Dae San Kim, Taekyun Kim
In this paper, we introduce the Lah-Bell numbers and their natural extensions, namely the Lah-Bell polynomials, and derive some basic properties of such numbers and polynomials by using elementary methods. In addition, we consider the degenerate Lah-Bell numbers and polynomials as degenerate versions of the Lah-Bell numbers and polynomials.
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The aim of this paper is to study some basic properties of the r$-extended Lah-Bell numbers and polynomials associated with r-Lah numbers and to show the connection between the r-extended Lah-Bell polynomials and the rising factorial moments of the Poisson random variable shifted by an even nonnegative integer 2r. Here the nth r-extended Lah-Bell number counts the number of ways a set of n+r elements can be partitioned into non-empty linearly ordered subsets such that r distinguished elements have to be in distinct linearly ordered subsets and the r-extended Lah-Bell polynomials are natural ex…
The aim of this paper is to study the Poisson random variables in relation to the Lah-Bell polynomials and the degenerate binomial and degenerate Poisson random variables in connection with the degenerate Lah-Bell polynomials. Among other things, we show that the rising factorial moments of the degenerate Poisson random variable with parameter α are given by the degenerate Lah-Bell polynomials evaluated at α. We also show that the probability-generating function of the degenerate Poisson random variable is equal to the generating function of the degenerate Lah-Bell polynomials. Also, we show s…
The nth r-extended Lah-Bell number is defined as the number of ways a set with $n+r$ elements can be partitioned into ordered blocks such that r distinguished elements have to be in distinct ordered blocks. The aim of this paper is to introduce incomplete r-extended Lah-Bell polynomials and complete $r$-extended Lah-Bell polynomials respectively as multivariate versions of $r$-Lah numbers and the r-extended Lah-Bell numbers and to investigate some properties and identities for these polynomials. From these investigations, we obtain some expressions for the r-Lah numbers and the r-extended Lah-…
Probabilistic Lah numbers and Lah-Bell polynomials
本文研究了矩母函数存在的随机变量Y所对应的概率Lah数与概率Lah-Bell多项式，推导了其性质、显式表达式、递推关系和恒等式，并探讨了Poisson与Bernoulli随机变量的特殊情形。
Spivey's type recurrence relation for Lah-Bell polynomials
本文利用满足交换关系DX-XD=1的算子X和D推导了Lah-Bell多项式及r-Lah-Bell多项式的Spivey型递推关系，并用其他方法得到了r-Lah-Bell多项式的λ模拟的相应递推关系。
In this paper, we take advantage of the Mellin type derivative to produce some new families of polynomials whose coefficients involve r-Lah numbers. One of these polynomials leads to rediscover many of the identities of r-Lah numbers. We show that some of these polynomials and hyperharmonic numbers are closely related. Taking into account of these connections, we reach several identities for harmonic and hyperharmonic numbers.
In this article we point out interesting connections among Lah numbers, Laguerre polynomials of order negative one, and exponential polynomials. We also discuss several different expressions for the higher order derivatives of exp (1/x). A new representation of these derivatives is given in terms of exponential polynomials
Probabilistic heterogeneous Stirling numbers and Bell polynomials
本文提出概率异质斯特林数和贝尔多项式，统一了多个经典和概率性的数学家族，并推导了显式公式和递推关系，应用于泊松和伯努利分布。
The aim of this paper is to introduce the degenerate generalized Laguerre polynomials as the degenerate version of the generalized Laguerre polynomials and to derive some properties related to those polynomials and Lah numbers, including an explicit expression, a Rodrigues' type formula and expressions for the derivatives. The novelty of the present paper is that it is the first paper on degenerate versions of orthogonal polynomials.
On a class of polynomials connected to Bell polynomials
本文研究一类与Bell多项式相关的多项式序列，探讨其显式表达式、导数关系、实零点及递推关系，并简化了若干已知结果。
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 using several different tools including generating functions, combinatorial methods, $p$-adic analysis, umbral calculus, special functions, differential equations and probability theory as well. \par The degenerate Poisson random variables are degenerate versions of the Poisson random variables. In [6], studied are the degenerate binomial and degenerate Poi…
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 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.
In this paper, we consider the problem of representing any polynomial in terms of the ordered Bell and degenerate ordered Bell polynomials, and more generally of the higher-order ordered Bell and higher-order degenerate ordered Bell polynomials. We derive explicit formulas with the help of umbral calculus and illustrate our results with some examples.
