Abstract
The aim of this paper is to study the degenerate Bell numbers and polynomials which are degenerate versions of the Bell numbers and polynomials. We derive some new identities and properties of those numbers and polynomials that are associated with the degenerate Stirling numbers of both kinds.
1. Introduction
The Bell number counts the number of partitions of a set with elements into disjoint nonempty subsets. The Bell polynomials , also called Touchard or exponential polynomials, are natural extensions of Bell numbers. The partial and complete Bell polynomials, which are multivariate generalizations of the Bell polynomials, have diverse applications not only in mathematics but also in physics and engineering as well (see [1]).
For instance, the following formula, due to Faà di Bruno formula:gives an explicit formula for higher derivatives of composite functions. Here, the partial Bell polynomials are defined bywhere the sum runs over all nonnegative integers , satisfying and (see [1], p. 133). Then, the complete Bell polynomials are given by , and .
As a degenerate version of those Bell polynomials and numbers, the degenerate Bell polynomials and numbers (see (17)) are introduced and studied under the different names of the partially degenerate Bell polynomials and numbers in [2]. Some interesting identities for them were obtained in connection with Stirling numbers of the first and second kinds [2]. We hope that we will be able to find many interesting applications of these polynomials and numbers in near future.
In [3], Carlitz initiated the exploration of degenerate Bernoulli and Euler polynomials, which are degenerate versions of the ordinary Bernoulli and Euler polynomials. Along the same line as Carlitz’s pioneering work, intensive studies have been done for degenerate versions of quite a few special polynomials and numbers (see [2–10] and the references therein). It is worthwhile to mention that these studies of degenerate versions have been done not only for some special numbers and polynomials but also for transcendental functions like gamma functions (see [8]). The studies have been carried out by various means like combinatorial methods, generating functions, differential equations, umbral calculus techniques, -adic analysis, and probability theory.
The aim of this paper is to further investigate the degenerate Bell polynomials and numbers by means of generating functions. In more detail, we derive several properties and identities of those numbers and polynomials which include recurrence relations for degenerate Bell polynomials (see Theorems 1, 3, 4, and 8), and expressions for them that can be derived from repeated applications of certain operators to the exponential functions (see Theorem 2, Proposition 1), the derivatives of them (Corollary 1), the antiderivatives of them (see Theorem 6), and some identities involving them (see Theorems 5, 9). For the rest of this section, we recall some necessary facts that are needed throughout this paper.
For any , the degenerate exponential functions are defined by(see [9]), where
When , we see use the notation .
In [3], Carlitz introduced the degenerate Bernoulli numbers given by
Note that , where are the ordinary Bernoulli numbers given by(see [1–14]).
From (5), we deduce thatfrom which we compute the first few values of as follows:
It is well known that the Stirling numbers of the first kind are defined by(see [14]), where .
As the inversion formula of (9), the Stirling numbers of the second kind are given by(see [14]).
The degenerate Stirling numbers of the first kind are defined by(see [5]), and the degenerate Stirling numbers of the second kind are given by(see [5, 7]).
Here, is the degenerate logarithm given by (18).
We also recall the degenerate absolute Stirling numbers of the first kind that are defined by(see [10]), where
It is well known that the Bell polynomials are defined by(see [12, 13]).
When are called the Bell numbers.
From (12), we note that(see [12, 13]).
In [2], the degenerate Bell polynomials are defined by
Note that . For , are called the degenerate Bell numbers.
The compositional inverse of is given by , namely, , where(see [5]).
Note that .
From (17), we note that(see [2]).
From (12), we can deduce the recurrence relation given byand the values
Now, we compute from (19), (3), and (21) the first few degenerate Bell polynomials as follows:
In Figure 1, we plot the shapes of Bell polynomial . The upper-left graph looks different from the others. However, of course they all go to infinity as tends to infinity.
2. Some New Properties on Degenerate Bell Polynomials
Let be a nonzero constant. First, we observe that
Therefore, by (23), we obtain the following lemma.
