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Recently, Masjed-Jamei-Beyki-Koepf studied the so called new type Euler polynomials without making use of Euler polynomials of complex variable. Here we study degenerate and type 2 versions of these new type Euler polynomials, namely the…

Number Theory · Mathematics 2019-08-30 Taekyun Kim , Dae san Kim , Lee-Chae Jang , Han-Young Kim

In this paper we construct $q$-Genocchi numbers and polynomials. By using these numbers and polynomials, we investigate the $q$-analogue of alternating sums of powers of consecutive integers due to Euler.

Number Theory · Mathematics 2007-05-23 Taekyun Kim

It is well known that ascents, descents and plateaux are equidistributed over the set of classical Stirling permutations. Their common enumerative polynomials are the second-order Eulerian polynomials, which have been extensively studied by…

Combinatorics · Mathematics 2025-06-27 Shi-Mei Ma , Jun-Ying Liu , Jean Yeh , Yeong-Nan Yeh

This paper sets out to introduce the generalized derangement polynomials of order $r $. It then proceeds to establish various identities associated with these polynomials, along with providing recurrence relations for derangement…

Combinatorics · Mathematics 2024-02-27 Ghania Guettai , Diffalah Laissaoui , Mourad Rahmani

The first aim of this paper is to construct new generating functions for the generalized {\lambda}-Stirling type numbers of the second kind, generalized array type polynomials and generalized Eulerian type polynomials and numbers, attached…

Number Theory · Mathematics 2018-11-19 Yilmaz Simsek

In this paper we consider the q-extension of the generating function for the higher-order generalized Genocchi numbers and polynomials attached to Dirichlet's character.

Number Theory · Mathematics 2010-08-10 T. Kim , Byungje Lee , C. S. Ryoo

Recently, the degenerate Stirling numbers of the first kind were introduced. In this paper, we give some formulas for the degenerate Stirling numbers of the first kind in the terms of the complete Bell polynomials with higher-order harmonic…

Number Theory · Mathematics 2018-02-06 Taekyun Kim , Dae San Kim

Recently, the degenerate harmonic and the degenerate hyperharmonic numbers are introduced respectively as degenerate versions of the harmonic and the hyperharmonic numbers. The aim of this paper is to introduce the degenerate…

Number Theory · Mathematics 2023-06-22 Dae San Kim , Taekyun Kim

In this paper, we consider sums of values of degenerate falling factorials and give a probabilistic proof of a recurrence relation for them. This may be viewed as a degenerate version of the recent probabilistic proofs on sums of powers of…

Number Theory · Mathematics 2024-09-13 Taekyun Kim , Dae san Kim

This paper is concerned with multivariate refinements of the gamma-positivity of Eulerian polynomials by using the succession and fixed point statistics. Properties of the enumerative polynomials for permutations, signed permutations and…

Combinatorics · Mathematics 2020-08-11 Shi-Mei Ma , Jun Ma , Jean Yeh , Yeong-Nan Yeh

In this paper, by using some families of special numbers and polynomials with their generating functions, we give various properties of these numbers and polynomials. These numbers are related to the well-known numbers and polynomials,…

Combinatorics · Mathematics 2023-02-24 Yilmaz Simsek

In recent years, studying degenerate versions of various special polynomials and numbers have attracted many mathematicians. Here we introduce degenerate type 2 Bernoulli polynomials, fully degenerate type 2 Bernoulli polynomials and…

Number Theory · Mathematics 2019-03-12 Dae San Kim , Han Young Kim , Sung-Soo Pyo , Taekyun Kim

The degenerate exponentials play an important role in recent study on degenerate versions of many special numbers and polynomials, the degenerate gamma function, the degenerate umbral calculus and the degenerate q-umbral calculus. The aim…

Number Theory · Mathematics 2023-01-10 Dae San Kim , Hye Kyung Kim , Taekyun Kim

The normal ordering of an integral power of the number operator in terms of boson operators is expressed with the help of the Stirling numbers of the second kind. As a `degenerate version' of this, we consider the normal ordering of a…

Number Theory · Mathematics 2022-04-07 Taekyun Kim , Dae san Kim , Hye Kyung Kim

In this work, we consider the generating function of Kim's q-Euler polynomials and introduce new generalization of q-Genocchi polynomials and numbers of higher order. Also, we give surprising identities for studying in Analytic Numbers…

Number Theory · Mathematics 2019-07-04 Serkan Araci , Mehmet Acikgoz , Jong Jin Seo

The aim of this paper is twofold. The first one is to find several relations between the type 2 higher-order degenerate Euler polynomials and the type 2 higher-order Changhee polynomials in connection with the degenerate stirling numbers of…

Number Theory · Mathematics 2020-04-28 Taekyun Kim , Dae San Kim

In ths paper we discuss the new concept of the q-extension of Genocchi numbers and give the some relations between q-Genocchi polynomials and q-Euler numbers.

Number Theory · Mathematics 2007-05-23 Taekyun Kim

Thw purpose of this paper is to present a systemic study of some families of the generalized q-Euler numbers and polynomials of higher order.

Number Theory · Mathematics 2009-12-25 Taekyun Kim

The aim of this paper is to study Jindalrae and Gaenari numbers and polynomials in connection with Jindalrae-Stirling numbers of the first and second kinds. For this purpose, we first introduce Jindalrae-Stirling numbers of the first and…

Number Theory · Mathematics 2020-04-09 Taekyun Kim , Dae San Kim , Lee-Chae jang , Hyunseok Lee

The purpose of this paper is to generalize this relation of symmetry between the power sum polynomials and the generalized Euler polynomials to the relation between the power sum polynomials and the generalized higher-order Euler…

Number Theory · Mathematics 2009-10-07 Taekyun Kim