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A probabilistic journey through the Newton-Girard identities

Jean-Christophe Pain

TL;DR

This work reframes the Newton–Girard identities through a probabilistic and integral lens by modeling $(X_i)$ as i.i.d. uniform variables on $(0,1)$. The key finding is that normalized power sums and elementary symmetric polynomials converge to simple constants, e.g. $ rac{S_eta^{(n)}}{S_1^{(n)}} o rac{2}{eta+1}$ and $ rac{ abla_k^{(n)}}{(S_1^{(n)})^k} o rac{1}{k!}$, yielding a limiting form of the Newton–Girard relations: $ rac{1}{(k-1)!}= obreak obreak obreak obreak obreak obreak extstyle\sum_{j=1}^k (-1)^{j-1} rac{2^j}{(k-j)!(j+1)}$. The paper then develops the Schwerdtfeger framework for matrix functions, introduces Frobenius covariants, and shows how the Le Verrier–Souriau–Faddeev algorithm computes characteristic polynomials via traces, linking spectral theory with probabilistic limits. Analytic extensions generalize these integral identities to trigonometric, multiplicative, and density-generalized forms, with convergence results and spectral interpretations via Lagrange–Sylvester interpolation. Collectively, the results unify combinatorial, probabilistic, and spectral methods and suggest practical probabilistic approximations for high-degree symmetric polynomials and connections to 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.

A probabilistic journey through the Newton-Girard identities

TL;DR

This work reframes the Newton–Girard identities through a probabilistic and integral lens by modeling as i.i.d. uniform variables on . The key finding is that normalized power sums and elementary symmetric polynomials converge to simple constants, e.g. and , yielding a limiting form of the Newton–Girard relations: . The paper then develops the Schwerdtfeger framework for matrix functions, introduces Frobenius covariants, and shows how the Le Verrier–Souriau–Faddeev algorithm computes characteristic polynomials via traces, linking spectral theory with probabilistic limits. Analytic extensions generalize these integral identities to trigonometric, multiplicative, and density-generalized forms, with convergence results and spectral interpretations via Lagrange–Sylvester interpolation. Collectively, the results unify combinatorial, probabilistic, and spectral methods and suggest practical probabilistic approximations for high-degree symmetric polynomials and connections to 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.
Paper Structure (20 sections, 6 theorems, 77 equations)

This paper contains 20 sections, 6 theorems, 77 equations.

Table of Contents

  1. Introduction
  2. A probabilistic model and integral limit theorems
  3. Asymptotics of elementary symmetric polynomials
  4. Normalized Newton-Girard identities and their limiting form
  5. Matrix functions and the Schwerdtfeger framework
  6. Frobenius covariants and spectral decomposition
  7. The Schwerdtfeger formula
  8. From spectral theory to characteristic polynomials
  9. Application: Le Verrier-Souriau-Faddeev algorithm
  10. Analytic extensions and convergence proofs
  11. Motivation and overview
  12. Convergence of trigonometric ratio and Selberg integrals
  13. Multiplicative numerators and geometric means
  14. Generalization to arbitrary power ratios
  15. Extension to arbitrary probability densities
  16. ...and 5 more sections

Key Result

Theorem 2.1

Let $\alpha \ge 1$. Then

Theorems & Definitions (13)

  • Theorem 2.1: Integral limit theorem
  • proof
  • Theorem 3.1: Normalized asymptotics
  • proof
  • Proposition 4.1: Limiting identity
  • proof
  • Example 6.1
  • Theorem 7.1
  • proof
  • Lemma C.1
  • ...and 3 more