Some Identities Involving Derangement Polynomials and Numbers and Moments of Gamma Random Variables

Lee-Chae Jang,Hyunseok Lee,Dae San Kim,Taekyun Kim,Ioan Rasa

doi:10.1155/2020/6624006

Lee-Chae Jang, Hyunseok Lee + Show 3 more

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https://doi.org/10.1155/2020/6624006

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Abstract

The problem of counting derangements was initiated by Pierre Rémond de Montmort in 1708. A derangement is a permutation that has no fixed points, and the derangement number D n 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.

Highlights

Introduction and PreliminariesThe problem of counting derangements was initiated by Pierre Rémond de Montmort in 1708
In the final section, we show that if X is the gamma random variable with parameters 1, 1, DnðpÞ, DðncÞðp, qÞ, DðnsÞðp, qÞ are given by the “moments” of some variants of X
We want to show that if X is the gamma random variable with parameters 1, 1, DnðpÞ, DðncÞðp, qÞ, DðnsÞðp, qÞ are given by the “moments” of some variants of X

Summary

Introduction and Preliminaries

The problem of counting derangements was initiated by Pierre Rémond de Montmort in 1708 (see [1, 2]). We show a recurrence relation for derangement polynomials. We derive identities involving derangement polynomials, Bell polynomials, and Stirling numbers of both kinds. We have an identity relating Bell polynomials, derangement polynomials, and Euler numbers. Among other things, their explicit expressions and recurrence relations. In the rest of this section, we recall the derangement numbers, especially their explicit expressions, generating function, and recurrence relations. We recall the gamma random variable with parameters α, λ along with their moments and the Bell polynomials. By comparing the coefficients on both sides of (43), we obtain the following theorem. DðncÞðx, yÞ as a polynomial in x, for each fixed y, and DnðxÞ are Appell sequences. By (35) and (37) and Theorem 13, we obtain the Comparing the coefficients on both sides of (64), we have following corollary.

Further Remarks

Conclusion