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In this paper, we consider the problem of representing any polynomial in terms of the degenerate Bernoulli polynomials and more generally of the higher-order degenerate Bernoulli polynomials. We derive explicit formulas with the help of…

Number Theory · Mathematics 2021-08-12 Dae san Kim , Taekyun Kim

The aim of this paper is to study probabilistic versions of the degenerate Whitney numbers of the second kind and those of the degenerate Dowling polynomials, namely the probabilistic degenerate Whitney numbers of the second kind associated…

Number Theory · Mathematics 2023-12-18 Taekyun Kim , Dae San Kim

We introduce poly-Bernoulli polynomials in two variables by using a generalization of Stirling numbers of the second kind that we studied in a previous work. We prove the bi-variate poly-Bernoulli polynomial version of some known results on…

Number Theory · Mathematics 2023-06-22 Claudio Pita-Ruiz

In this paper, we study the degenerate Eulerian polynomials and numbers and give some new and interesting identities associated with several special numbers and polynomials.

Number Theory · Mathematics 2017-05-04 Taekyun Kim , Dae san kim

Recently, introduced are the generalized Euler-Genocchi and generalized degenerate Euler-Genocchi polynomials. The aim of this note is to study the multi-Euler-Genocchi and degenerate multi-Euler-Genocchi polynomials which are defined by…

Number Theory · Mathematics 2023-01-18 Taekyun Kim , Dae San Kim , Jin-Woo park , Jongkyum Kwon

In this paper, by introducing the degenerate Fubini-type polynomials, we give several relations with the help of the Fa\`a di Bruno formula and some properties of Bell polynomials, and generating function methods. Also, we derive some new…

Number Theory · Mathematics 2021-04-20 Muhammet Cihat Dağli

In this paper, we consider degenerate Carlitz's type q-Euler polynmials and numbers and we investigate some identities arising from the fermionic p-adic integral equations and the generating function of thoe polynomials.

Number Theory · Mathematics 2015-07-17 Dmitry V. Dolgy , Taekyun Kim , Jin-Woo Park , Jong-Jin Seo

In this paper, we introduce the degenerate multiple polyexponential functions which are multiple versions of the degenerate modified polyexponential functions. Then we consider the degenerate multi-poly-Genocchi polynomials which are…

Number Theory · Mathematics 2020-05-18 Taekyun Kim , Dae San Kim , Han Young kim , Jongkyum Kwon

The main aim of this paper is to define and investigate a new class of the degenerate poly-Frobenius-Genocchi polynomials with the help of the polyexponential functions. In this paper, we define the degenerate poly-Frobenius-Genocchi…

Number Theory · Mathematics 2020-07-17 Burak Kurt

Let Y be a random variable such that the moment generating function of Y exists in a neighborhood of the origin. The aim of this paper is to study probabilistic versions of the degenerate Fubini polynomials and the degenerate Fubini…

Probability · Mathematics 2024-01-08 Rongrong Xu , Taekyun Kim , Dae San Kim , Yuankui Ma

Carlitz [2] initiated a study on degenerate versions of Bernoulli and Euler numbers which has been extended recently to the researches on various degenerate versions of quite a few special numbers and polynomials. They have been explored by…

Number Theory · Mathematics 2021-06-28 Taekyun Kim , Dae san Kim , Hyunseok Lee , Seong Ho Park , Jongkyum Kwon

This paper introduces a degenerate version of the Euler-Seidel method by incorporating a parameter lambda into the classical recurrence relation. We define a degenerate Euler-Seidel matrix associated with an initial sequence and establish…

Number Theory · Mathematics 2025-12-24 Taekyun Kim , Dae San Kim , Hyunseok Lee , Kyo-Shin Hwang

In combinatorics, a derangement is a permutation of the elements of a set, such that no element appears in its original position. The number of derangement of an n-element set is called the nth derangement number. Recently, the degenerate…

Number Theory · Mathematics 2024-10-15 Taekyun Kim , Dae San Kim

We introduce the $B$-Stirling numbers of the first and second kind, which are the coefficients of the potential polynomials when we express them in terms of the monomials and the falling factorials, respectively. These numbers include, as…

Combinatorics · Mathematics 2024-10-17 José A. Adell , Beáta Bényi

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

We establish new operational formulae of Burchnall type for the complex disk polynomials (generalized Zernike polynomials). We then use them to derive some interesting identities involving these polynomials. In particular, we establish…

Classical Analysis and ODEs · Mathematics 2015-04-03 Bouchra Aharmim , Amal El Hamyani , Fouzia El Wassouli , Allal Ghanmi

Spivey's combinatorial method revealed an important identity for Bell numbers, involving Stirling numbers of the second kind. This paper extends his work by deriving Spivey-type recurrence relations for fully degenerate Bell polynomials and…

Number Theory · Mathematics 2025-09-09 Taekyun Kim , Dae San Kim

Recently, several types of degenerate Bell polynomials have been introduced as degenerate versions of the ordinary Bell polynomials. The aim of this paper is to study some identities for the degenerate Bell polynomials and their related…

Number Theory · Mathematics 2021-12-17 Taekyun Kim , Dae san kim

Let Y be a random variable whose moment generating function exists in a neighborhood of the origin. The aim of this paper is to study the probabilistic degenerate poly-Bell polynomials associated with the random variable Y, arising from the…

Number Theory · Mathematics 2025-03-11 Pengxiang Xue , Yuankui Ma , Taekyun Kim , Dae San Kim , Wenpeng Zhang

The aim of this paper is to investigate generating functions for modification of the Milne-Thomson's polynomials, which are related to the Bernoulli polynomials and the Hermite polynomials. By applying the Umbral algebra to these generating…

Classical Analysis and ODEs · Mathematics 2018-11-19 Rahime Dere , Yilmaz Simsek