An Algebraic Approach to Degenerate Bernoulli Numbers

TL;DR

This paper introduces a new algebraic framework for degenerate Bernoulli numbers and polynomials, revealing surprising links with matrices and fields, and highlighting their algebraic and computational properties.

Contribution

It develops an algebraic approach to degenerate Bernoulli numbers, uncovering new properties and connections with matrices and Galois fields.

Findings

New algebraic variant of degenerate Bernoulli polynomials introduced

Links established with circulant matrices and Galois fields

Properties enabling efficient computations identified

Abstract

In this work we study the properties of a new algebraic variant of the degenerate Bernoulli polynomial $\tilde{\beta}_{k}(m,x)$ and study the corresponding degenerate Bernoulli number $\tilde{\beta}_{k}(m,1)=m^{k}\beta_{k}(1/m)$, where $\beta_{k}(\lambda), \lambda\neq 0$ is the standard degenerate Bernoulli number. Our approach relies on a new algebraic framework for generating functions and the action of a symbolic evaluation function on powers of polynomials. We show that $\tilde{\beta}_{k}(m,x)$ displays surprising links with other mathematical objects (such as Circulant matrices and Galois fields) and enjoys many interesting algebraic, symmetric and dynamical properties that could be deployed to perform efficient computations.