Representations of degenerate poly-Bernoulli polynomials

As is well-known, poly-Bernoulli polynomials are defined in terms of polylogarithm functions. Recently, as degenerate version of such functions and polynomials, degenerate polylogarithm functions were introduced and degenertae poly-Bernoulli polynomials were defined by means of the degenerate polylogarithm functions, and some their properties were investigated. The aim of this paper is to furthur study some properties of the degenertae poly-Bernoulli polynomials by using three formulas coming from the recently developed lambda-umbral calculus. In more detail, among other things, we represent the degenerate poly-Bernoulli polynomials by higher-order degenertae Bernoulli polynomials and by higher-order degenerate derangements polynomials.


INTRODUCTION
Carlitz is the first one who initiated the study of degenerate versions of some special numbers and polynomials, namely the degenerate Bernoulli and Euler polynomials and numbers (see [1]). In recent years, studying degenerate versions of some special numbers and polynomials regained interests of some mathematicians with their interests not only in combinatorial and arithmetic properties but also in applications to differential equations, identities of symmetry and probability theory (see [5,6,9,10,13,15,16] and the references therein). It is noteworthy that studying degenerate versions is not only limited to polynomials but also can be extended to transcendental functions like gamma functions (see [8]).
The Rota's theory of umbral calculus is based on linear functionals and differential operators (see [2][3][4][17][18][19][20][21]). The Sheffer sequences occupy the central position in the theory and are characterized by the generating functions where the usual exponential function enters. The motivation for the paper [6] starts from the question that what if the usual exponential function is replaced by the degenerate exponential functions (see (2)). As it turns out, it corresponds to replacing the linear functional by the family of λ -linear functionals (see (12)) and the differential operator by the family of λ -differential operators (see (14)). Indeed, these replacements lead us to define λ -Sheffer polynomials and degenerate Sheffer polynomials (see (16)).
As is well-known, poly-Bernoulli polynomials are defined in terms of polylogarithm functions. Recently, as degenerate versions of such functions and polynomials, degenerate polylogarithm functions were introduced and degenerate poly-Bernoulli polynomials were defined by means of the degenerate polylogarithm functions, and some properties of the degenerate poly-Bernoulli polynomials were investigated (see [13]).
The aim of this paper is to further study the degenerate poly-Bernoulli polynomials, which is a λ -Sheffer sequence and hence a degenerate Sheffer sequence, by using the above-mentioned λlinear functionals and λ -differential operators. In more detail, these polynomials are investigated by three different tools, namely a formula about representing a λ -Sheffer sequence by another (see (19)), a formula coming from the generating functions of λ -Sheffer sequences (see Theorem 1) and a formula arising from the definitions for λ -Sheffer sequences (see Theorems 6,7). Then, among other things, we represent the degenerate poly-Bernoulli polynomials by higher-order degenerate Bernoulli polynomials and by higher-order degenerate derangement polynomials. The rest of this section is devoted to recalling the necessary facts that are needed throughout the paper, which includes the 'λ -umbral calculus'.
For k ∈ Z, and 0 = λ ∈ R, the degenerate polylogarithm functions are defined by For any λ ∈ R, the degenerate exponential functions are given by (1) n,λ n! t n , (see [8]).
In [5], the degenerate poly-Bernoulli polynomials are defined by Kim-Kim as n,λ (0) are called the degenerate poly-Bernoulli numbers. It is well known that Carlitz's degenerate Bernoulli polynomials of order r are defined by t n n! , (see [1]).
As the inversion formula of (8), the degenerate Stirling numbers of first kind appear as the coefficients in the expansion [5]).
In view of (6), the degenerate derangement polynomials of order r(∈ N) are defined by t n n! , (see [10]).
We remark that the umbral calculus has long been studied by many people (see [2,3,4,6,[18][19][20]). For the rest of this section, we will recall the necessary facts on the λ -linear functionals, λ -differential operators and λ -Sheffer sequences, and so on. The details on these can be found in the recent paper [6].
Here S n,λ (x) is called the λ -Sheffer sequence for g(t), f (t) , which is denoted by S n,λ (x) ∼ g(t), f (t) λ . The sequence S n,λ (x) is the λ -Sheffer sequence for (g(t), f (t)) if and only if e y λ f (t) = ∞ ∑ n=0 S n,λ (y) t n n! , (see [6]), Then, from Theorem 16 of [6], we recall that In this paper, we study the properties of degenerate poly-Bernoulli polynomial arising from degenerate polylogarithmic function and give some identities of those polynomials associated with special polynomials which are derived from the properties of λ -Sheffer sequences.

From (20), we note that
Therefore, by (21), we obtain the following theorem.
Now, we observe that Therefore, by (25), we obtain the following theorem.
By (6) and (17), we note that For n ≥ 0, we let P n = p(x) ∈ C[x] deg p(x) ≤ n . Then P n is an (n + 1)-dimensional vector space over C.