On the error in a certain interpolation formula and in the Gaussian integration formula

We first prove a basic theorem that if a set of polynomials satisfies an orthogonality relation with respect to integration, the set also satisfies an orthogonality relation with respect to summation. This result is then used to derive the Gaussian quadrature formula. The orthogonality relations give rise to interpolation formulas and a connection between the coefficients in these interpolation formulas is established. Finally, the analysis is used to get an estimate of the error in the Gaussian quadrature formula. Some error coefficients are evaluated in the cases where the orthogonal polynomials are those of Jacobi, Laguerre, Hermite and Bessel.