A Note on the Poly-Bernoulli Polynomials of the Second Kind

In the book Ars Conjectandi, Bernoulli introduced the Bernoulli number terms of the sum of powers of consecutive integers (see [1, 2]). In [3], Luo and Srivastava defined the Apostol-Bernoulli polynomials and obtained an explicit series representation for their polynomials involving the Gaussian hypergeometric function as well as an explicit series representation involving the Hurwitz function. Frappier defined a generalized Bernoulli polynomials by using the Bessel function of the first kind and found a generalization of a well-known Fourier series representation of Bernoulli polynomials in [4]. In [5], Natalini and Bernardini defined a new class of generalized Bernoulli polynomials and showed that if a differential equation with these polynomials is of order n, then all the considered families of polynomials can be viewed as solutions of differential operators of infinite order. In [6], Kaneko defined the poly-Bernoulli polynomials and found an explicit formula and a duality theorem for those numbers. Khan et al. defined Laguerre-based Hermite-Bernoulli polynomials and derived summation formulas and related bilateral series associated with the newly introduced generating function in [7]. In [8], Jang and Kim defined type 2 degenerate Bernoulli polynomials and showed that these polynomials could be represented linear combinations of the Stirling numbers of the first and the second kinds, Bernoulli polynomials, and those numbers. Moreover, in [9], the degenerate type 2 poly-Bernoulli numbers and polynomials as degenerate versions of such numbers and polynomials were defined, and several explicit expressions and some identities for those numbers and polynomials were derived. As is well known, Bernoulli polynomials of order r are defined by the generating function to be


Introduction
In the book Ars Conjectandi, Bernoulli introduced the Bernoulli number terms of the sum of powers of consecutive integers (see [1,2]). In [3], Luo and Srivastava defined the Apostol-Bernoulli polynomials and obtained an explicit series representation for their polynomials involving the Gaussian hypergeometric function as well as an explicit series representation involving the Hurwitz function. Frappier defined a generalized Bernoulli polynomials by using the Bessel function of the first kind and found a generalization of a well-known Fourier series representation of Bernoulli polynomials in [4]. In [5], Natalini and Bernardini defined a new class of generalized Bernoulli polynomials and showed that if a differential equation with these polynomials is of order n, then all the considered families of polynomials can be viewed as solutions of differential operators of infinite order. In [6], Kaneko defined the poly-Bernoulli polynomials and found an explicit formula and a duality theorem for those numbers. Khan et al. defined Laguerre-based Hermite-Bernoulli polynomials and derived summation formulas and related bilateral series associated with the newly introduced generating function in [7]. In [8], Jang and Kim defined type 2 degenerate Bernoulli polynomials and showed that these polynomials could be represented linear combinations of the Stirling numbers of the first and the second kinds, Bernoulli polynomials, and those numbers. Moreover, in [9], the degenerate type 2 poly-Bernoulli numbers and polyno-mials as degenerate versions of such numbers and polynomials were defined, and several explicit expressions and some identities for those numbers and polynomials were derived.
As is well known, Bernoulli polynomials of order r are defined by the generating function to be (see [1,5,9,10]). In particular, if r = 1, B n ðxÞ = B ð1Þ n ðxÞ are the ordinary Bernoulli polynomials. When x = 0, B ðrÞ n = B ðrÞ n ð0Þ are called the Bernoulli numbers of order r. In [1], the relationship between the Bernoulli numbers and zeta functions was studied, and in [2,8,[10][11][12], generalized Bernoulli numbers were defined, and the properties of those numbers and polynomials were investigated.
The Bernoulli polynomials of the second kind (or the Cauchy polynomials) are defined by the generating function to be (see [13][14][15]). When x = 0, b n = b n ð0Þ are called the Bernoulli numbers of the second kind.
In [17,19], the authors defined the generalized Stirling numbers of the first and second kinds and generalized binomial coefficients and showed that degenerated special polynomials are represented by linear combinations of those numbers.
In this paper, we define poly-Bernoulli polynomials of the second kind with the polyexponential function and derive some interesting identities between the Stirling numbers of the first kind or the second kind, Bernoulli numbers, Bernoulli numbers of the second kind, and those polynomials. In addition, we define unipoly-Bernoulli polynomials of the second kind and derive some interesting identities of those polynomials.

The Poly-Bernoulli Polynomials of the Second Kind
By the definition of the Bernoulli polynomials of the second kind and (7), we define the poly-Bernoulli polynomials of the second kind by the generating function to be In particular, if x = 0, b ðkÞ n = b ðkÞ n ð0Þ are called the poly-Bernoulli numbers of the second kind. By (8), we know that for each nonnegative integer n, are the Bernoulli polynomials of the second kind. Note that Hence, by (10), we obtain the following theorem. where By replacing t by e t − 1 in (8), we get Journal of Function Spaces and by (6), we have Therefore, by (12) and (13), we obtain the following theorem.
In particular, we have From (4) and (8), we derive Thus, by (16), we obtain the following theorem.
By (9) and Theorem 3, we get the following corollary.
In Corollary 4, we have Therefore, we obtain the following corollary.

Journal of Function Spaces
In particular, if we put k = 2 in (23), then by (23) and (24), we have Therefore, by (25), we obtain the following theorem.
Theorem 6. For a nonnegative integer n, we have

The Unipoly-Bernoulli Polynomials of the Second Kind
Let p be an arithmetic function which is a real or complex valued function defined on ℕ. In [21], Kim and Kim defined the unipoly function attached to polynomial pðxÞ by In particular, if pðxÞ = 1, then is an ordinary polylogarithm function. Note that by (27), we get for k ≥ 2. In addition, it is well known that (see [9,11,12,19,21,23,25]).
In the viewpoint of (8), we define the unipoly-Bernoulli polynomials of the second kind as From (31), we derive the following equation: and thus, by (32), we obtain the following theorem.
In the definition of unipoly-Bernoulli polynomials of the second kind, if x = 0, then we get