Stieltjes- and Geronimus-type polynomials

摘要

This work is an extension to the formal case of a previous work by Monegato [5].A Stieltjes-type polynomial is a polynomial of degree (n + 1), En+1, orthogonal with respect to the functional \s{;c̃i = c(Pn(x)xi), i = 0, 1, 2,…\s}, where c is a functional defined by the series f(t) = ∑∞i=0citi. En+1 is of important use in the estimation of the error in Padé approximation. An iteration of the construction of En+1 is attempted. (Sections 1, 2, 3, 4).In the last section, we study the properties of the polynomial Sn associated with En+1 with respect to another functional c̄. Sn is called a Geronimus-type polynomial. It is shown that Sn verifies the system c(PnSnGk) = 0 for k = 1,…,n, where the Gk's are orthogonal with respect to c̄.