Representing polynomials by degenerate Bernoulli polynomials

TL;DR

This paper develops explicit formulas for representing any polynomial using degenerate Bernoulli and higher-order degenerate Bernoulli polynomials, utilizing umbral calculus techniques.

Contribution

It introduces new explicit formulas for polynomial representation in terms of degenerate Bernoulli polynomials, expanding their applicability.

Findings

Derived explicit formulas for polynomial representation

Utilized umbral calculus for formula derivation

Provided illustrative examples of the representations

Abstract

In this paper, we consider the problem of representing any polynomial in terms of the degenerate Bernoulli polynomials and more generally of the higher-order degenerate Bernoulli polynomials. We derive explicit formulas with the help of umbral calculus and illustrate our results with some examples.