Lemma 1. For , the th derivative of is given by
Let in (23). Then, we have
Let
Then, we note from (19) that we have
The generating function of is given by
Indeed, this can be seen from the following:
Taking the derivative with respect to on both sides of (30), we have
Thus, by comparing the coefficients on both sides of (31) and from (28), we obtain the following theorem.
Theorem 1. For , the following recurrence relation holds:
Assume that the following identity holds:
Then, we have
This together with (26) gives the next result.
Theorem 2. For , the following relations hold true:
From the first equality in (35) and (27), we see that we have
Clearly, . We can check that
By taking in the second equality of (35), on the one hand, we have
On the other hand, we also havewhere .
From (38) and (39) and Theorem 2, we note that
Therefore, by (40) and Theorem 2, we obtain the following theorem.
Theorem 3. For , the following identity holds:where .
From (17), we note that
Thus, by comparing the coefficients on both sides of (42), we get
Taking the derivative with respect to on both sides of (17), we have
On the other hand,
Therefore, by (44) and (45), we obtain the following theorem.
Theorem 4. . For , the following recurrence relation is valid:
Remark 1. Theorems 3 and 4 and (43) give us the following:This implies that the following identity must hold true:the validity of which follows from Theorem 4.
From Theorem 3, we note that
Therefore, by (49), we obtain the following corollary.
Corollary 1. For , we have the following identity:
We observe that
Thus, by (51), we get
From (17), we have
Therefore, by comparing the coefficients on both sides of (53), we obtain the following theorem.
Theorem 5. For , the following binomial identity holds:
From (17), we note that
On the other hand, we also have
Therefore, by (55) and (56), we obtain the following theorem.
Theorem 6. For , the antiderivative of is given bywhere are Carlitz’s degenerate Bernoulli numbers given by .
For , by (12), we get
By comparing the coefficients on both sides of (58), we have
Let , and let . As , we have
Thus, we have
Therefore, by (61), we obtain the following proposition.
Proposition 1. For , we have the following operational formula:where .
From (12), we note that
By (63), we getwhere .
We prove the next theorem by induction on .
Theorem 7. Assume that is an infinitely differentiable function. Then, for , the following operational formula holds:where .
Proof. The statement is obviously true for . Assume that it is true for .Let . Then, we haveObserve that, for any , we haveBy the Leibniz rule, we getFrom Theorem 2, we note thatBy (69) and (70), we getOn the other hand,By (71) and (72), we getTherefore, by comparing the coefficients on both sides of (73), we obtain the following theorem.
Theorem 8. For , we have the following expression:
Taking in (74), we have
From (19) and (68), we note that
On the other hand, by Leibniz rule (69) and Theorem 2, we get
By (68) and (69) and Theorem 2, we easily get
Therefore, by (76) and (79), we obtain the following theorem.
Theorem 9. For , the following identity holds true.
By (11) and (13), we easily get
Indeed,
Therefore, by (82), we get
Replacing by in (17), we get
Thus, from (81) and (84), we get
From (13), we note that
Thus, by comparing the coefficients on both sides of (86), we get
3. Conclusion
Here, we studied by means of generating functions the degenerate Bell polynomials which are degenerate versions of the Bell polynomials. In more detail, we derived recurrence relations for degenerate Bell polynomials (see Theorems 1, 3, 4, and 8), and expressions for them that can be derived from repeated applications of certain operators to the exponential functions (see Theorem 2 and Proposition 1), the derivatives of them (Corollary 1), the antiderivatives of them (see Theorem 6), and some identities involving them (see Theorems 5 and 9).
As one of our future projects, we would like to continue to study degenerate versions of certain special polynomials and numbers and their applications to physics, science, and engineering as well as to mathematics. An earlier version of this paper has been presented as preprint in [15].
Data Availability
No data were used to support this study.
Disclosure
An earlier version of the paper has been presented as preprint in the following link: https://arxiv.org/abs/2108.06260.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
This work was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (No. 2020R1F1A1A01071564).