Degenerate Stirling Numbers Research Articles

Type 2 degenerate modified poly-Bernoulli polynomials arising from the degenerate poly-exponential functions

<abstract><p>We present a new type of degenerate poly-Bernoulli polynomials and numbers by modifying the polyexponential function in terms of the degenerate exponential functions and degenerate logarithm functions. Also, we introduce a new variation of the degenerate unipoly-Bernoulli polynomials by the similar modification. Based on these polynomials, we investigate some properties, new identities, and their relations to the known special functions and numbers such as the degenerate type 2-Bernoulli polynomials, the type 2 degenerate Euler polynomials, the degenerate Bernoulli polynomials and numbers, the degenerate Stirling numbers of the first kind, and $ \lambda $-falling factorial sequence. In addition, we compute some of the proposed polynomials and present their zeros and behaviors for different variables in specific cases.</p></abstract>

New Properties on Degenerate Bell Polynomials

The aim of this paper is to study the degenerate Bell numbers and polynomials which are degenerate versions of the Bell numbers and polynomials. We derive some new identities and properties of those numbers and polynomials that are associated with the degenerate Stirling numbers of both kinds.

Degenerate Derangement Polynomials and Numbers

In this paper, we consider a new type of degenerate derangement polynomial and number, which shall be called the degenerate derangement polynomials and numbers of the second kind. These concepts are motivated by Kim et al.’s work on degenerate derangement polynomials and numbers. We investigate some properties of these new degenerate derangement polynomials and numbers and explore their connections with the degenerate gamma distributions for the case λ∈(−1,0). In more detail, we derive their explicit expressions, recurrence relations, and some identities involving our degenerate derangement polynomials and numbers and other special polynomials and numbers, which include the fully degenerate Bell polynomials, the degenerate Fubini polynomials, and the degenerate Stirling numbers of the first and the second kinds. We also show that those polynomials and numbers are connected with the moments of some variants of the degenerate gamma distributions. Moreover, we compare the degenerate derangement polynomials and numbers of the second kind to those of Kim et al.

Generalized degenerate Bernoulli numbers and polynomials arising from Gauss hypergeometric function

A new family of p-Bernoulli numbers and polynomials was introduced by Rahmani (J. Number Theory 157:350–366, 2015) with the help of the Gauss hypergeometric function. Motivated by that paper and in the light of the recent interests in finding degenerate versions, we construct the generalized degenerate Bernoulli numbers and polynomials by means of the Gauss hypergeometric function. In addition, we construct the degenerate type Eulerian numbers as a degenerate version of Eulerian numbers. For the generalized degenerate Bernoulli numbers, we express them in terms of the degenerate Stirling numbers of the second kind, of the degenerate type Eulerian numbers, of the degenerate p-Stirling numbers of the second kind and of an integral on the unit interval. As to the generalized degenerate Bernoulli polynomials, we represent them in terms of the degenerate Stirling polynomials of the second kind.

A note on degenerate derangement polynomials and numbers

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 explicit expressions, recurrence relations and some identities involving the degenerate derangement polynomials and numbers and other special polynomials and numbers, which include the fully degenerate Bell polynomials, the degenerate Fubini polynomials and the degenerate Stirling numbers of both kinds. We also show that those polynomials and numbers are connected with the moments of some variants of the degenerate gamma distributions.

Note on the Type 2 Degenerate Multi-Poly-Euler Polynomials

Kim and Kim (Russ. J. Math. Phys. 26, 2019, 40-49) introduced polyexponential function as an inverse to the polylogarithm function and by this, constructed a new type poly-Bernoulli polynomials. Recently, by using the polyexponential function, a number of generalizations of some polynomials and numbers have been presented and investigated. Motivated by these researches, in this paper, multi-poly-Euler polynomials are considered utilizing the degenerate multiple polyexponential functions and then, their properties and relations are investigated and studied. That the type 2 degenerate multi-poly-Euler polynomials equal a linear combination of the degenerate Euler polynomials of higher order and the degenerate Stirling numbers of the first kind is proved. Moreover, an addition formula and a derivative formula are derived. Furthermore, in a special case, a correlation between the type 2 degenerate multi-poly-Euler polynomials and degenerate Whitney numbers is shown.

Degenerate poly-Bernoulli polynomials arising from degenerate polylogarithm

Recently, degenerate polylogarithm functions were introduced by Kim and Kim. In this paper, we introduce degenerate poly-Bernoulli polynomials by means of the degenerate polylogarithm functions and investigate some their properties. In more detail, we find certain explicit expressions for those polynomials in terms of the Carlitz degenerate Bernoulli polynomials and the degenerate Stirling numbers of the second kind. Furthermore, we obtain some expressions for differences of the degenerate poly-Bernoulli polynomials.

New type of degenerate Daehee polynomials of the second kind

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.

A note on degenerate poly-Genocchi numbers and polynomials

Recently, some mathematicians have been studying a lot of degenerate versions of special polynomials and numbers in some arithmetic and combinatorial aspects. Our research is also interested in this field. In this paper, we introduce a new type of the degenerate poly-Genocchi polynomials and numbers, based on Kim and Kim’s (J. Math. Anal. Appl. 487(2):124017, 2020) modified polyexponential function. The paper is divided into two parts. In Sect. 2, we consider a new type of the degenerate poly-Genocchi polynomials and numbers constructed from the modified polyexponential function. We also show several combinatorial identities related to the degenerate poly-Genocchi polynomials and numbers. Some of them include the degenerate and other special polynomials and numbers such as the Stirling numbers of the first kind, the degenerate Stirling numbers of the second kind, degenerate Euler polynomials, degenerate Bernoulli polynomials and Bernoulli numbers of order α, etc. In Sect. 3, we also introduce the degenerate unipoly Genocchi polynomials attached to an arithmetic function by using the degenerate polylogarithm function. We give some new explicit expressions and identities related to degenerate unipoly Genocchi polynomials and special numbers and polynomials.

