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 is the number of fixed point free permutations on an 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.
1. Introduction and Preliminaries
The problem of counting derangements was initiated by Pierre Rémond de Montmort in 1708 (see [1, 2]). A derangement is a permutation of the elements of a set, such that no element appears in its original position. In other words, a derangement is a permutation that has no fixed points. The derangement number is the number of fixed point free permutations on an element set.
The aim of this paper is to study 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. Here, the derangement polynomials are natural extensions of the derangement numbers.
The outline of our main results is as follows. We show a recurrence relation for derangement polynomials. Then, we derive identities involving derangement polynomials, Bell polynomials, and Stirling numbers of both kinds. In addition, we also have an identity relating Bell polynomials, derangement polynomials, and Euler numbers. Next, we introduce the two variable polynomials, namely, cosine-derangement polynomials and sine-derangement polynomials , in a natural manner by means of derangement polynomials. We obtain, among other things, their explicit expressions and recurrence relations. Lastly, in the final section, we show that if is the gamma random variable with parameters , then are given by the “moments” of some variants of .
In the rest of this section, we recall the derangement numbers, especially their explicit expressions, generating function, and recurrence relations. Also, we give the derangement polynomials and give their explicit expressions. Then, we recall the gamma random variable with parameters along with their moments and the Bell polynomials. Finally, we give the definitions of the Stirling numbers of the first and second kinds.
As before, let denote the derangement number for , and let . Then, the first few derangement numbers are 1, 0, 1, 2, 9, 44, 265, 1854, 14833, 133496, 1334961, . For , the derangement numbers are given by [3–5]
From (1), we note that [1–4, 6, 7]
By (2), we get
From (3), we can easily derive the following recurrence relation [5, 8–11]:
Now, we consider the derangement polynomials which are given by [10]
From (5), we have
By comparing the coefficients on both sides of (6), we get [10]
On the other hand,
From (6), (7), and (8), we have
A continuous random variable whose density function is given by [12–14]
for some and is said to be the gamma random variable with parameter which is denoted by .
For , the -th moment of is given by
It is well known that the Bell polynomials are defined by [15]
When , are called the Bell numbers.
The Stirling numbers of the first kind are defined as [16, 17]
where .
As an inversion formula of (13), the Stirling numbers of the second kind are defined by [16–18]
2. Derangement Polynomials and Numbers
From (5), we have
On the other hand,
Therefore, by (15) and (16), we obtain the following lemma.
Lemma 1. For , we have
Replacing by in (5), we get
From (18), we have
It is easy to show that
Replacing by in (20), we get
Therefore, by (19) and (22), we obtain the following theorem.
Theorem 2. For , we have
Corollary 3. For , we have
Replacing by in (5), we get
On the other hand, we have
where are the ordinary Euler numbers.
Therefore, by (25) and (26), we obtain the following theorem.
Theorem 4. For , we have
Now, we observe that
where is a positive integer.
On the other hand,
Therefore, by (28) and (29), we obtain the following proposition.
Proposition 5. For , we have
It is well known that [16, 18, 19]
From (5), we note that
By (9), (32), and (33), we get
From (34) and (35), we can derive the following equations:
We define cosine-derangement polynomials and sine-derangement polynomials, respectively, by
Thus, we have
Therefore, we obtain the following theorem.
Theorem 6. For , we have
Before proceeding further, we recall that
From (38)and (42), we note that
Therefore, by comparing the coefficients on both sides of (43), we obtain the following theorem.
Theorem 7. For , we have
Corollary 8. For , we have
By (38), we get
Thus, we have
Therefore, by (47) and (42), we obtain the following theorem.
Theorem 9. For , we have
By (38), we get
On the other hand,
Therefore, by (49) and (50), we obtain the following theorem.
Theorem 10. For , we have
It is not difficult to show that
where is a positive integer.
By comparing the coefficients on both sides of (47), we get
Now, we observe that
Form (54), we note that
Therefore, we obtain the following theorem.
Theorem 11. For , we have
In particular,
Corollary 12. as a polynomial in , for each fixed , and are Appell sequences.
Before proceeding further, we recall that
From (39)and (58), we note that
Therefore, by (59), we obtain the following theorem.
Theorem 13. For , we have
By (35) and (37) and Theorem 13, we obtain the following corollary.
Corollary 14. For , we have
By (59), we see that
Therefore, by (62) and (58), we obtain the following theorem.
Theorem 15. For , we have
It is easy to show that . However, is not an Appell sequence, since .
We observe that
Comparing the coefficients on both sides of (64), we have the following theorem.
Theorem 16. For , we have
For , we have
Thus, we obtain
3. Further Remarks
As applications, we want to show that if is the gamma random variable with parameters , then are given by the “moments” of some variants of . We let the reader refer to the papers [20–22] for some recent papers related to this section.
Let be a gamma random variable with parameters which is denoted by . Then, we observe that
where is the density function of and .
From (10) and (68), we can derive the following equation:
On the other hand, by Taylor expansion, we get
Therefore, by (69) and (70), we obtain the following theorem.
Theorem 17. For , , the moment of is given by
When , , .
Thus, we note that
For , we note that the moment of is given by , .
Therefore, by (72), we obtain the following corollary.
Corollary 18. For , , we have
For , we have
where .
From (74), we note that
On the other hand, by Taylor expansion, we get
Therefore, by (76) and (77), we obtain the following theorem.
Theorem 19. For , , we have
It is easy to show that
where .
Thus, we have
where .
4. Conclusion
The introduction of derangement numbers goes back to as early as 1708 when Pierre Rémond de Montmort considered some counting problem on derangements. In this paper, we dealt with derangement polynomials which are natural extensions of the derangement numbers. We showed a recurrence relation for derangement polynomials. We derived identities involving derangement polynomials, Bell polynomials, and Stirling numbers of both kinds. In addition, we also obtained an identity relating Bell polynomials, derangement polynomials, and Euler numbers. Next, we introduced the cosine-derangement polynomials and sine-derangement polynomials , by means of derangement polynomials. Then, we derived, among other things, their explicit expressions and recurrence relations. Lastly, as applications, we showed that if is the gamma random variable with parameters , then are given by the “moments” of some variants of .
We have witnessed that the study of some special numbers and polynomials was done intensively by using several different means, which include generating functions, combinatorial methods, umbral calculus, -adic analysis, probability theory, special functions, and differential equations. Moreover, the same has been done for various degenerate versions of quite a few special numbers and polynomials in recent years with their interests not only in combinatorial and arithmetical properties but also in their applications to symmetric identities, differential equations, and probability theories. It would have been nicer if we were able to find abundant applications in other disciplines.
It is one of our future projects to continue to investigate many ordinary and degenerate special numbers and polynomials by various means and find their applications in physics, science, engineering, and mathematics.
Data Availability
No data were used to support this study.
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this paper.