A new approach to Bell and poly-Bell numbers and polynomials

The aim of this paper is to introduce Bell polynomials and numbers of the second kind and poly-Bell polynomials and numbers of the second kind, and to derive their explicit expressions, recurrence relations and some identities involving those polynomials and numbers. We also consider degenerate versions of those polynomials and numbers, namely degenerate Bell polynomials and numbers of the second kind and degenerate poly-Bell polynomials and numbers of the second kind, and deduce their similar results.


INTRODUCTION
There are various ways of studying special numbers and polynomials, to mention a few, generating functions, combinatorial methods, p-adic analysis, umbral calculus, differential equations, probability theory, special functions and analytic number theory.
The aim of this paper is to introduce several special polynomials and numbers, and to study their explicit expressions, recurrence relations and identities involving those polynomials and numbers by using generating functions.
Indeed, we introduce Bell polynomials and numbers of the second kind (see (15), (17)) and poly-Bell polynomials and numbers of the second kind (see (36)). The generating function of Bell numbers of the second kind is the compositional inverse of the generating function of Bell numbers minus the constant term. Then Bell polynomials of the second kind are natural extensions of those numbers. The poly-Bell polynomials of the second kind, which are defined with the help of polylogarithm, become the Bell polynomials of the second kind up to sign when the index of the polylogarithm is k = 1.
We also consider degenerate versions of those numbers and polynomials, namely degenerate Bell numbers and polynomials of the second (see (28), (30)) and degenerate poly-Bell numbers and polynomials (see (40)), and derive similar results. It is worthwhile to note that degenerate versions of many special numbers and polynomials have been explored in recent years with aforementioned tools and many interesting arithmetical and combinatorial results have been obtained (see [8,9,12,13,18]). In fact, studying degenerate versions can be done not only for polynomials and numbers but also for transcendental functions like gamma functions. For the rest of this section, we recall the necessary facts that are needed throughout this paper.
The Stirling numbers of the first kind, S 1 (n, k), are given by [5,17]), As the inversion formula of (1), the Stirling numbers of the second kind, S 2 (n, k), are given by It is well known that the Bell polynomials are defined as [17]).
In view of (2), the degenerate Stirling numbers of the second kind are defined by , (see [9]).
In [11], the degenerate Bell polynomials are defined by Bel n,λ (x) t n n! .

BELL POLYNOMIALS OF THE SECOND KIND
From (4), we note that Bel n t n n! .
Let f (t) = e e t −1 − 1. Then the compositional inverse of f (t) is given by We consider the new type Bell numbers, called Bell numbers of the second kind, defined by Now, we observe that Therefore, by (15) and (16), we obtain the following theorem.
Theorem 1. For n ≥ 1, we have where n k are the unsigned Stirling numbers of the first kind. Also, we consider the new type Bell polynomials, called Bell polynomials of the second kind, defined by (17) bel From (17), we can derive the following equation.
Thus the generating function of Bell polynomials of the second kind is given by Note here that bel n = bel n (1). From (19), we note that Replacing t by e t − 1 in (20), we get where d n is the derangement number ( [13]). Therefore, by comparing the coefficients on both sides of (22), we obtain the following theorem. Replacing t by e e t −1 − 1 in (15), we get bel k S 2 ( j, k)S 2 (n, j) t n n! .
Thus we obtain following theorem. Replacing t by e t − 1 in (19), we get On the other hand, Therefore, by (24) and (25), we obtain the following theorem.
Theorem 4. For n ≥ 1, we have In particular, bel k S 2 (n, k).

DEGENERATE BELL POLYNOMIALS OF THE SECOND KIND
From (3), we note that Bel n,λ t n n! .
Let f λ (t) = e λ e λ (t) − 1 − 1. Then the compositional inverse of f λ (t) is given by . We consider the new type degenerate Bell numbers, called degenerate Bell numbers of the second kind, defined by bel n,λ t n n! .
Now, we observe that Therefore, by (28) and (29), we obtain the following theorem.
Also, we define the degenerate Bell polynomials of second kind by Note that bel n,λ = bel n,λ (1). From (30), we note that Thus the generating function of bel n,λ (x) is given by Replacing t by e λ (t) − 1 in (32), we get On the other hand, Therefore, by (33) and (34), we obtain the following theorem.
Therefore, by comparing the coefficients on both sides of (35), we obtain the following theorem.

POLY-BELL POLYNOMIALS OF THE SECOND KIND
Now, we consider the poly-Bell polynomials of the second kind which are defined as t n n! .
n (1) are called the poly-Bell numbers of the second kind. From (11), we note that Therefore, by (36) and (37), we obtain the following theorem.

DEGENERATE POLY-BELL POLYNOMIALS OF THE SECOND KIND
We define the degenerate poly-Bell polynomials of the second kind by n,λ (x) t n n! .
n,λ (1) are called the degenerate poly-Bell numbers of the second. From (13), we note that Therefore, by (40) and (41), we obtain the following theorem.

CONCLUSION
By means of various different tools, degenerate versions of many special polynomials and numbers have been studied in recent years. Here we introduced Bell polynomials of the second kind, poly-Bell polynomials of the second kind and their degenerate versions, namely degenerate Bell polynomials of the second kind and degenerate poly-Bell polynomials of the second kind. By using generating functions, we explored their explicit expressions, recurrence relations and some identities involving those polynomials and numbers.
It is one of our future projects to continue this line of research, namely to explore many special numbers and polynomials and their degenerate versions with the help of various different tools.

Acknowledgments:
Funding: Availability of data and materials: Not applicable.
Competing interests: The authors declare no conflict of interest.