Polynomials arising in factoring generalized Vandermonde determinants II: a condition of monicity

In our previous paper [1], we observed that generalized Vandermonde determinants of the form V_{n;u}(x(1),...,x(s)) = \x(i)(muk)\, i less than or equal to i, k less than or equal to n, where the x(i) are distinct points belonging to an interval [a, b] of the real line, the index n stands for the order, the sequence mu consists of ordered integers 0 less than or equal to mu(1) < mu(2) <...< mu(n), can be factored as a product of the classical Vandermoncle determinant and a Schur function. On the other hand, we showed that when x = x,, the resulting polynomial in x is a Schur function which can be factored as a two-factors polynomial: the first is the constant Pi(i=1)(n-1) x(i)(mu1) times the monic polynomial Pi(i=1)(n-1)(x - x(i)), while the second is a polynomial P-M(x) of degree M=m(n-1)-n+1. In this paper, we first present a typical application in which these factorizations arise and then we discuss a condition under which the polynomial P-M(x) is monic.