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 New type of degenerate Daehee polynomials of the second kind

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Sunil Kumar Sharma ◽  
Waseem A. Khan ◽  
Serkan Araci ◽  
Sameh S. Ahmed
Keyword(s):  
Generating Function ◽  
Special Class ◽  
Stirling Numbers ◽  
Bernoulli Numbers ◽  
Higher Order ◽  
Bernoulli Numbers And Polynomials ◽  
New Type ◽  
Order Degenerate ◽  
Math Phys

Abstract Recently, Kim and Kim (Russ. J. Math. Phys. 27(2):227–235, 2020) have studied new type degenerate Bernoulli numbers and polynomials by making use of degenerate logarithm. Motivated by (Kim and Kim in Russ. J. Math. Phys. 27(2):227–235, 2020), we consider a special class of polynomials, which we call a new type of degenerate Daehee numbers and polynomials of the second kind. By using their generating function, we derive some new relations including the degenerate Stirling numbers of the first and second kinds. Moreover, we introduce a new type of higher-order degenerate Daehee polynomials of the second kind. We also derive some new identities and properties of this type of polynomials.

scholarly journals Two-Variable Type 2 Poly-Fubini Polynomials

Mathematics ◽  
2021 ◽  
Vol 9 (3) ◽  
pp. 281
Author(s):  
Ghulam Muhiuddin ◽  
Waseem Ahmad Khan ◽  
Ugur Duran
Keyword(s):  
Integral Representation ◽  
Bernoulli Polynomials ◽  
Stirling Numbers ◽  
Higher Order ◽  
Variable Type ◽  
Summation Formulas

In the present work, a new extension of the two-variable Fubini polynomials is introduced by means of the polyexponential function, which is called the two-variable type 2 poly-Fubini polynomials. Then, some useful relations including the Stirling numbers of the second and the first kinds, the usual Fubini polynomials, and the higher-order Bernoulli polynomials are derived. Also, some summation formulas and an integral representation for type 2 poly-Fubini polynomials are investigated. Moreover, two-variable unipoly-Fubini polynomials are introduced utilizing the unipoly function, and diverse properties involving integral and derivative properties are attained. Furthermore, some relationships covering the two-variable unipoly-Fubini polynomials, the Stirling numbers of the second and the first kinds, and the Daehee polynomials are acquired.

scholarly journals Notes on the Poly-Korobov polynomials and related polynomials

Filomat ◽  
2020 ◽  
Vol 34 (2) ◽  
pp. 469-474
Author(s):  
Burak Kurt
Keyword(s):  
Stirling Numbers ◽  
Bernoulli Numbers ◽  
Euler Polynomials ◽  
Changhee Polynomials

In recent years, many mathematicians ([2], [7], [8], [9], [15], [16], [21]) introduced and investigated for the Korobov polynomials. They gave some identities and relations for the Korobov type polynomials. In this work, we give some relations for the first kind Korobov polynomials and Korobov type Changhee polynomials. Further, wegive two relations between the poly-Changhee polynomials and the poly-Korobov polynomials. Also, we give a relation among the poly-Korobov type Changhee polynomials, the Stirling numbers of the second kind, the Euler polynomials and the Bernoulli numbers.

Higher-order degenerate Euler polynomials

2015 ◽  
Vol 9 ◽  
pp. 57-73 ◽  
Author(s):  
Dae San Kim ◽  
Taekyun Kim
Keyword(s):  
Higher Order ◽  
Euler Polynomials ◽  
Order Degenerate

On applications for Mahler expansion associated with p-adic q-integrals

2019 ◽  
Vol 15 (01) ◽  
pp. 67-84 ◽  
Author(s):  
Ugur Duran ◽  
Mehmet Acikgoz
Keyword(s):  
Explicit Formula ◽  
Gamma Function ◽  
Recurrence Relations ◽  
Stirling Numbers ◽  
Euler Polynomials ◽  
Symmetric Relation ◽  
Euler Constant ◽  
Basic Properties ◽  
Changhee Polynomials

In this paper, we primarily consider a generalization of the fermionic [Formula: see text]-adic [Formula: see text]-integral on [Formula: see text] including the parameters [Formula: see text] and [Formula: see text] and investigate its some basic properties. By means of the foregoing integral, we introduce two generalizations of [Formula: see text]-Changhee polynomials and numbers as [Formula: see text]-Changhee polynomials and numbers with weight [Formula: see text] and [Formula: see text]-Changhee polynomials and numbers of second kind with weight [Formula: see text]. For the mentioned polynomials, we obtain new and interesting relationships and identities including symmetric relation, recurrence relations and correlations associated with the weighted [Formula: see text]-Euler polynomials, [Formula: see text]-Stirling numbers of the second kind and Stirling numbers of first and second kinds. Then, we discover multifarious relationships among the two types of weighted [Formula: see text]-Changhee polynomials and [Formula: see text]-adic gamma function. Also, we compute the weighted fermionic [Formula: see text]-adic [Formula: see text]-integral of the derivative of [Formula: see text]-adic gamma function. Moreover, we give a novel representation for the [Formula: see text]-adic Euler constant by means of the weighted [Formula: see text]-Changhee polynomials and numbers. We finally provide a quirky explicit formula for [Formula: see text]-adic Euler constant.

An identity of symmetry for the degenerate Frobenius-Euler Polynomials

2018 ◽  
Vol 68 (1) ◽  
pp. 239-243 ◽  
Author(s):  
Taekyun Kim ◽  
Dae San Kim
Keyword(s):  
Recurrence Relations ◽  
Higher Order ◽  
Euler Polynomials ◽  
Type Result ◽  
Order Degenerate ◽  
Multiplication Theorem

Abstract In this paper, we prove an identity of symmetry for the higher-order degenerate Frobenius-Euler polynomials and derive the recurrence relations and multiplication theorem type result for the degenerate Frobenius-Euler polynomials.

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.

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