Representations of degenerate poly-Bernoulli polynomials

TL;DR

This paper explores the properties of degenerate poly-Bernoulli polynomials, introducing new representations using higher-order degenerate Bernoulli and derangement polynomials via lambda-umbral calculus.

Contribution

It provides novel representations of degenerate poly-Bernoulli polynomials using higher-order degenerate Bernoulli and derangement polynomials, expanding their theoretical framework.

Findings

Represented degenerate poly-Bernoulli polynomials through higher-order degenerate Bernoulli polynomials.

Expressed degenerate poly-Bernoulli polynomials using higher-order degenerate derangement polynomials.

Utilized lambda-umbral calculus to derive new properties and formulas.

Abstract

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.