On the Bernoulli Numbers via the Newton-Girard Identities

2019/12/12 by Mario DeFranco, DeFranco, Mario
Mathematics · Physics and Astronomy · #Advanced Combinatorial Mathematics #Advanced Mathematical Identities #Advanced Mathematical Theories and Applications #FOS: Mathematics #Number Theory (math.NT) #math.NT

paper · pdf · doi:10.48550/arxiv.1912.06009

arxiv created 2019/12/12 · openalex publication_date 2019/12/12 · arxiv updated 2019/12/13 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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.

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