Degenerate binomial and degenerate Poisson random variables

TL;DR

This paper explores the properties of degenerate binomial and Poisson random variables, linking them to Lah-Bell polynomials and their degenerate counterparts, revealing new connections between these polynomials and moments of these variables.

Contribution

It introduces the degenerate Lah-Bell polynomials and establishes their relationship with moments and generating functions of degenerate Poisson and binomial variables.

Findings

Rising factorial moments of degenerate Poisson variables are given by degenerate Lah-Bell polynomials.

The probability-generating function of degenerate Poisson variables equals the generating function of degenerate Lah-Bell polynomials.

Similar results are shown for Poisson random variables with Lah-Bell polynomials.

Abstract

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 {\alpha} are given by the degenerate Lah-Bell polynomials evaluated at {\alpha}. 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 similar results for the Poisson random variables. Here the nth Lah-Bell number counts the number of ways a set of n elements can be partitioned into non-empty linearly ordered subsets, the Lah-Bell polynomials are natural extensions of the Lah-Bell numbers and the degenerate Lah-Bell polynomials are degenerate versions of the Lah-Bell polynomials.