New Properties on Degenerate Bell Polynomials

+eBell number Beln counts the number of partitions of a set with n elements into disjoint nonempty subsets. +e Bell polynomials Beln(x), also called Touchard or exponential polynomials, are natural extensions of Bell numbers. +e 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:


Introduction
e Bell number Bel n counts the number of partitions of a set with n elements into disjoint nonempty subsets. e Bell polynomials Bel n (x), also called Touchard or exponential polynomials, are natural extensions of Bell numbers. e 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]).
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][3][4][5][6][7][8][9][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]). e studies have been carried out by various means like combinatorial methods, generating functions, differential equations, umbral calculus techniques, p-adic analysis, and probability theory. e 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 eorems 1, 3, 4, and 8), and expressions for them that can be derived from repeated applications of certain operators to the exponential functions (see eorem 2, Proposition 1), the derivatives of them (Corollary 1), the antiderivatives of them (see eorem 6), and some identities involving them (see eorems 5,9). For the rest of this section, we recall some necessary facts that are needed throughout this paper.
We also recall the degenerate absolute Stirling numbers of the first kind that are defined by (see [10]), where 2 Complexity It is well known that the Bell polynomials are defined by (see [12,13]). When x � 1, Bel n � Bel n (1) are called the Bell numbers.
From (12), we can deduce the recurrence relation given by and the values Now, we compute from (19), (3), and (21) the first few degenerate Bell polynomials as follows: Complexity 3 In Figure 1, we plot the shapes of Bell polynomial Bel k,λ (x). e upper-left graph looks different from the others. However, of course they all go to infinity as x tends to infinity.

Some New Properties on Degenerate Bell Polynomials
Let a be a nonzero constant. First, we observe that erefore, by (23), we obtain the following lemma.
e generating function of S n,λ is given by  Indeed, this can be seen from the following: Bel n,λ (1) t n n! � ee e λ (t)− 1 � e e λ (t) .
Taking the derivative with respect to t on both sides of (30), we have us, by comparing the coefficients on both sides of (31) and from (28), we obtain the following theorem.
Assume that the following identity holds: en, we have is together with (26) gives the next result.
By taking x(d/dx) in the second equality of (35), on the one hand, we have On the other hand, we also have where Bel n,λ ′ (x) � (d/dx)Bel n,λ (x). From (38) and (39) and eorem 2, we note that erefore, by (40) and eorem 2, we obtain the following theorem.
From (17), we note that us, by comparing the coefficients on both sides of (42), we get (n ≥ 1).

(43)
Taking the derivative with respect to t on both sides of (17), we have On the other hand, erefore, by (44) and (45), we obtain the following theorem.
is implies that the following identity must hold true: the validity of which follows from eorem 4.

Corollary 1.
For n ≥ 1, we have the following identity: We observe that 6 Complexity us, by (51), we get From (17), we have erefore, by comparing the coefficients on both sides of (53), we obtain the following theorem.
From (17), we note that On the other hand, we also have Complexity erefore, by (55) and (56), we obtain the following theorem.

Proposition 1.
For n ≥ 0, we have the following operational formula: x nλ x 1− λ D n e ax p � p n Bel n,(λ/p) ax p e ax p , where D � (d/dx).
Theorem 7. Assume that f is an infinitely differentiable function. en, for n ≥ 0, the following operational formula holds: where D � (d/dx).
Proof. e statement is obviously true for n � 0. Assume that it is true for n, (n ≥ 0). (67) Observe that, for any α, we have 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
e authors declare that they have no conflicts of interest.