Some Relations of Two Type 2 Polynomials and Discrete Harmonic Numbers and 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.

Jindalrae and Gaenari numbers and polynomials in connection with Jindalrae\u2013Stirling numbers

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 second kinds as extensions of the notions of the degenerate Stirling numbers of the first and second kinds, and deduce several relations involving those special numbers. Then we introduce Jindalrae and Gaenari numbers and polynomials and obtain some explicit expressions and identities associated with those numbers and polynomials. In addition, we interpret our results by using umbral calculus.

Some Identities on Type 2 Degenerate Bernoulli Polynomials of the Second Kind

In recent years, many mathematicians studied various degenerate versions of some special polynomials for which quite a few interesting results were discovered. In this paper, we introduce the type 2 degenerate Bernoulli polynomials of the second kind and their higher-order analogues, and study some identities and expressions for these polynomials. Specifically, we obtain a relation between the type 2 degenerate Bernoulli polynomials of the second and the degenerate Bernoulli polynomials of the second, an identity involving higher-order analogues of those polynomials and the degenerate Stirling numbers of the second kind, and an expression of higher-order analogues of those polynomials in terms of the higher-order type 2 degenerate Bernoulli polynomials and the degenerate Stirling numbers of the first kind.

CLOSED FORMS FOR DEGENERATE BERNOULLI POLYNOMIALS

Qi and Chapman [‘Two closed forms for the Bernoulli polynomials’, J. Number Theory159 (2016), 89–100] gave a closed form expression for the Bernoulli polynomials as polynomials with coefficients involving Stirling numbers of the second kind. We extend the formula to the degenerate Bernoulli polynomials, replacing the Stirling numbers by degenerate Stirling numbers of the second kind.

Some Identities of Degenerate Bell Polynomials

The new type degenerate of Bell polynomials and numbers were recently introduced, which are a degenerate version of Bell polynomials and numbers and are different from the previously introduced partially degenerate Bell polynomials and numbers. Several expressions and identities on those polynomials and numbers were obtained. In this paper, as a further investigation of the new type degenerate Bell polynomials, we derive several identities involving those degenerate Bell polynomials, Stirling numbers of the second kind and Carlitz’s degenerate Bernoulli or degenerate Euler polynomials. In addition, we obtain an identity connecting the degenerate Bell polynomials, Cauchy polynomials, Bernoulli numbers, Stirling numbers of the second kind and degenerate Stirling numbers of the second kind.

Ordinary and degenerate Euler numbers and polynomials

In this paper, we study some identities on Euler numbers and polynomials, and those on degenerate Euler numbers and polynomials which are derived from the fermionic p-adic integrals on mathbb{Z}_{p}. Specifically, we obtain a recursive formula for alternating integer power sums and representations of alternating integer power sum polynomials in terms of Euler polynomials and Stirling numbers of the second kind, as well as various properties about Euler numbers and polynomials. In addition, we deduce representations of degenerate alternating integer power sum polynomials in terms of degenerate Euler polynomials and degenerate Stirling numbers of the second kind, as well as certain properties on degenerate Euler numbers and polynomials.

Degenerate central factorial numbers of the second kind

In this paper, we introduce the degenerate central factorial polynomials and numbers of the second kind which are degenerate versions of the central factorial polynomials and numbers of the second kind. We derive some properties and identities for those polynomials and numbers. We obtain, among other things, recursive formulas for the degenerate central factorial polynomials and numbers of the second kind. Recently, Dolgy and Kim (Proc Jangjeon Math Soc 21(2):309–317, 2018) obtained some explicit formulas of degenerate Stirling numbers associated with the degenerate special numbers and polynomials. This paper motivated our to do this research.

Some identities involving special numbers and moments of random variables

In this paper, we derive some identities involving special numbers and moments of random variables by using the generating functions of the moments of certain random variables. Here, the related special numbers are Stirling numbers of the first and second kinds, degenerate Stirling numbers of the first and second kinds, derangement numbers, higher-order Bernoulli numbers and Bernoulli numbers of the second kind.

Explicit relations for the modified degenerate Apostol-type polynomials

Recently, the degenerate Bernoulli numbers and polynomials and the degenerate Euler numbers and polynomials have been studied by several authors. In this paper, we consider the modified Apostol-Bernoulli polynomials and the modified Apostol-Euler polynomials. We give explicit relation for the modified degenerate Bernoulli polynomials and the modified degenerate Euler polynomials. Also, we prove some identities between the modified Apostol-Bernoulli polynomials and the degenerate Stirling numbers of two kinds.

Degenerate Bernstein polynomials

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 their applications’ (Simsek in Filomat 30(7):1683–1689, 2016, Math Methods Appl Sci 1–12, 2018) and Carlitz’s degenerate Bernoulli polynomials. We derived their generating function, symmetric identities, recurrence relations, and some connections with generalized falling factorial polynomials, higher-order degenerate Bernoulli polynomials and degenerate Stirling numbers of the second kind.

Generalizations of poly-Bernoulli and poly-Cauchy numbers

In this paper we consider some generalizations of poly-Bernoulli and poly-Cauchy numbers. The first is by means of the Hurwitz–Lerch zeta function. The second generalization is via weighted Stirling numbers. The third one is given with the help of degenerate Stirling numbers. All these generalizations lead to symmetries between various types of Stirling numbers, and enable us to investigate and expand algebraic properties of poly-Bernoulli and poly-Cauchy numbers. We also combine these generalizations and derive numerous combinatorial and arithmetical identities.