On Gautschi’s conjecture for generalized Gauss–Radau and Gauss–Lobatto formulae

Recently, Gautschi introduced so-called generalized Gauss–Radau and Gauss–Lobatto formulae which are quadrature formulae of Gaussian type involving not only the values but also the derivatives of the function at the endpoints. In the present note we show the positivity of the corresponding weights; this positivity has been conjectured already by Gautschi.As a consequence, we establish several convergence theorems for these quadrature formulae.