On the Bernoulli Numbers via the Newton-Girard Identities

TL;DR

This paper derives formulas for Bernoulli numbers using Newton-Girard identities, connecting them to the Riemann zeta function and combinatorial structures like plane trees, with proofs involving symmetric groups.

Contribution

It introduces a novel combinatorial approach to Bernoulli numbers via Newton-Girard identities and provides new formulas and properties for related polynomials.

Findings

Established positivity of polynomial coefficients.

Derived a combinatorial formula for Bernoulli numbers as sums over plane trees.

Provided a combinatorial proof of Newton-Girard identities using symmetric groups.

Abstract

We prove formulas for the Bernoulli numbers by using the Newton-Girard identities to evaluate the Riemann zeta function at positive even integers. To do this, we define a sequence of positive integers, a sequence of polynomials, and a sequence of linear operators on the space of functions. We prove properties of these polynomials, such as the positivity of their coefficients, and present a combinatorial formula for the Bernoulli numbers as a positive sum over plane trees which can be generalized as a transform of sequences. We also combinatorially prove the Newton-Girard identities using the symmetric group.