Probabilistic derangement numbers and polynomials
arxiv(2024)
摘要
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 α,e̱ṯa̱ > 0.
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