Explicit expressions of the generalized Stieltjes polynomial

The existence and uniqueness of a Kronrod type extension to the well-known Gauss-Turan quadrature formulas were proved by Li (1994, pp.71-83). For the generalized Chebyshev weight functions and for the Gori-Micchelli weight function, we found explicit formulas of the corresponding generalized Stieltjes polynomials. General real Kronrod extensions of the Gaussian quadrature formulas with multiple nodes are introduced. In some cases, the explicit expressions of the polynomials, whose zeros are the nodes of the considered quadratures, are determined.