New interpolatory quadrature formulae with Gegenbauer abscissae. (Q1421200)

The author studies quadrature formulae of the following forms: \[ \int\limits^1_{-1} f(t) dt=w_0^{(+)}f(1)+ \sum^n_{\nu=1} w_\nu^{(+)}f(\tau_\nu)+R^{(+)}_n(f) \] \[ \int\limits^1_{-1}f(t)dt=\sum^m_{\nu=1}w_n^{(-)}f(\tau_\nu)+ w_{n+1}^{(-)}f(-1)+R_n^{(-)}(f) \] where \(\tau_\nu\) are the zeros of the Gegenbauer polynomial \(P^{(\lambda)}_n\), \(\lambda>-\frac{1}{2}\); and \(n\) is a given even positive integer. The convergence or non convergence of the above formulae is established, the precise degree of exactness is determined and also optimal error bounds are obtained.