Degenerate Versions Research Articles

Some identities on degenerate harmonic and degenerate higher-order harmonic numbers

The harmonic numbers and higher-order harmonic numbers appear frequently in several areas which are related to combinatorial identities, many expressions involving special functions in analytic number theory, and analysis of algorithms. The aim of this paper is to study the degenerate harmonic and degenerate higher-order harmonic numbers, which are respectively degenerate versions of the harmonic and higher-order harmonic numbers, in connection with the degenerate zeta and degenerate Hurwitz zeta function. Here the degenerate zeta and degenerate Hurwitz zeta function are respectively degenerate versions of the Riemann zeta and Hurwitz zeta function. We show that several infinite sums involving the degenerate higher-order harmonic numbers can be expressed in terms of the degenerate zeta function. Furthermore, we demonstrate that an infinite sum involving finite sums of products of the degenerate harmonic numbers can be represented by using the degenerate Hurwitz zeta function.

Probabilistic degenerate Stirling polynomials of the second kind and their applications

ABSTRACT The aim of this paper is to further study probabilistic versions of the degenerate Stirling polynomials of the second kind, namely the probabilistic degenerate Stirling polynomials of the second kind associated with Y , which are also degenerate versions of the probabilistic Stirling polynomials of the second kind investigated earlier. Here Y is a random variable whose moment generating function exists in some neighbourhood of the origin. We derive some properties, explicit expressions and certain identities for those polynomials. In addition, we apply our results to the Bernoulli random variable with probability of success p , the normal random variable with parameters μ = 0 , σ 2 = 1 , and to the uniform random variable on ( 0 , 1 ) .

Study on discrete degenerate Bell distributions with two parameters

Abstract Recently, Freud and Rodriguez proposed a new counting process which is called the Bell–Touchard process and based on the Bell–Touchard probability distribution. This process was developed to solve the problem of rare events hypothesis which is one of the limitations of the Poisson process. In this paper, we consider the discrete degenerate Bell distributions and the degenerate Bell process which are “degenerate versions” of the Bell–Touchard probability distributions and the Bell–Touchard process, respectively. We investigate several properties of the degenerate Bell distribution. We introduce the degenerate Bell process by giving two equivalent definitions and show one method of constructing a new infinite family of degenerate Bell process out of a given infinite family of degenerate Bell process.

Some identities of degenerate harmonic and degenerate hyperharmonic numbers arising from umbral calculus

Abstract Hyperharmonic numbers were introduced by Conway and Guy (The Book of Numbers, Copernicus, New York, 1996), whereas harmonic numbers have been studied since antiquity. Recently, the degenerate hyperharmonic and degenerate harmonic numbers were introduced as their respective degenerate versions. The aim of this article is to further investigate some properties and identities involving the degenerate hyperharmonic numbers and degenerate harmonic numbers. Especially, we derive some identities by making use of the transfer formula for associated sequences from umbral calculus.

Generalized degenerate Stirling numbers arising from degenerate Boson normal ordering

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 -Stirling numbers of the second and their natural extensions to polynomials, namely the generalized degenerate -Bell polynomials, arising from certain ‘degenerate boson normal ordering.’ We derive some properties, explicit expressions and generating functions for those numbers and polynomials. The generalized degenerate -Stirling numbers of the second and the degenerate boson normal ordering are respectively degenerate versions of the generalized -Stirling numbers of the second and the boson normal ordering studied earlier by Blasiak-Person-Solomon.

A note on infinite series whose terms involve truncated degenerate exponentials

The degenerate exponentials are degenerate versions of the ordinary exponential and the truncated degenerate exponentials are obtained from the Taylor expansions of them by truncating the first finitely many terms. The degenerate exponentials play an important role in recent studies 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 of this note is to consider infinite series whose terms involve truncated degenerate exponentials together with binomial coefficients, the generalized falling factorials and the Stirling numbers of the second kind, and to find either their values or some other expressions of them as finite sums.

