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arXiv · 2008.03769

Degenerate binomial and degenerate Poisson random variables

Dae San Kim·Taekyun Kim
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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.

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Dae San Kim, Taekyun Kim. 2020-08-09. Degenerate binomial and degenerate Poisson random variables. https://arxiv.org/abs/2008.03769

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