A probabilistic journey through the Newton-Girard identities

TL;DR

This paper offers a novel probabilistic interpretation of the Newton-Girard identities, connecting classical algebraic relations with limits of integrals and moments of probability distributions, and exploring their spectral implications.

Contribution

It introduces a probabilistic framework for understanding Newton-Girard identities, linking integrals, moments, and spectral theory in a unified approach.

Findings

Coefficients interpreted as limits of integrals over the unit cube

Ratios of moments of probability distributions relate to identities

Spectral implications discussed via Random Matrix Theory

Abstract

This article presents a pedagogical probabilistic exploration of the Newton-Girard identities. We show that the coefficients in these classical relations between power sums and elementary symmetric polynomials can be interpreted as the stable limits of integrals over the unit cube, and as ratios of moments of simple probability distributions. Several classes of integrals are studied, including trigonometric and multiplicative forms. In addition, we discuss the spectral implications via the Le Verrier-Souriau-Faddeev algorithm and Random Matrix Theory, providing a unified framework for the asymptotic algebraic behavior of these identities. While the identities are classical, the probabilistic interpretation of the limits of their normalized forms is the specific focus of the present work.