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Many works have been done in recent years as to explorations for degenerate versions of some special polynomials and numbers, which began with the introduction of the degenerate Bernoulli and degenerate Euler polynomials by Carlitz. The aim…

Number Theory · Mathematics 2022-12-13 Taekyun Kim , Dae San Kim , Jongkyum Kwon

The degenerate Stirling numbers of the second kind and of the first kind, which are respectively degenerate versions of the Stirling numbers of the second kind and of the first kind, appear frequently when we study various degenerate…

Number Theory · Mathematics 2022-06-10 Taekyun Kim , Dae san Kim , Hye Kyung Kim

The present paper deals with multiplication formulas for the Apostol-Genocchi polynomials of higher order and deduces some explicit recursive formulas. Some earlier results of Carlitz and Howard in terms of Genocchi numbers can be deduced.…

Number Theory · Mathematics 2012-10-23 Hassan Jolany , Hesam Sharifi

In this paper, we consider the degenerate Carlitz q-Bernoulli numbers and polynomials and we investigate some properties of those polynomials.

Number Theory · Mathematics 2015-07-20 Taekyun Kim

In a recent work, the degenerate Stirling polynomials of the second kind were studied by T. Kim. In this paper, we investigate the extended degenerate Stirling numbers of the second kind and the extended degenerate Bell polynomials…

Number Theory · Mathematics 2017-06-30 Taekyun Kim , Dae San Kim

The aim of this paper is to represent any polynomial in terms of the degenerate Frobenius-Euler polynomials and more generally of the higher-order degenerate Frobenius-Euler polynomials. We derive explicit formulas with the help of umbral…

Number Theory · Mathematics 2021-09-29 Taekyun Kim , Dae San Kim

In this paper we study the genralized q-Euler numbers and polynomials. From our results, we derive some interesting congruences related tothe generalized q-Euler numbers.

Number Theory · Mathematics 2009-10-15 Kyoung-Ho Park , Young-Hee Kim , Taekyun Kim

The main object of this paper is to investigate a new class of the generalized Hurwitz type poly-Bernoulli numbers and polynomials from which we derive some algorithms for evaluating the Hurwitz type poly-Bernoulli numbers and polynomials.…

Combinatorics · Mathematics 2023-10-05 Mohamed Amine Boutiche , Mohamed Mechacha , Mourad Rahmani

The aim of this paper is to study the degenerate Bell numbers and polynomials which are degenerate version of the Bell numbers and polynomials. we derive some new identities and properties of those numbers and polynomials that are…

Number Theory · Mathematics 2021-08-16 Taekyun Kim , Dae San Kim , Hyunseok Lee , Seongho Park

It is remarkable that, in recent years, intensive studies have been done for degenerate versions of many special polynomials and numbers and have yielded many interesting results. The aim of this paper is to study the generalized degenerate…

Number Theory · Mathematics 2023-05-09 Taekyun Kim , Dae San Kim , Hye Kyung Kim

In this paper, we consider the degenerate Changhee numbers and polynomials of the second kind which are different from the previously introduced degenerate Changhee numbers and polynomials by Kwon-Kim-Seo (see [11]). We investigate some…

Number Theory · Mathematics 2017-08-01 Taekyun Kim , Dae San Kim

In this paper, some formulae for Genoochi polynomials of higher order are derived using the fact that sets of Bernoulli and Euler polynomials of higher order form basis for the polynomial space.

Classical Analysis and ODEs · Mathematics 2020-09-09 Cristina B. Corcino , Roberto B. Corcino , Joy Ann A. Canete

This paper is concerned with the generalized Euler polynomial matrix $\E^{(\alpha)}(x)$ and the Euler matrix $\E$. Taking into account some properties of Euler polynomials and numbers, we deduce product formulae for $\E^{(\alpha)}(x)$ and…

Number Theory · Mathematics 2018-11-06 Yamilet Quintana , William Ramírez , Alejandro Urieles

In this paper, we introduce the new fully degenerate poly-Bernoulli numbers and polynomials and investigate some properties of these polynomials and numbers. From our properties, we derive some identities for the fully degenerate…

Number Theory · Mathematics 2015-05-27 Dae San Kim , Taekyun Kim

In this paper, we introduce The 2-variable unified family of generalized Apostol-Euler, Bernoulli and Genocchi polynomials and derive some implicit summation formulae and general symmetry identities. The result extend some known summations…

Classical Analysis and ODEs · Mathematics 2018-11-16 Beih S. El-Desouky , Rabab S. Gomaa , Alia M. Magar

In this paper, we will deal with some new formulae for two product Genocchi polynomials together with both Euler polynomials and Bernoulli polynomials. We get some applications for Genocchi polynomials. Our applications possess a number of…

Number Theory · Mathematics 2019-07-04 Serkan Araci , Mehmet Acikgoz , Erdoğan Şen

We study two generalizations of the gamma-expansion of Eulerian polynomials from the viewpoint of the decompositions of statistics. We first present an expansion formula of the trivariate Eulerian polynomials, which are the enumerators for…

Combinatorics · Mathematics 2021-11-18 Shi-Mei Ma , Jun Ma , Jean Yeh , Yeong-Nan Yeh

The goal of this paper is twofold. First, we review the recently developed geometric approach to the combinatorics of the median Genocchi numbers. The Genocchi numbers appear in this context as Euler characteristics of the degenerate flag…

Combinatorics · Mathematics 2012-06-12 Evgeny Feigin

According to generalized Mellin derivative (Kargin), we introduce a new family of polynomials called higher order generalized geometric polynomials. We obtain some properties of them.We discuss their connections to degenerate Bernoulli and…

Classical Analysis and ODEs · Mathematics 2019-08-01 Levent Kargin , Bayram Çekim

We introduce degenerate Hermite polynomials as a degenerate version of the ordinary Hermite polynomials. Then, among other things, by using the formula about representing one lambda-Sheffer polynomial in terms of other lambda-Sheffer…

Number Theory · Mathematics 2020-10-29 Taekyun Kim , Dae San Kim , Lee-Chae Jang , Hyunseok Lee , Hanyoung Kim