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

We consider generalized Vandermonde determinants of the form V-s;mu(x(1),...x(s)) = /x(i)(muk)/, 1 less than or equal to i, k less than or equal to s, 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 mu consists of ordered integers 0 less than or equal to mu (1) < mu (2) < ... < mu (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 Pi (s-1)(i=1) x(i)(mu1) times the (monic) polynomial Pi (s-)(i=1)1 (x -x(i)): 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 PM(s). We present an algorithm for the computation of the coefficients of P-M based on the Jacobi-Trudi identity for Schur function