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I recent years, studying degenerate versions of some special polynomials, which was initiated by Carlitz in an investigation of the degenerate Bernoulli and Euler polynomials, regained lively interest of mant mathematicains. In this paper,…

Number Theory · Mathematics 2020-02-10 Taekyun Kim , Dae San Kim

The aim of this paper is to introduce Bell polynomials and numbers of the second kind and poly-Bell polynomials and numbers of the second kind, and to derive their explicit expressions, recurrence relations and some identities involving…

Number Theory · Mathematics 2021-06-29 Dae San Kim , Dmitry V. Dolgy , Hye-Kyung Kim , Hyunseok Lee , Taekyun Kim

As is well-known, poly-Bernoulli polynomials are defined in terms of polylogarithm functions. Recently, as degenerate version of such functions and polynomials, degenerate polylogarithm functions were introduced and degenertae…

Number Theory · Mathematics 2020-12-14 Taekyun Kim , Dae San Kim , Jongkyum Kwon , Hyunseok Lee

The aim of this paper is to introduce truncated degenerate Bell polynomials and numbers and to investigate some of their properties. In more detail, we obtain explicit expressions, identities involving other special polynomials, integral…

Number Theory · Mathematics 2020-12-10 Taekyun Kim , Dae san Kim

In this paper, we study the degenerate derangement polynomials and numbers, investigate some properties of those polynomials and numbers and explore their connections with the degenerate gamma distributions. In more detail, we derive their…

Number Theory · Mathematics 2020-11-18 Taekyun Kim , Dae san Kim , Hyunseok Lee , Lee-Chae Jang

In this paper, we introduce two variable degenerate Bell polynomials as a generalization of the degenerate Bell polynomials which were studied earlier. Among other things, we show their connections with Poisson degenerate central moments…

Number Theory · Mathematics 2020-06-25 Dae San Kim , Taekyun Kim , Han-Young Kim , Hynseok Lee

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

Studying degenerate versions of various special polynomials have become an active area of research and yielded many interesting arithmetic and combinatorial results. Here we introduce a degenerate version of polylogarithm function, called…

Number Theory · Mathematics 2020-02-12 Taekyun Kim , Dae San Kim

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

In this paper, we consider degenerate poly-Bernoulli numbers and polynomials associated with polylogarithmic function and p-adic invariant integral on Zp. By using umbral calculus, we derive some identities of those numbers and polynomials

Number Theory · Mathematics 2015-06-11 Dae San Kim , Taekyun Kim

The aim of this paper is to further study some properties and identities on the degenerate Fubini and the degenerate Bell polynomials which are degenerate versions of the Fubini and the Bell polynomials, respectively. Especially, we find…

Number Theory · Mathematics 2022-03-02 Taekyun Kim , Dae san Kim

In this paper, we introduce a new class of polynomials, called probabilistic q-Bernstein polynomials, alongside their generating function. Assuming Y is a random variable satisfying moment conditions, we use the generating function of these…

Classical Analysis and ODEs · Mathematics 2025-04-01 Ayse Karagenc , Mehmet Acikgoz , Serkan Araci

In this paper, we study degenerate ordered Bell polynomials with the viewpoint of Carlitz's degenerate Bernoulli and Euler polynomials and derive by using umbral calculus some properties and new identities for the degenerate ordered Bell…

Number Theory · Mathematics 2017-04-25 Taekyun Kim , Dae san Kim

Recently, Kim-Kim introduced the degenerate r-Bell polynomials and investigated some results which are derived from umbral calculus. The aim of this paper is to study some properties of the degenerate r-Bell polynomials and numbers via…

Number Theory · Mathematics 2022-08-11 Taekyun Kim , Dae san Kim , Hye Kyung 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

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

This paper aims to construct a new family of numbers and polynomials which are related to the Bell numbers and polynomials by means of the confluent hypergeometric function. We give various properties of these numbers and polynomials…

Number Theory · Mathematics 2018-12-12 Ghania Guettai , Diffalah Laissaoui , Mourad Rahmani , Madjid Sebaoui

Here we consider the degenerate Bernstein polynomials as a degenerate version of Bernstein polynomials, which are motivated by Simsek's recent work 'Generating functions for unification of the multidimensional Bernstein polynomials and…

Number Theory · Mathematics 2018-06-19 Taekyun Kim , Dae san Kim

In this paper, we consider the degenerate poly-Bernoulli polynomials and present new and explicit formulas for computing them in terms of the degenerate Bernoulli polynomials and Stirling numbers of the second kind.

Number Theory · Mathematics 2015-03-31 Dae San Kim , Taekyun Kim

In this paper we study Appell polynomials by connecting them to random variables. This probabilistic approach yields, e.g., the mean value property which is fundamental in the sense that many other properties can be derived from it. We also…

Probability · Mathematics 2013-11-21 Bao Quoc Ta