Explicit calculation of some polynomials introduced by W. Gautschi (Q1282226)

In his investigation of the numerical condition of methods for constructing orthogonal polynomials from modified moments, \textit{W. Gautschi} [Numer. Math. 48, 369-382 (1986; Zbl 0613.65014)] introduced a map \(G_n : {\mathbb R}^{2n} \to {\mathbb R}^{2n}\), from the set of modified moments \(m=(m_0, \dots, m_{2n-1})^T\) to the set of vectors \(\gamma=(\vec{\sigma},\vec{\tau})\) of weights and nodes of Gauss-Christoffel quadrature rules with respect to the measure of orthogonality. Gautschi also showed that the Frobenius norm of the Jacobian \(G_n'\) of \(G_n\) can be expressed in terms of the \(L^2\)-norm of a polynomial \(g_n\) of degree \(4n-2\) constructed from \(\gamma\). In this paper the author computes \(g_n\) in some special cases. First, for the Jacobi weight on \([-1,1]\) some identities and estimates are established. Then, formulas for Chebyshev weights of the first, second and third kind (i.e., \(|\alpha|=|\beta|=1/2\), \(\alpha, \beta >-1\)) are obtained.