In this paper, we consider central complete and incomplete Bell polynomials which are generalizations of the recently introduced central Bell polynomials and central analogues for the complete and incomplete Bell polynomials. We investigate some properties and identities for these polynomials. Especially, we give explicit formulas for the central complete and incomplete Bell polynomials related to central factorial numbers of the second kind.
In this paper, we study the degenerate central complete and incomplete Bell polynomials which are degenerate versions of the recently introduced central complete and incomplete Bell polynomials and also central analogues for the degenerate complete and incomplete Bell polynomials. We investigate some properties and identities for these polynomials.
Recently, several authors have studied the Stirling numbers of the second kind and Bell polynomials. In this paper, we study the extended Stirling polynomials of the second kind and the extended Bell polynomials associated with the Stirling numbers of the second kind. In addition, we note that the extended Bell polynomials can be expressed in terms of the moments of the Poisson random variable with parameter
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 polynomials associated with special polynomials.
Recently, several authors have studied the degenerate Bernoulli and Euler polynomials and given some intersting identities of those polynomials. In this paper, we consider the degenerate Bell numbers and polynomials and derive some new identities of those numbers and polynomials associated with special numbers and polynomials. In addition, we investigate some properties of the degenerate Bell polynomials which are derived by using the notion of composita. From our investigation, we give some new relations between the degenerate Bell polynomials and the special polynomials.
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 and Charlier polynomials.
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 identities by making use of series transformation formula obtained by Boyadzhiev. In particular, this answers to the natural question about the relationship between two different types of degenerate Bell polynomials.
Probabilistic generalization of Spivey-type relation for degenerate Bell polynomials
本文在Spivey发现贝尔数递推关系及Kim和Kim建立退化贝尔与道林多项式Spivey型关系的基础上，推导了退化贝尔多项式和退化r-贝尔多项式的Spivey型递推关系的概率推广。
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 (generating functions, explicit formulas, integral representations, recurrence relations, probabilistic representation,...). We also derive some combinatorial sums including the generalized Bernoulli polynomials, lower incomplete gamma function, generalized Bell polynomials. Finally, by applying Cauchy formula for repeated integration, we introduce poly-Bell numbe…
The relations between the Bernoulli and Eulerian polynomials of higher order and the complete Bell polynomials are found that lead to new identities for the Bernoulli and Eulerian polynomials and numbers of higher order. General form of these identities is considered and generating function for polynomials satisfying this general identity is found.
The aim of this paper is to study the degenerate Bell numbers and polynomials which are degenerate version 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 the both kinds.
For finite sums of non-negative powers of arithmetic progressions the generating functions (ordinary and exponential ones) for given powers are computed. This leads to a two parameter generalization of Stirling and Eulerian numbers. A direct generalization of Bernoulli numbers and their polynomials follows. On the way to find the Faulhaber formula for these sums of powers in terms of generalized Bernoulli polynomials one is led to a one parameter generalization of Bernoulli numbers and their polynomials. Generalized Lah numbers are also considered.
Spivey-type recurrence relation for fully degenerate Bell polynomials
本文基于第二类退化Stirling数与a阶双变量退化Fubini多项式，推导了完全退化Bell多项式和退化Fubini多项式的Spivey型递推关系，拓展了Spivey关于Bell数恒等式的相关研究。
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 several expressions for the generating function of the sum of the values of the generalized falling factorials at positive consecutive integers.
In a recent work, the degenerate Stirling polynomials of the second kind were studied by T. Kim. In this paper, we investigate the extended degenerate Stirling numbers of the second kind and the extended degenerate Bell polynomials associated with them. As results, we give some expressions, identities and properties about the extended degener- ate Stirling numbers of the second kind and the extended degenerate Bell polynomials.
