New interpolatory quadrature formulae with Gegenbauer abscissae

摘要

We study interpolatory quadrature formulae, relative to the Legendre weight function on [−1,1], having as nodes the zeros of the nth degree Gegenbauer polynomial plus one of the points 1 or −1. In particular, we establish the convergence or nonconvergence for continuous and Riemann integrable functions on [−1,1], we determine the precise degree of exactness, we obtain asymptotically optimal error bounds, and we examine the definiteness or nondefiniteness of these formulae. In addition, we try, numerically, to investigate the question of positivity of all quadrature weights and to fill a gap regarding the definiteness or nondefiniteness property. The paper concludes by comparing the results derived here for the quadrature formulae in question with those that have been previously obtained for the corresponding open and closed interpolatory formulae with Gegenbauer abscissae.