Error bounds for Gaussian quadrature formulae with Legendre weight function for analytic integrands

In this paper we are concerned with a method for the numerical evaluation of the error terms in Gaussian quadrature formulae with the Legendre weight function. Inspired by the work of H. Wang and L. Zhang [J. Sci. Comput, 75 (2018), pp. 457-477] and applying the results of S. Notaris [Math. Comp., 75 (2006), pp. 1217-1231], we determine an explicit formula for the kernel. This explicit expression is used for finding the points on ellipses where the maximum of the modulus of the kernel is attained. Effective error bounds for this quadrature formula for analytic integrands are derived. Key words. Gauss quadrature formulae, Legendre polynomials, remainder term for analytic function, error bound AMS subject classifications. 65D32, 65D30, 41A55