Polynomials arising in factoring generalized Vandermonde determinants II: A condition for monicity Original Research Article

In our previous paper [1], we observed that generalized Vandermonde determinants of the form Vn;ν(x1,…,xs) = |xiμk|, 1 ≤ i, k ≤ n, where the xi are distinct points belonging to an interval [a, b] of the real line, the index n stands for the order, the sequence μ consists of ordered integers 0 ≤ μ1 < μ2 < … < μn, can be factored as a product of the classical Vandermonde determinant and a Schur function. On the other hand, we showed that when x = xn, the resulting polynomial in x is a Schur function which can be factored as a two-factors polynomial: the first is the constant ∏i=1n−1xiμ1 times the monic polynomial ∏i=1n−1 (x − xi, while the second is a polynomial PM(x) of degree M = mn−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 PM (x) is monic.