Some identities involving derangement polynomials and numbers
Taekyun Kim, Dae San Kim, Lee-Chae Jang, Hyunseok Lee
TL;DR
This paper explores derangement polynomials and numbers, their relationships with cosine and sine derangement polynomials, and their applications to moments of gamma-related random variables.
Contribution
It introduces new connections between derangement polynomials and trigonometric variants, and applies these to probabilistic moments.
Findings
Derangement polynomials relate to cosine and sine derangement polynomials.
Connections established between derangement polynomials and gamma distribution moments.
New identities involving derangement numbers and polynomials derived.
Abstract
The problem of counting derangements was initiated by Pierre Remonde de Motmort in 1708. A derangement is a permutation that has no fixed points and the derangement number Dn is the number of fixed point free permutations on an n element set. Furthermore, the derangement polynomials are natural extensions of the derangement numbers. In this paper, we study the derangement polynomials and numbers, their connections with cosine-derangement polynomials and sine-derangement polynomials and their applications to moments of some variants of gamma random variables.
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Taxonomy
TopicsMathematical functions and polynomials · Advanced Mathematical Identities · Bayesian Methods and Mixture Models