Novel identities for elementary and complete symmetric polynomials with diverse applications

Abstract

<p>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.</p>