Two closed forms for the Bernoulli polynomials

TL;DR

This paper presents two new closed-form expressions for Bernoulli polynomials involving Stirling numbers and determinants, enhancing the understanding of their combinatorial structure.

Contribution

It introduces novel closed-form formulas for Bernoulli polynomials using Stirling numbers and determinants, which were not previously known.

Findings

Derived two explicit formulas for Bernoulli polynomials.

Connected Bernoulli polynomials to Stirling numbers and determinants.

Provided new combinatorial representations of Bernoulli numbers.

Abstract

In the paper, the authors find two closed forms involving the Stirling numbers of the second kind and in terms of a determinant of combinatorial numbers for the Bernoulli polynomials and numbers.