Some identities related to degenerate r-Bell and degenerate Fubini polynomials

Many works have been done in recent years as to explorations for degenerate versions of some special polynomials and numbers, which began with the pioneering work of Carlitz on the degenerate Bernoulli and degenerate Euler polynomials. This paper is focused on the study of some properties, recurrence relations and identities related to the degenerate r-Bell polynomials, the two variable degenerate Fubini polynomials and the degenerate r-Stirling numbers of the second kind. Especially, we express the power series in terms of the degenerate r-Bell polynomials, of the degenerate r-Stirling numbers of the second kind and of the degenerate Fubini polynomials.

Combinatorial identities involving degenerate harmonic and hyperharmonic numbers

In recent years, some degenerate versions of quite a few special numbers and polynomials are introduced and investigated by means of various methods. The aim of this paper is to study some results on degenerate harmonic numbers, degenerate hyperharmonic numbers, degenerate Fubi polynomials and degenerate r-Fubini polynomials from a general identity which is valid for any two formal power series and involves the degenerate r-Stirling numbers of the second kind.

Some identities of degenerate higher-order Daehee polynomials based on $ \lambda $-umbral calculus

<abstract><p>The degenerate versions of special polynomials and numbers, initiated by Carlitz, have regained the attention of some mathematicians by replacing the usual exponential function in the generating function of special polynomials with the degenerate exponential function. To study the relations between degenerate special polynomials, $ \lambda $-umbral calculus, an analogue of umbral calculus, is intensively applied to obtain related formulas for expressing one $ \lambda $-Sheffer polynomial in terms of other $ \lambda $-Sheffer polynomials. In this paper, we study the connection between degenerate higher-order Daehee polynomials and other degenerate type of special polynomials. We present explicit formulas for representations of the polynomials using $ \lambda $-umbral calculus and confirm the presented formulas between the degenerate higher-order Daehee polynomials and the degenerate Bernoulli polynomials, for example. Additionally, we investigate the pattern of the root distribution of these polynomials.</p></abstract>

Some identities on degenerate hyperbolic functions arising from $ p $-adic integrals on $ \mathbb{Z}_p $

<abstract><p>The aim of this paper is to introduce several degenerate hyperbolic functions as degenerate versions of the hyperbolic functions, to evaluate Volkenborn and the fermionic $ p $-adic integrals of the degenerate hyperbolic cosine and the degenerate hyperbolic sine functions and to derive from them some identities involving the degenerate Bernoulli numbers, the degenerate Euler numbers and the Cauchy numbers of the first kind.</p></abstract>

Some identities of degenerate multi-poly-Changhee polynomials and numbers

<abstract><p>Recently, many researchers studied the degenerate multi-special polynomials as degenerate versions of the multi-special polynomials and obtained some identities and properties of the those polynomials. The aim of this paper was to introduce the degenerate multi-poly-Changhee polynomials arising from multiple logarithms and investigate some interesting identities and properties of these polynomials that determine the relationship between multi-poly-Changhee polynomials, the Stirling numbers of the second kind, degenerate Stirling numbers of the first kind and falling factorial sequences. In addition, we investigated the phenomenon of scattering the zeros of these polynomials.</p></abstract>

Degenerate $ r $-truncated Stirling numbers

<abstract><p>For any positive integer $ r $, the $ r $-truncated (or $ r $-associated) Stirling number of the second kind $ S_{2}^{(r)}(n, k) $ enumerates the number of partitions of the set $ \{1, 2, 3, \dots, n\} $ into $ k $ non-empty disjoint subsets, such that each subset contains at least $ r $ elements. We introduce the degenerate $ r $-truncated Stirling numbers of the second kind and of the first kind. They are degenerate versions of the $ r $-truncated Stirling numbers of the second kind and of the first kind, and reduce to the degenerate Stirling numbers of the second kind and of the first kind for $ r = 1 $. Our aim is to derive recurrence relations for both of those numbers.</p></abstract>

Study on r-truncated degenerate Stirling numbers of the second kind

Abstract 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 versions of some special numbers and polynomials. The aim of this article is to consider the r-truncated degenerate Stirling numbers of the second kind, which reduce to the degenerate Stirling numbers of the second for r = 1, and to investigate their explicit expressions, some properties and related identities, in connection with several other degenerate special numbers and polynomials.

