Representations of degenerate poly-Bernoulli polynomials

Abstract

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.