Degenerate polyexponential functions and type 2 degenerate poly-Bernoulli numbers and polynomials

The polyexponential functions were introduced by Hardy and rediscovered by Kim, as inverses to the polylogarithm functions. Recently, the type 2 poly-Bernoulli numbers and polynomials were defined by means of the polyexponential functions. In this paper, we introduce the degenerate polyexponential functions and the degenerate type 2 poly-Bernoulli numbers and polynomials, as degenerate versions of such functions and numbers and polynomials. We derive several explicit expressions and some identities for those numbers and polynomials.


Introduction
For k ∈ Z, the polyexponential function is defined by x n (n -1)!n k (see [10]). (1) By (1), we see that Ei 1 (x) = e x -1. The polyexponential function was first introduced by Hardy and is given by x n (n + a) s n! Re(a) > 0 .
Recently, the degenerate polylogarithm function was defined by Kim-Kim as where x n n k = Li k (x) is the polylogarithm of index k. For k ∈ Z, the degenerate poly-Bernoulli numbers are defined by n,λ t n n! (see [17]).
As an inversion formula of (7), the degenerate Stirling numbers of the first kind are defined by [23]).
Kaneko defined the poly-Bernoulli numbers by making use of the polylogarithm functions and Kim-Kim-Kim-Jang studied degenerate poly-Bernoulli numbers and polynomials by using polyexponential function [18]. The polyexponential functions were first introduced by Hardy and rediscovered recently by Kim-Kim [10], as inverses to the polylogarithm functions. In addition, the type 2 poly-Bernoulli numbers and polynomials were defined by means of the polyexponential functions. In this paper, we study the degenerate polyexponential functions and the degenerate type 2 poly-Bernoulli polynomials and numbers, as degenerate versions of such functions and numbers and polynomials. We derive several explicit expressions and some identities for those numbers and polynomials.

Type degenerate poly-Bernoulli numbers and polynomials
The degenerate polyexponential function is defined in [15]. In the light of (1), we now consider the degenerate modified polyexponential function given by Note that Ei 1,λ (x) = e λ (x) -1.
From (11), we note that For k ≥ 2, by (12), we have In view of (2) and using the degenerate modified polyexponential function, we define the type 2 degenerate poly-Bernoulli polynomials by When n,λ (0) are called type 2 degenerate poly-Bernoulli numbers. It is well known that the degenerate Bernoulli polynomials of the second kind are defined by [17]).
From (14), we note that Therefore, by comparing the coefficients on both sides of (21), we obtain the following theorem.
Theorem 2 For n ≥ 0, we have Thus, by (4) On the other hand, Therefore, by (23) and (24), we obtain the following theorem. From (14), we note that On the other hand, Therefore, by (25) and (26), we obtain the following theorem.
On the other hand, Therefore, by (28) and (29), we obtain the following theorem. where δ n,k is the Kronecker delta.
Note that Thus, by Theorems 4 and 5, we get From (14), we note that n,λ t n n! .
By replacing t by e λ (t) -1, we get On the other hand, Therefore, by (32) and (33), we obtain the following theorem.

Further remark
The higher-order degenerate Bernoulli polynomials are defined by Carlitz and given by t n n! (see [2,3]), where r is a positive integer.