Extraction of the $i^{th}$ Elementary Symmetric Polynomial from a Product in Binomial Form ${{m_1+...+m_n}\choose i}$

TL;DR

This paper presents a method to extract the $i^{th}$ elementary symmetric polynomial from a binomial product expansion using an alternating sum, providing a new algebraic approach.

Contribution

It introduces a novel algebraic technique to isolate elementary symmetric polynomials directly from binomial products.

Findings

Derived a formula for the $i^{th}$ elementary symmetric polynomial

Simplified extraction process from binomial expansions

Enhanced understanding of polynomial relationships

Abstract

The $i^{th}$ elementary symmetric polynomial of the set of $n$ variables $\mathcal{R}=\{m_1,m_2,m_3,...,m_n\}$ is isolated from the expansion of the $i^{th}$ binomial product ${{m_1+...+m_n}\choose i}$ via an alternating sum.