On Gauss-type quadrature formulas with prescribed nodes anywhere on the real line

In this paper, quadrature formulas on the real line with the highest degree of accuracy, with positive weights, and with one or two prescribed nodes anywhere on the interval of integration are characterized. As an application, the same kind of rules but with one or both (finite) endpoints being fixed nodes and one or two more prescribed nodes inside the interval of integration are derived. An efficient computation of such quadrature formulas is analyzed by considering certain modified Jacobi matrices. Some numerical experiments are finally presented.