Interpolatory quadrature formulae with Chebyshev abscissae of the third or fourth kind

We consider interpolatory quadrature formulae, relative to the Legendre weight function on [−1, 1], having as nodes the zeros of the nth-degree Chebyshev polynomial of the third or fourth kind. Szegö has shown that the weights of these formulae are all positive. We derive explicit formulae for the weights, and subsequently use them to establish the convergence of the quadrature formulae for functions having a monotonic singularity at one or both endpoints of [−1, 1]. Moreover, we generate two new quadrature formulae, by adding 1, −1 to the sets of nodes considered previously, and show that these new formulae have almost all weights positive, exceptions occurring only among the weights corresponding to 1, −1. Also, we determine the precise degree of exactness of all the quadrature formulae in consideration, we obtain asymptotically optimal error bounds for these formulae, and show that almost all of them are nondefinite, exceptions occurring only among the formulae with a small number of nodes.