. Equ. 2021:30 2021) introduced the poly-Dedekind type DC sums by replacing the Euler function appearing in Dedekind sums, and they were shown to satisfy a reciprocity relation. In this paper, we consider two kinds of new generalizations of the poly-Dedekind type DC sums. One is a unipoly-Dedekind type DC sum associated with the type 2 unipoly-Euler functions expressed in the type 2 unipoly-Euler polynomials using the modified polyexponential function, and we study some identities and the reciprocity relation for these unipoly-Dedekind type DC sums. The other is a unipoly-Dedekind sums type DC associated with the poly-Euler functions expressed in the unipoly-Euler polynomials using the polylogarithm function, and we derive some identities and the reciprocity relation for those unipoly-Dedekind type DC sums.

scholarly journals Some Relations of Two Type 2 Polynomials and Discrete Harmonic Numbers and Polynomials

Symmetry ◽  
2020 ◽  
Vol 12 (6) ◽  
pp. 905 ◽  
Author(s):  
Taekyun Kim ◽  
Dae San Kim
Keyword(s):  
Zeta Functions ◽  
Stirling Numbers ◽  
Higher Order ◽  
High Order ◽  
Euler Polynomials ◽  
Harmonic Numbers ◽  
Order Degenerate ◽  
Riemann Zeta ◽  
Changhee Polynomials

Harmonic numbers appear, for example, in many expressions involving Riemann zeta functions. Here, among other things, we introduce and study discrete versions of those numbers, namely the discrete harmonic numbers. The aim of this paper is twofold. The first is to find several relations between the Type 2 higher-order degenerate Euler polynomials and the Type 2 high-order Changhee polynomials in connection with the degenerate Stirling numbers of both kinds and Jindalrae–Stirling numbers of both kinds. The second is to define the discrete harmonic numbers and some related polynomials and numbers, and to derive their explicit expressions and an identity.

scholarly journals On the type 2 poly-Bernoulli polynomials associated with umbral calculus

2021 ◽  
Vol 19 (1) ◽  
pp. 878-887
Author(s):  
Taekyun Kim ◽  
Dae San Kim ◽  
Dmitry V. Dolgy ◽  
Jin-Woo Park
Keyword(s):  
Bernoulli Polynomials ◽  
Higher Order ◽  
Umbral Calculus ◽  
Euler Polynomials ◽  
Special Polynomials

Abstract Type 2 poly-Bernoulli polynomials were introduced recently with the help of modified polyexponential functions. In this paper, we investigate several properties and identities associated with those polynomials arising from umbral calculus techniques. In particular, we express the type 2 poly-Bernoulli polynomials in terms of several special polynomials, like higher-order Cauchy polynomials, higher-order Euler polynomials, and higher-order Frobenius-Euler polynomials.

scholarly journals Construction of the Type 2 Degenerate Multi-Poly-Euler Polynomials and Numbers

2020 ◽  
Author(s):  
Waseem Ahmad Khan ◽  
Mehmet Acikgoz ◽  
Ugur Duran
Keyword(s):  
Linear Combination ◽  
Stirling Numbers ◽  
Euler Polynomials ◽  
Addition Formula ◽  
New Class ◽  
Whitney Numbers ◽  
Derivative Formula ◽  
Euler Polynomials And Numbers ◽  
Special Case

In this paper, we consider a new class of polynomials which is called the multi-poly-Euler polynomials. Then, we investigate their some properties and relations. We provide that the type 2 degenerate multi-poly-Euler polynomials equals a linear combination of the degenerate Euler polynomials of higher order and the degenerate Stirling numbers of the first kind. Moreover, we provide an addition formula and a derivative formula. Furthermore, in a special case, we acquire a correlation between the type 2 degenerate multi-poly-Euler polynomials and degenerate Whitney numbers.

scholarly journals On type 2 degenerate Bernoulli and Euler polynomials of complex variable

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
Taekyun Kim ◽  
Dae San Kim ◽  
Lee-Chae Jang ◽  
Han-Young Kim
Keyword(s):  
Complex Variable ◽  
Bernoulli Polynomials ◽  
Affirmative Answer ◽  
Euler Polynomials ◽  
The Real ◽  
New Type ◽  
Explicit Expressions

AbstractRecently, Masjed-Jamei, Beyki, and Koepf studied the so-called new type Euler polynomials without using Euler polynomials of complex variable. Here we study the type 2 degenerate cosine-Euler and type 2 degenerate sine-Euler polynomials, which are type 2 degenerate versions of these new type Euler polynomials, by considering the degenerate Euler polynomials of complex variable and by treating the real and imaginary parts separately. In addition, we investigate the corresponding ones for Bernoulli polynomials in the same manner. We derive some explicit expressions for those new polynomials and some identities relating to them. Here we note that the idea of separating the real and imaginary parts separately gives an affirmative answer to the question asked by Hacène Belbachir.

