An asymptotic expansion for the weights of Gaussian quadrature formulae
This paper is concerned with a complete asymptotic expansion of the weights \(a_\nu \equiv a^{(\lambda)}_{\nu, n}\), \(\nu = 1, 2, \dots, n\), of the Gaussian quadrature formula \[ Q_n^{(\lambda)} [f] = \sum^n_{\nu = 1} a_\nu f(x_\nu), \] based on the zeros \(x_\nu \equiv x_{\nu, n}^{(\lambda)}\), \(\nu = 1, 2, \dots, n\), of the \(n\)-th ultraspherical polynomial \(P_n^{(\lambda)} (x)\). The first result, which holds only for integer values of \(\lambda\), is given by an interesting representation of \(Q_n^{(\lambda)}\) as a simple elementary function of the nodes \(x_\nu\). The generalization of this representation to noninteger values of \(\lambda > - 1/2\) yields a very important result which states that, by setting \[ a_\mu (\lambda) = \left( {(2 \mu)! \over 2^\mu \mu!} \right)^2 {\lambda - 1 + \mu \choose 2 \mu}, \quad N = n + \lambda , \] we have, as \(n \to \infty\) and for arbitrary \(m\), \[ a_\nu = {\pi \over N} (1 - x^2_\nu)^\lambda \left[ 1 + \sum^{m - 1}_{\mu = 1} {a_\mu (\lambda) \over (1 - x_\nu^2)^\mu} \prod^\mu_{k = 1} {1 \over N^2 - k^2} + O \bigl( n^{- 2m} (1 - x^2_n u)^{-m} \bigr) \right]. \]
scientific article; zbMATH DE number 878117