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Asymptotic representation of the weights in the Gauss quadrature formula. (English. Russian original) Zbl 0733.41033
Math. Notes 47, No. 4, 354-358 (1990); translation from Mat. Zametki 47, No. 4, 63-68 (1990).
Consider the quadrature formula of Gauss-Jacobi for Legendre’s weight dx on [0,1]. Given n, let $$\lambda_{n,j}$$, $$j=1,2,...,n$$, be the Christoffel coefficient corresponding to the zero $$x_{n,j}$$, $$j=1,2,...,n$$, of the nth Legendre orthogonal polynomial. Set cos $$\theta$$ $${}_{n,j}=x_{n,j}$$. It is proved that $\lambda_{n,j}=\frac{\pi}{2n+1}\sin \theta_{n,j}+O(\frac{1}{n^ 3\sin \theta_{n,j}}).$