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 Type 2 Degenerate Poly-Euler Polynomials

Symmetry ◽  
2020 ◽  
Vol 12 (6) ◽  
pp. 1011 ◽  
Author(s):  
Dae Sik Lee ◽  
Hye Kyung Kim ◽  
Lee-Chae Jang
Keyword(s):  
Final Section ◽  
Bernoulli Polynomials ◽  
Stirling Numbers ◽  
Euler Polynomials ◽  
Special Polynomials ◽  
Special Numbers And Polynomials ◽  
Euler Polynomials And Numbers ◽  
New Type ◽  
Explicit Expressions

In recent years, many mathematicians have studied the degenerate versions of many special polynomials and numbers. The polyexponential functions were introduced by Hardy and rediscovered by Kim, as inverses to the polylogarithms functions. The paper is divided two parts. First, we introduce a new type of the type 2 poly-Euler polynomials and numbers constructed from the modified polyexponential function, the so-called type 2 poly-Euler polynomials and numbers. We show various expressions and identities for these polynomials and numbers. Some of them involving the (poly) Euler polynomials and another special numbers and polynomials such as (poly) Bernoulli polynomials, the Stirling numbers of the first kind, the Stirling numbers of the second kind, etc. In final section, we introduce a new type of the type 2 degenerate poly-Euler polynomials and the numbers defined in the previous section. We give explicit expressions and identities involving those polynomials in a similar direction to the previous section.

scholarly journals Note on the Type 2 Degenerate Multi-Poly-Euler Polynomials

Symmetry ◽  
2020 ◽  
Vol 12 (10) ◽  
pp. 1691
Author(s):  
Waseem Ahmad Khan ◽  
Mehmet Acikgoz ◽  
Ugur Duran
Keyword(s):  
Bernoulli Polynomials ◽  
Stirling Numbers ◽  
Euler Polynomials ◽  
Addition Formula ◽  
Whitney Numbers ◽  
Derivative Formula ◽  
New Type ◽  
Special Case ◽  
Math Phys

Kim and Kim (Russ. J. Math. Phys. 26, 2019, 40-49) introduced polyexponential function as an inverse to the polylogarithm function and by this, constructed a new type poly-Bernoulli polynomials. Recently, by using the polyexponential function, a number of generalizations of some polynomials and numbers have been presented and investigated. Motivated by these researches, in this paper, multi-poly-Euler polynomials are considered utilizing the degenerate multiple polyexponential functions and then, their properties and relations are investigated and studied. That the type 2 degenerate multi-poly-Euler polynomials equal a linear combination of the degenerate Euler polynomials of higher order and the degenerate Stirling numbers of the first kind is proved. Moreover, an addition formula and a derivative formula are derived. Furthermore, in a special case, a correlation between the type 2 degenerate multi-poly-Euler polynomials and degenerate Whitney numbers is shown.

scholarly journals A Note on Type 2 Degenerate q-Euler Polynomials

Mathematics ◽  
2019 ◽  
Vol 7 (8) ◽  
pp. 681 ◽  
Author(s):  
Taekyun Kim ◽  
Dae San Kim ◽  
Han Young Kim ◽  
Sung-Soo Pyo
Keyword(s):  
Generating Function ◽  
Euler Polynomials ◽  
Integer Power ◽  
Power Sums

Recently, type 2 degenerate Euler polynomials and type 2 q-Euler polynomials were studied, respectively, as degenerate versions of the type 2 Euler polynomials as well as a q-analog of the type 2 Euler polynomials. In this paper, we consider the type 2 degenerate q-Euler polynomials, which are derived from the fermionic p-adic q-integrals on Z p , and investigate some properties and identities related to these polynomials and numbers. In detail, we give for these polynomials several expressions, generating function, relations with type 2 q-Euler polynomials and the expression corresponding to the representation of alternating integer power sums in terms of Euler polynomials. One novelty about this paper is that the type 2 degenerate q-Euler polynomials arise naturally by means of the fermionic p-adic q-integrals so that it is possible to easily find some identities of symmetry for those polynomials and numbers, as were done previously.

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