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

We consider generalized Vandermonde determinants of the form where the xi 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 ≤ μ1 < μ2 < ⋯ < μs. 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 = xs 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 PM(x) of degree M = ms−1 − s + 1. in result is then the computation of the coefficients of the monic polynomial PM(x). We present an algorithm for the computation of the coefficients of PM based on the Jacobi-Trudi identity for Schur functions.