Recently, Kim-Kim introduced the lambda-umbral calculus, in which the lambda-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 oflambda-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…
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, as a degenerate version of polyexponential functions introduced by Hardy, we study degenerate polyexponential functions and derive various properties of them. Also, we introduce new type degenerate Bell polynomials, which are different from the previous studied partially degenerate Bell polynomials and arise naturally in the recent study of degenerate zero-t…
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 representations, Dobinski-like formula and expressions of the generating function in terms of differential operators and linear incomplete gamma function. In addition, we introduce truncated degenerate modified Bell polynomials and numbers and get similar results for those polynomials. As an application of our results, we show that the truncated degenerate Bel…
Many authors studied mix type polynomials and their properties. For their numerous uses in Statistics, Combinatorial analysis and Number theory. In the present article, we define a generating function of mix type Bell based Apostol type polynomials and numbers of order a. Furthermore, we derive some elementary properties and identities of Bell based Apostol type polynomials of order a such as correlation formulas, Implicit summation formulas, derivative formulas and their special cases. Also, we derive some relation with Stirling numbers.
In this paper, we introduce the extended r-central factorial numbers of the second and first kinds and the extended r-central Bell polynomials, as extended versions and central analogues of some previously introduced numbers and polynomials. Then we study various properties and identities related to these numbers and polynomials and also their connections.
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 boson operators. In particular, we obtain two expressions for the generating function of the degenerate r-Bell polynomials in |z| , and a recurrence relation and Dobinski-like formula for the degenerate r-Bell numbers. These are derived from the degenerate normal ordering of a degenerate integral power of the number operator in terms of boson operators where the…
Many works have been done in recent years as to explorations for degenerate versions of some special polynomials and numbers, which began with the introduction of the degenerate Bernoulli and degenerate Euler polynomials by Carlitz. The aim of this paper is to study 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.
Probabilistic central Bell polynomials
本文将中心 Bell 多项式推广到随机变量情形，研究相关概率中心 Bell 多项式、第二类概率中心阶乘数和概率中心 Fubini 多项式，并给出其性质、显式表达式、恒等式与递推关系。
Asymptotic for the rightmost zeros of Bell and Eulerian polynomials
本文证明了 Bell 多项式最右负零点的渐近公式，并将方法推广至 Eulerian 多项式等族，得到类似结果。
Spivey's type recurrence relation for degenerate Bell polynomials
本文利用满足对易关系 DX−XD=1 的算子 X 与 D，推导退化 Bell 多项式的 Spivey 型递推关系，并进一步得到退化 r-Bell 多项式的递推关系。
Probabilistic degenerate r-Stirling numbers of the second and probabilistic degenerate r-Bell polynomials
该研究针对矩生成函数在原点邻域存在的随机变量Y，提出其对应的概率退化第二类r-Stirling数与概率退化r-Bell多项式，作为已有同类概念的概率扩展，并通过生成函数推导其性质、显式表达式与相关恒等式。
In this paper, we investigate the integral of $x^n\log^m(\sin(x))$ for natural numbers $m$ and $n$. In doing so, we recover some well-known results and remark on some relations to the log-sine integral $\operatorname{Ls}_{n+m+1}^{(n)}(θ)$. Later, we use properties of Bell polynomials to find a closed expression for the derivative of the central binomial and shifted central binomial coefficients in terms of polygamma functions and harmonic numbers.
Bell Transforms of Arithmetic Functions: Euler Products, Congruences, and Polynomial Sequences
本文提出一个统一的代数框架，利用形式贝尔变换将算术函数的狄利克雷卷积与无限欧拉型乘积的组合结构联系起来，通过分析指数生成函数的对数导数，建立了贝尔指数与莫比乌斯逆运算之间的显式映射，并应用于推导经典序列的精确消失性质和同余继承性质，包括拉马努金的tau函数和素数着色分区。此外，证明了逆贝尔变换能够无缝恢复经典分区递推关系，并提供了一个离散的组合引擎来生成特殊多项式族，包括经典的Appell和Sheffer序列。
Central factorial numbers associated with sequences of polynomials
本文借助umbral calculus技术，统一研究了与任意多项式序列P关联的第一、二类中心阶乘数，揭示了其正交性与反演关系，并通过实例验证。
Generalized Bell polynomial operators arising from generalized normal ordering
本文研究广义Heisenberg代数中的变形组合结构，通过真空本征值定义量子算子阶乘和广义量子指数函数，构造广义相干态并证明量子算子幂展开为广义下降阶乘，相干态期望值由广义Bell多项式算子显式给出。
Probabilistic bivariate Bell polynomials
本文研究了矩母函数存在的随机变量相关的概率二元Bell多项式与概率二元r-Bell多项式，并推导出对应于Zheng和Li所得递推关系的递推关系。
Our paper deals about identities involving Bell polynomials. Some identities on Bell polynomials derived using generating function and successive derivatives of binomial type sequences. We give some relations between Bell polynomials and binomial type sequences in first part, and, we generalize the results obtained in [4] in second part.
