r-extended Lah-Bell numbers and polynomials associated with r-Lah numbers

Lah-Bell numbers and polynomials

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.

Degenerate binomial and degenerate Poisson random variables

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…

Complete and incomplete Bell polynomials associated with Lah-Bell numbers and polynomials

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-…

概率Lah数与Lah-Bell多项式

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。

Lah-Bell多项式的Spivey型递推关系

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。

概率异质斯特林数和贝尔多项式

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。