Representations of degenerate poly-Bernoulli polynomials

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 of their properties were investigated. The aim of this paper is to further study some properties of the degenerate poly-Bernoulli polynomials by using three formulas coming from the recently developed ‘λ-umbral calculus’. In more detail, among other things, we represent the degenerate poly-Bernoulli polynomials by higher-order degenerate Bernoulli polynomials and by higher-order degenerate derangement 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 [2]). 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 [9, 10, 12-14, 17, 19, 21] 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 [13]). The Rota's theory of umbral calculus is based on linear functionals and differential operators (see [3-7, 20, 23-27]). 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 [10] starts from the question that what if the usual exponential function is replaced with the degenerate exponential functions (see (2)). As it turns out, it corresponds to replacing the linear functional with the family of λlinear functionals (see (12)) and the differential operator with the family of λ-differential operators (see (14)). Indeed, these replacements lead us to defining 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 [17]).
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 λ-linear 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' .
In view of (6), the degenerate derangement polynomials of order r(∈ N) are defined by t n n! (see [12,19]).
We remark that the umbral calculus has long been studied by many people (see [3-7, 20, 23-27]). For the rest of this section, we recall the necessary facts on the λ-linear functionals, λ-differential operators, λ-Sheffer sequences, and so on. The details on these can be found in the recent paper [10].
Let C be the field of complex numbers, and let By (12), we get where δ n,k is the Kronecker symbol. The λ-differential operators on P are defined by For f (t) = ∞ k=0 a k t k k! ∈ F , and by (14), we get e y λ (t) λ (x) n,λ = (x + y) n,λ (n ≥ 0), (see [10]).
Let f (t) be a delta series, and let g(t) be an invertible series. Then there exists a unique sequence S n,λ (x)(deg S n,λ (x) = n) of polynomials satisfying the orthogonality conditions 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 S n,λ (y) t n n! (see [10]) (17) for all y ∈ C, where f (t) is the compositional inverse of f (t) such that f (f (t)) = f (f (t)) = t. Let S n,λ (x) ∼ (g(t), f (t)) λ . Then, from Theorem 16 of [10], we recall that For S n,λ (x) ∼ (g(t), f (t)) λ , r n,λ (x) ∼ (h(t), l(t)) λ , we have S n,λ (x) = n k=0 C n,k r k,λ (x), (n ≥ 0), (see [10]), where Finally, we note that λ-umbral calculus has some merit over umbral calculus when dealing with λ-Sheffer sequences. As one example, we illustrate this with the problem of representing the degenerate Bernoulli polynomial β n,λ (x) in terms of the degenerate falling factorials (x) k,λ . As before, let f (t) and g(t) be respectively a delta series and an invertible series. First, we recall that S n (x) is Sheffer for (g(t), f (t)) denoted by S n (x) ∼ (g(t), f (t)) if and only if 1 g(f (t)) e xf (t) = ∞ n=0 S n (x) t n n! (see [24]).
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 (5) and (17), we have By Theorem 1 applied to (25), we get the following corollary.

Corollary 2
For n ≥ 0, we have Here we remark that Corollary 2 can also be obtained by combining (15) and the first line of (27).