Some Identities of Degenerate Bell Polynomials

: The new type degenerate of Bell polynomials and numbers were recently introduced, which are a degenerate version of Bell polynomials and numbers and are different from the previously introduced partially degenerate Bell polynomials and numbers. Several expressions and identities on those polynomials and numbers were obtained. In this paper, as a further investigation of the new type degenerate Bell polynomials, we derive several identities involving those degenerate Bell polynomials, Stirling numbers of the second kind and Carlitz’s degenerate Bernoulli or degenerate Euler polynomials. In addition, we obtain an identity connecting the degenerate Bell polynomials, Cauchy polynomials, Bernoulli numbers, Stirling numbers of the second kind and degenerate Stirling numbers of the second kind.

Recently, the new type degenerate Bell polynomials are introduced by the generating function as (see [10]).
Note that lim λ→0 Bel n,λ (x) = Bel n (x), (n ≥ 0). We see that the middle term in (13) is equal to and hence we get Studying degenerate versions of some special polynomials has been very fruitful and regained lively interest of many mathematicians in recent years. In [1], Carlitz initiated a study of degenerate versions of some special polynomials, namely the degenerate Bernoulli and Euler polynomials and numbers.
In the recent paper [10], the new type degenerate Bell polynomials Bel n,λ (x) (see (13)) were introduced and some interesting results about them were obtained, which are different from the previously defined partially degenerate Bell polynomials (see [9]) and a degenerate version of the ordinary Bell polynomials Bel n (x) (see (8)).
As a further study of the new type degenerate Bell polynomials, we will obtain two expressions involving these degenerate Bell polynomials, Carlitz's degenerate Bernoulli polynomials and the Stirling numbers of the second kind, two identities involving those degenerate Bell polynomials, degenerate Euler polynomials and the Stirling numbers of the second kind. In additon, we will be able to find an identity involving those degenerate Bell polynomials, Cauchy polynomials, Bernoulli numbers, Stirling numbers of the second kind and degenerate Stirling numbers of the second kind.

Some Identities of Degenerate Bell Polynomials
The following equation can be easily derived from (13): where n ≥ 0. Indeed, we see that the middle term in (13) is equal to By using (16), we can show that By replacing t by e t − 1 in (1), we get On the other hand, Therefore, by comparing the coefficients on both sides of (18) and (19), we obtain the following theorem.
The left hand side of (20) is also given by Therefore, from (20) and (21), we obtain the following theorem.
Replacing t by e t − 1 in (22), we get The right hand side of (23) is equal to On the other hand, the left hand side of (23) is equal to Therefore, by equating (24) and (25), we obtain the following theorem. From (3), we note that By (26), we get By comparing the coefficients on both sides of (27), we get Therefore, by (28), we obtain the following theorem.
In other words, for n ≥ 0, we have Let us replace t by e λ (e t − 1) − 1 in (12). Then we have As is well known, the degenerate Stirling numbers of second kind are defined by (see [8]) By (29) and (30), we have Hence, by (29) and (31), we get On the other hand, where B n are the ordinary Bernoulli numbers. Therefore, by (32) and (33), we obtain the following theorem.

Conclusions
As a further study of the new type degenerate Bell polynomials, we obtained two expressions involving these degenerate Bell polynomials, Carlitz's degenerate Bernoulli polynomials and the Stirling numbers of the second kind, two identities involving those degenerate Bell polynomials, degenerate Euler polynomials and the Stirling numbers of the second kind. In additon, we were able to find an identity involving those degenerate Bell polynomials, Cauchy polynomials, Bernoulli numbers, Stirling numbers of the second kind and degenerate Stirling numbers of the second kind.
In our previous works related to this paper, we studied various degenerate versions of many special polynomials. They have been investigated by using several different means, such as generating functions, combinatorial methods, umbral calculus techniques, probability theory, p-adic analysis, differential equations, and so on. Further, q-analogues of those degenerate versions of special polynomials were also introduced by using bosonic and fermionic p-adic q-integrals, and their number theoretic and combinatorial properties were investigated.
It is one of our research projects to continue this line of study. Namely, we would like to study various degenerate versions of special polynomials and numbers and also their q-analogues, and investigate their possible applications to physics and engineering, as well as to mathematics.