Novel identities for elementary and complete symmetric polynomials with diverse applications

Abstract

This article aims to present novel identities for elementary and complete symmetric polynomials and explore their applications, particularly to generalized Vandermonde and special tri-diagonal matrices. It also extends existing results on Jacobi polynomials $ P_n^{(\alpha, \beta)}(x) $ and introduces an explicit formula based on the zeros of $ P_{n-1}^{(\alpha, \beta)}(x) $. Several illustrative examples are included.

Keywords

• elementary symmetric polynomials
• complete symmetric polynomials
• schur convexity
• vandermonde matrix
• tri-diagonal matrix
• jacobi polynomials