Complete Homogeneous Symmetric Polynomials with Repeating Variables

In this paper, we consider complete homogeneous symmetric polynomials evaluated for variables repeated with given multiplicities; in other words, we consider polynomials obtained from complete homogeneous polynomials by identifying some subsets of their variables. We represent such polynomials as linear combinations of the powers of the variables, where all exponents are equal to the degree of the original polynomial. We give two proofs for the proposed formulas: the first proof uses the decomposition of the generating function into partial fractions, and the second involves the inverse of the confluent Vandermonde matrix. We also discuss the computational feasibility of the proposed formulas.