Some Identities on Degenerate $$r$$-Stirling Numbers via Boson Operators

Broder introduced the r-Stirling numbers of the first kind and of the second kind which enumerate restricted permutations and respectively restricted partitions, the restriction being that the first r elements must be in distinct cycles and respectively in distinct subsets. Kim-Kim-Lee-Park constructed the degenerate r-Stirling numbers of both kinds as degenerate versions of them. The aim of this paper is to derive some identities and recurrence relations for the degenerate r-Stirling numbers of the first kind and of the second kind via boson operators. In particular, we obtain the normal ordering of a degenerate integral power of the number operator multiplied by an integral power of the creation boson operator in terms of boson operators where the degenerate r-Stirling numbers of the second kind appear as the coefficients.

SOME IDENTITIES OF FULLY DEGENERATE DOWLING AND FULLY DEGENERATE BELL POLYNOMIALS ARISING FROM λ-UMBRAL CALCULUS

Recently, Kim–Kim introduced the [Formula: see text]-umbral calculus, in which the [Formula: see text]-Sheffer sequences occupy the central position. In this paper, we introduce the fully degenerate Bell and the fully degenerate Dowling polynomials, and investigate some properties and identities relating to those polynomials with the help of [Formula: see text]-umbral calculus. Here, we note that the fully degenerate Bell poynomials and the fully degenerate Dowling polynomials are, respectively, degenerate versions of the Bell polynomials and the Dowling polynomials, of which the latters are the natural extension of the Whitney numbers of the second kind.

Some identities related to degenerate Stirling numbers of the second kind

Abstract The degenerate Stirling numbers of the second kind were introduced as a degenerate version of the ordinary Stirling numbers of the second kind. They appear very frequently when one studies various degenerate versions of some special numbers and polynomials. The aim of this article is to further study some identities and properties related to the degenerate Stirling numbers of the second kind, in connection with the degenerate Bell polynomials, the degenerate Fubini polynomials, the degenerate Bernoulli polynomials, and the degenerate Euler polynomials.

Poisson degenerate central moments related to degenerate Dowling and degenerate r-Dowling polynomials

ABSTRACT Degenerate Dowling and degenerate r-Dowling polynomials were introduced earlier as degenerate versions and further generalizations of Dowling and r-Dowling polynomials. The aim of this paper is to show their connections with Poisson degenerate central moments for a Poisson random variable with a certain parameter and with Charlier polynomials.

Degenerate r-Whitney numbers and degenerate r-Dowling polynomials via boson operators

Dowling showed that the Whitney numbers of the first kind and of the second kind satisfy Stirling number-like relations. Recently, Kim-Kim introduced the degenerate r-Whitney numbers of the first kind and of the second kind, as degenerate versions and further generalizations of the Whitney numbers of both kinds. 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. In this paper, it is noted that the normal ordering of a certain quantity involving the number operator is expressed in terms of the degenerate r-Whitney numbers of the second kind. We derive some properties, recurrence relations, orthogonality relations and several identities on those numbers from such normal ordering. In addition, we consider the degenerate r-Dowling polynomials as a natural extension of the degenerate r-Whitney numbers of the second kind and investigate their properties.

Representations of degenerate Hermite polynomials

The study on degenerate versions of some special numbers and polynomials, which began with Carlitz's pioneering work, has regained recent interests of some mathematicians. Motivated by this, we introduce degenerate Hermite polynomials as a degenerate version of the ordinary Hermite polynomials. Recently, introduced was λ-umbral calculus where the usual exponential function appearing in the generating function of Sheffer sequence is replaced by the degenerate exponential function. Then, among other things, by using the formula of the λ-umbral calculus about expressing one λ-Sheffer polynomial in terms of another λ-Sheffer polynomials we represent the degenerate Hermite polynomials in terms of the higher-order degenerate Bernoulli, Euler, and Frobenius-Euler polynomials and vice versa.

A new approach to Bell and poly-Bell numbers and polynomials

<abstract><p>Bell polynomials are widely applied in many problems arising from physics and engineering. The aim of this paper is to introduce new types of special polynomials and numbers, namely 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 those polynomials and numbers. We also consider degenerate versions of those polynomials and numbers, namely degenerate Bell polynomials and numbers of the second kind and degenerate poly-Bell polynomials and numbers of the second kind, and deduce their similar results.</p></abstract>