New quadrature formulas based on the zeros of the Chebyshev polynomials of the second kind
Quadrature formulas are derived for integrals \[ \int^{+1}_{-1} f(x)w(x)\,dx,\quad w(x) = (1 - x^2)^{1/2},\text{ or }w(x) = (1 - x^2)^{-1/2} \] which are accurate for polynomials of respective degree \(2(s+1)n +2s -1\), \(2(s+1)n+ 2s+1\), \(s,n\in\mathbb{N}\). The formulas employ divided differences at the zeros of \((1-x^2)Q_n(x)\) where \(Q_n(x)\) is an \(n\)-th degree Chebyshev polynomial of the second kind. When \(s = 0\) and \(n\) is replaced by \(2n -1\) the second formula reduces to a formula given by \textit{A. K. Varma} and \textit{E. Landau} [ibid. 30, No. 3--6, 213--220 (1995; Zbl 0833.41027)] exact for polynomials of degree \(4n -1\). The formulas are related to the Gauss-Turán quadrature formula.
scientific article; zbMATH DE number 2082283