Probabilistic derangement numbers and polynomials

Taekyun Kim; Dae San Kim

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arXiv · 2401.03986

Probabilistic derangement numbers and polynomials

Taekyun Kim·Dae San Kim

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Abstract

Let Y be a random variable such that the moment generating function of Y exists in a neighborhood of the origin. The aim of this paper is to study probabilistic versions of the derangement polynomials, the derangement polynomials of type 2 and the r-derangement numbers, namely the probabilistic derangement polynomials associated with Y, the probabilistic derangement polynomials of type 2 associated with Y and the probabilistic r-derangement numbers associated Y, respectively. We derive some properties, explicit expressions, certain identities and recurrence relations for those polynomials and numbers. In addition, we consider the special case that Y is the gamma random variable with parameters {\alpha},\b{eta} > 0.

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Taekyun Kim, Dae San Kim. 2024-01-08. Probabilistic derangement numbers and polynomials. https://arxiv.org/abs/2401.03986

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