The remainder term for analytic functions of Gauss-Lobatto quadratures

For analytic functions the remainder term of Gauss-Lobatto quadrature rules can be represented as a contour integral with a complex kernel. In this paper the kernel is studied on elliptic contours for the Chebyshev weight functions of the second, third, and fourth kind. Starting from explicit expressions of the corresponding kernels the location of their maximum modulus on ellipses is determined. This gives an answer to Gautschiʹs (1991) conjectures on the location of the maximum point for these kernels. Finally, some extensions to Gaussian quadrature rules as well as numerical examples are given.