Complete homogeneous symmetric polynomials with repeating variables
Luis Angel González-Serrano, Egor A. Maximenko
TL;DR
The paper addresses the problem of evaluating and expanding complete homogeneous polynomials with repeating variables $h_m(y^{[\varkappa]})$. It develops two equivalent expansion frameworks: one via a partial fraction decomposition yielding coefficients $A_{y,\varkappa,s,r}$ and another via the inverse of a confluent Vandermonde matrix yielding coefficients $B_{y,\varkappa,s,r}$. The main results express $h_m(y^{[\varkappa]})$ as sums over $s$ and $r$ of these coefficients times powers of $y_s$, with $m$-dependent binomial factors, and they connect these expansions to Schur polynomials with repeating variables. The work provides explicit combinatorial, determinant-based, and derivative-based methods, offering computational schemes and insights into the structure of repeated-variable symmetric polynomials, with potential applications to banded Toeplitz matrices and the theory of confluent Vandermonde inverses.
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.