Polynomials arising in factoring generalized Vandermonde determinants: an algorithm for computing their coefficients

Abstract

We consider generalized Vandermonde determinants of the form ▪ where the x i are distinct points belonging to an interval [ a, b] of the real line, the index s stands for the order, the sequence μ consists of ordered integers 0 ≤ μ 1 < μ 2 < ⋯ < μ s . These determinants can be factored as a product of the classical Vandermonde determinant and a homogeneous symmetric function of the points involved, that is, a Schur function. On the other hand, we show that when x = x s in the resulting polynomial, depending on the variable x, the Schur function can be factored as a two-factors polynomial: the first is the constant ▪ times the (monic) polynomial ▪, while the second is a polynomial P M ( x) of degree M = m s−1 − s + 1. Our main result is then the computation of the coefficients of the monic polynomial P M ( x). We present an algorithm for the computation of the coefficients of P M based on the Jacobi-Trudi identity for Schur functions.