On a generalization of derangement polynomials and numbers

1 Introduction

Let X be a set with n elements, say X = 1,2 , … , n . A permutation of X with no fixed point is called a derangement, and the number of derangements of X is called the nth derangement number and denoted by D _n . By the inclusion-exclusion principle, the nth derangement number is (see [1], [2], [3], [4])

By (1.1), we see the generating function of the derangement numbers are (see [1], [2], [3], [4])

As a natural extension, the derangement polynomials are defined by the generating function to be (see [5], [6], [7])

When x = 0, D _n (0) = D _n is the nth derangement numbers.

The derangement numbers and polynomials are interesting things in the special function theory, combinatorics, applied mathematics. In [8], Garsia-Remmel generalized the derangement numbers to the q-derangement numbers, and Clarke-Sved found a relationship between Bell numbers and derangement numbers in [9]. Briggs-Remmel defined the p, q-analogue of the derangement numbers, extending some work of Garsia and Remmel and showed that these p, q-analogues satisfy recurrences that naturally extend the classical recurrences for the derangement numbers (see [10]). In [7], Kim-Kim-Lee-Jang defined the degenerate version of the derangement polynomials and found some interesting identities of those polynomials and numbers, and Kim-Kim defined the probabilistic derangement polynomials which was a generalization of those polynomials and derived some properties of those polynomials and numbers (see [11]).

From now on, we list some important numbers or polynomials in the combinatorics that we will need for this paper.

For nonzero integers n and k with n ≥ k, the Stirling numbers of the first kind and the second kind S ₁(n, k), S ₂(n, k), respectively, are defined by generating function to be (see [2],12])

where (x)₀ = 1, (x) _n = x(x − 1)…(x − n + 1), (n ≥ 1).

By (1.4), we see that for each nonnegative integer k,

(see [2],12]).

The Bell polynomials are defined by the generating function to be (see [12])

In the special case x = 1, Bel _n = Bel _n (1) are called the Bell numbers which counts the possible partitions of given set with n elements.

The Fubini polynomials are defined as follows (see [13]):

When x = 1, F _n = F _n (1) are called the Fubini numbers which counts the ordered partitions of a set with n elements.

For λ ≠ 0, the Apostol-Bernoulli polynomials are defined by the generating function to be (see [14],15])

In the special case x = 0, B _n (λ) = B _n (0|λ) are called the Apostol-Bernoulli numbers. Note that if we put λ = 1, then B _n (x|λ) = B _n (x) are the B. polynomials. By the definition of Apostol-Bernoulli polynomials, if λ ≠ 1, then we see that B ₀(x|λ) = 0, B 1 ( x | λ ) = 1 λ − 1 .

Let Y be a random variable with the moment generating function of Y.

exists for some r > 0 (see [11],16]). Note that if Y = 1, then E e Y t = e t .

The Poisson random variable indicates how many events occurred within a given period of time. A random variable Y, taking on one of the values 0, 1, 2…, is said to be the Poisson random variable with parameter α > 0, if the probability mass function of Y is given by (see [17],18])

In [16], Adell-B e ́ nyi defined the probabilistic Stirling numbers of the first kind associated with random variable Y and the second kind S 1 Y ( n , k ) and S 2 Y ( n , k ) , respectively as follows:

where n, k are nonnegative integers with n ≥ k ≥ 0.

In this paper, we defined the probabilistic derangement polynomials and numbers which are different from the one defined by Kim-Kim, and find relationships between probabilistic Stirling numbers of the first and the second kind, Stirling numbers of the first and the second kind, Bell polynomials, Fubini polynomials and those polynomials and numbers. In addition, we give some explicit formulas of those polynomials and numbers.

2 Probabilistic derangement polynomials and numbers

Throughout this paper, let Y be a random variable with the moment generating function of Y (1.5).

In viewpoint of (1.6) or (1.7), we define the probabilistic derangement polynomials as follows:

When x = 0, d n Y = d n Y ( 0 ) are called the probabilistic derangement numbers. Note that if we put Y = 1, then d n Y ( x ) = D n ( x ) .

By (2.1), we see that.

In addition,

when log ⁡ E e Y t < 1 . By (2.2) and (2.3), we obtain the following theorem.

Note that, by (1.7), we get.

where E e Y t < 1 . By (2.1) and (2.4), we have

and

Hence, by (2.5) and (2.6), we obtain the following theorem.

Note that, by (1.6),

By (2.7), we obtain the following theorem.

Note that

and, by (1.7), we get

when E e Y t − 1 < 1 . Since E e Y t = ∑ n = 0 ∞ E Y n t n n ! , by (2.8) and (2.9), we obtain the following theorem.

Let Y be the Poisson random variable with parameter α > 0. Then

and by (2.10), we see that

where α ( e t − 1 ) < 1 . In addition, by (2.10), we know that

By (2.11) and (2.12), we obtain the following theorem.

A continuous random variable Y whose density function defined by

for some α, β > 0 is called the gamma random variable with parameter α, β, and denoted by Y ∼Γ(α, β).

Let Y ∼Γ(1, 1). Then

and thus, by (2.13), we get

Replacing t bt 1 − e ^t in (2.14), we have

and

By (2.15) and (2.16), we see that

Since S ₂(0, m) = S ₂(m − 1, m) = 0 for each m ∈ N , by (2.17), we see that

By (2.18), we obtain the following theorem.

Let Y be Bernoulli random variable with probability success p. Then.

and by (2.19), we see that

and so we have

Note that