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A New Class of Relations for Homogeneous Symmetric Polynomials

Boris Y. Rubinstein

TL;DR

This work extends a novel class of relations from scalar partitions to general homogeneous symmetric polynomials expressible via complete Bell polynomials. It develops a unified Bell-polynomial framework, establishing vanishing conditions and rational structures for the derived polynomials $Z_n$ and $Y_{n-m}$, and applying them to Bernoulli polynomials and other families. A key result is the emergence of nonlinear relations among Bernoulli numbers, obtained by forcing $Z_n=0$ under a Bernoulli-inspired parameterization. The paper also outlines a power-sum–product approach that characterizes coefficients leading to universal vanishing, revealing both general structure and concrete low-degree solutions.

Abstract

Recently we introduced a new class of relations for Bernoulli symmetric polynomials. This manuscript shows that these relations are valid for arbitrary homogeneous symmetric polynomial. Analysis of these relations leads to the discovery of a new type of nonlinear relations for the Bernoulli numbers.

A New Class of Relations for Homogeneous Symmetric Polynomials

TL;DR

This work extends a novel class of relations from scalar partitions to general homogeneous symmetric polynomials expressible via complete Bell polynomials. It develops a unified Bell-polynomial framework, establishing vanishing conditions and rational structures for the derived polynomials and , and applying them to Bernoulli polynomials and other families. A key result is the emergence of nonlinear relations among Bernoulli numbers, obtained by forcing under a Bernoulli-inspired parameterization. The paper also outlines a power-sum–product approach that characterizes coefficients leading to universal vanishing, revealing both general structure and concrete low-degree solutions.

Abstract

Recently we introduced a new class of relations for Bernoulli symmetric polynomials. This manuscript shows that these relations are valid for arbitrary homogeneous symmetric polynomial. Analysis of these relations leads to the discovery of a new type of nonlinear relations for the Bernoulli numbers.

Paper Structure

This paper contains 11 sections, 54 equations.

Table of Contents

  1. Introduction
  2. Bernoulli polynomials as complete Bell polynomials
  3. Relations for complete Bell symmetric polynomials
  4. Structure of polynomials $Z_{n}({\bf x}^m)$
  5. Nonlinear relations for the Bernoulli numbers
  6. Relation for power sum products
  7. Expressions for $Z_{n}({\bf x}^m)$ with $n=3,4$
  8. Expressions for $Y_{n-2,\bf k}({\bf x}^2)$
  9. Expressions for $Y_{n-3,\bf k}({\bf x}^3)$
  10. Expressions for $Y_{n-4,\bf k}({\bf x}^4)$
  11. Coefficients $C_{n,\bf k}$ for $Y_{n-m}({\bf x}^m)=0$

Theorems & Definitions (3)

  • Conjecture 3.1
  • Conjecture 3.2
  • Conjecture 6.1