In this note, by the umbra calculus method, the Sun and Zagier's congruences involving the Bell numbers and the derangement numbers are generalized to the polynomial cases. Some special congruences are also provided.
Concise and elegant proofs of three formulas for complete Bell polynomials
利用完整Bell多项式的生成函数等技术，给出了完整Bell多项式三个公式的简洁优雅证明。
In the present article, we study Bell based Euler polynomial of order α and investigate some useful correlation formula, summation formula and derivative formula. Also, we introduce some relation of string number of the second kind. Moreover, we drive several important formulae of bell based Euler polynomial by using umbral calculus.
In a previous paper, Mihoubi et al. introduced the $(r_{1},...,r_{p}) $-Stirling numbers and the $(r_{1},...,r_{p}) $-Bell polynomials and gave some of their combinatorial and algebraic properties. These numbers and polynomials generalize, respectively, the $r$-Stirling numbers of the second kind introduced by Broder and the $r$-Bell polynomials introduced by Mező. In this paper, we prove that the $(r_{1},...,r_{p}) $-Stirling numbers of the second kind are log-concave. We also give generating functions and generalized recurrences related to the $(r_{1},...,r_{p}) $-Bell polynomials.
We survey a family of polynomials that are very useful in all kinds of power series manipulations, and appearing more frequently in the literature. Applications to formal power series, generating functions and asymptotic expansions are described, and we discuss the related work of De Moivre, Arbogast and Bell.
Recurrence relations for degenerate Bell and Dowling polynomials via Boson operators
本文利用满足特定对易关系的玻色子湮灭和产生算子，导出了退化Bell多项式、退化Dowling多项式以及r-Dowling多项式的Spivey型递推关系。
In this paper special values of Bell polynomials are given by using the power series solution of the equation $y^{(k)}=e^{ay}$. In addition, complete and partial exponential autonomous functions, exponential autonomous polynomials, autonomous polynomials and $(k,a)$-autonomous coefficients are defined. Finally, we show the relationship between various numbers counting combinatorial objects and binomial coefficients, Stirling numbers of the second kind and autonomous coefficients.
In this paper, we present new explicit simultaneous rational approximations converging sub-exponentially to the values of Bell polynomials at the points of the form $(γ, 1! (2a+1)ζ(2), 2!ζ(3),..., (m-1)!(a+1+(-1)^ma)ζ(m)),$ $m=1,2,...,a,$ $a\in{\mathbb N}.$
In this paper, by the generalized Bell umbra and Rolle's theorem, we give some results on the real rootedness of polynomials. Some applications on partition polynomials and the sigma polynomials of graphs are given.
Probabilistic degenerate poly-Bell polynomials associated with random variables
本文研究与随机变量相关的概率退化多贝尔多项式，推导其显式表达式与恒等式，并讨论伯努利和伽马随机变量的特殊情形。
A Generalized Recurrence for fully degenerate Bell polynomials
针对现有全退化Bell多项式的Spivey型递推中不自然出现双变量退化Fubini多项式的问题，本文定义新的全退化Bell多项式及其双变量、r拓展形式，利用算子方法推导自然的Spivey型递推及多种表达式。
In this paper, applying the Faà di Bruno formula and some properties of Bell polynomials, several closed formulas and determinantal expressions involving Stirling numbers of the second kind for higher-order Bernoulli and Euler polynomials are presented.
Moments of partition statistics, Bell polynomials and Eisenstein-type series
本文通过完全Bell多项式及其反演公式和Faà di Bruno公式，发展了一种将组合统计量矩的生成函数表示为划分迹的系统方法，并应用于整数划分的最大部分、划分crank和rank以及单峰序列rank等统计量。
A fast algorithm for computing Bell polynomials based on index break-downs using prime factorization
提出一种基于素数分解的索引拆分快速算法，利用Bell多项式与有理卷积幂的联系，显著减少计算部分Bell多项式的算术运算量。
In the paper, the authors first inductively establish explicit formulas for derivatives of the arc sine function, then derive from these explicit formulas explicit expressions for a family of Bell polynomials related to the square function, and finally apply these explicit expressions to find explicit formulas for derivatives of some elementary functions.
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