Some identities of type 2 Degenerate Bernoulli polynomials of the second kind

I recent years, many mathematicians studied various degenerate version of some spcial polynomials of which quite a few interesting results were discovered. In this paper, we introduce the type 2 degenerate Bernoulli polynomials of the second kind and their higher-order analogues, and study some identities and expressions for these polynomials. Specially, we obtain a relation between the type 2 degenerate Bernoulli polynomials of the second kind and degenerate Bernoulli polynomials of the second kind, and identity involving hihger-order analogues of those polynomials and the degenerate stirling number of the second kin, and and expression of higher-order analogues of those polynomials in terms of the higher-order type 2 degenerate Bernoulli polynomials and the degenerate stirling number of the first kind.


INTRODUCTION
As is known, the type 2 Bernoulli polynomials are defined by the generating function (1) t e t − e −t e xt = ∞ ∑ n=0 B * n (x) t n n! , (see [7]).
The central factorial numbers of the second kind are defined as (5) x n = n ∑ k=0 T (n, k)x [k] , (see [6]), or equivalently as . It is well known that the Daehee polynomials are defined by (see [14,15]).
When x = 0, D n = D n (0) are called the Daehee numbers.
The Bernoulli polynomials of the second kind of order r are defined by , (see [13]).
It is known that the Stirling numbers of the second kind are defined by (see [13]), and the Stirling numbers of the first kind by (see [13]).
For any nonzero λ ∈ R, the degenerate exponential function is defined by In particular, we let In [1,2], Carlitz introduced the degenerate Bernoulli polynomials which are given by the generating function (14) t Also, he considered the degenerate Euler polynomials given by (see [1,2]).
Recently, Kim-Kim considered the degenerate central factorial numbers of the second kind given by Note that lim λ →0 T λ (n, k) = T (n, k), (see [12]).
In this paper, we introduce the type 2 degenerate Bernoulli polynomials of the second kind and their higher-order analogues, and study some identities and expressions for these polynomials. Specifically, we obtain a relation between the type 2 degenerate Bernoulli polynomials of the second and the degenerate Bernoulli polynomials of the second, an identity involving higher-order analogues of those polynomials and the degenerate Stirling numbers of second kind, and an expression of higher-order analogues of those polynomials in terms of the higher-order type 2 degenerate Bernoulli polynomials and the degenerate Stirling numbers of the first kind.

TYPE 2 DEGENERATE BERNOULLI POLYNOMIALS OF THE SECOND KIND
Let log λ t be the compositional inverse of e λ (t) in (13). Then we have Note that lim λ →0 log λ t = logt. Now, we define the degenerate Daehee polynomials by Note that lim λ →0 D n,λ (x) = D n (x), (n ≥ 0). In view of (8), we also consider the degenerate Bernoulli polynomials of the second kind of order α given by For α = r ∈ N, and replacing t by e 2t − 1 in (20), we get On the other hand, From (21) and (22), we have Now, we define the type 2 degenerate Bernoulli polynomials of the second kind by When x = 0, b * n,λ = b * n,λ (0) are called the type 2 degenerate Bernoulli numbers of the second kind. Note that lim λ →0 b * n,λ (x) = b * n (x), where b * n (x) are the type 2 Bernoulli polynomials of the second kind given by From (19) and (24), we note that Therefore, we obtain the following theorem.
where r is a positive integer.
Let α = k ∈ N. Then we have By replacing t by e λ (t) − 1 in (29), we get where S 2,λ (n, l) are the degenerate Stirling numbers of the second kind given by (see [10]).
On the other hand, we also have Therefore, by (30) and (32), we obtain the following theorem.
For α ∈ R, we recall that the type 2 degenerate Bernoulli polynomials of order α are defined by n,λ (x) t n n! , (see [6,12]).
For k ∈ N, let us take α = −k and replace t by log λ (1 + t) in (33). Then we have

Conclusions
In [1,2], Carlitz initiated study of the degenerate Bernoulli and Euler polynomials. In recent years, many mathematicians have investigated various degenerate versions of some old and new polynomials and numbers, and found quite a few interesting results [3,4,6,10,11]. It is remarkable that studying degenerate versions is not only limited to polynomials but also can be applied to transcendental functions. Indeed, the degenerate gamma functions were introduced and studied in [8,9].
In this paper, we introduced the type 2 degenerate Bernoulli polynomials of the second kind and their higher-order analogues, and studied some identities and expressions for these polynomials. Specifically, we obtained a relation between the type 2 degenerate Bernoulli polynomials of the second and the degenerate Bernoulli polynomials of the second, an identity involving higher-order analogues of those polynomials and the degenerate Stirling numbers of second kind, and an expression of higher-order analogues of those polynomials in terms of the higher-order type 2 degenerate Bernoulli polynomials and the degenerate Stirling numbers of the first kind.
In addition, we obtained an identity involving the higher-order degenerate Bernoulli polynomials of the second kind, the type 2 Bernoulli polynomials and Stirling numbers of the second kind, and an identity involving the degenerate central factorial numbers of the second kind, the degenerate Stirling numbers of the first kind and the higher-order degenerate Bernoulli polynomials of the second kind.