Complete homogeneous symmetric polynomials with repeating variables

TL;DR

This paper derives a new representation for complete homogeneous symmetric polynomials with repeated variables using partial fraction decomposition and Vandermonde matrix inversion, providing explicit formulas and alternative proofs.

Contribution

It introduces a novel explicit formula for these polynomials and offers an alternative proof via the inverse of the confluent Vandermonde matrix.

Findings

Derived a sum representation involving binomial coefficients and coefficients independent of m

Provided an alternative proof using the inverse of the confluent Vandermonde matrix

Enhanced understanding of symmetric polynomials with repeated variables

Abstract

We consider polynomials of the form $\operatorname{h}_m(y_1^{[\varkappa_1]},\ldots,y_n^{[\varkappa_n]})$, where $\operatorname{h}_m$ is the complete homogeneous polynomial of degree $m$ and $y_j^{[\varkappa_j]}$ denotes $y_j$ repeated $\varkappa_j$ times. Using the decomposition of the generating function into partial fractions we represent such polynomials in the form \[ \operatorname{h}_m(y_1^{[\varkappa_1]},\ldots,y_n^{[\varkappa_n]}) =\sum_{j=1}^n \sum_{r=1}^{\varkappa_j} \binom{r+m-1}{r-1} A_{y,\varkappa,j,r} y_j^m, \] where $A_{y,\varkappa,j,r}$ are some coefficients that do not depend on $m$. We also provide an alternative proof using the inverse of the confluent Vandermonde matrix.