Polynomials arising in factoring generalized Vandermonde determinants Ⅲ : computation of their roots

摘要

Determinants of the form V_α(x) = |x_i~(α_j)|, i,j = 1, ..., N where x = (x_1,..., x_N) is formed by N distinct points belonging to some interval [a, b] of the real line and the α_j are ordered integers α_1 > α_2 > … > α_N > 0 are known as generalized Vandermonde determinants. These determinants were considered by Heineman at the end of the 1920s. The paper presents some results concerning univariate polynomials arising from V_α(x), by considering one of the x_i as an unknown. In particular we shall consider the problem of computing their roots by means of a family of iteration functions having a symmetric structure which is connected to the structure of our polynomials.