Computing the coefficients of a recurrence formula for numerical integration by moments and modified moments

To evaluate the class of integrals ∫1−1e−αxƒ(x) dx, where R † and the function f(x) is known only approximately in a tabular form, we wish to use a Gaussian quadrature formula. Nodes and weights have to be computed using the family of monic orthogonal polynomials, with respect to the weight function e−αx, obtained through the three-term recurrence relation Pk+1(x) = (x + Bk+1)Pk(x) − Ck+1Pk−1(x). To guarantee a good precision, we must evaluate carefully the values for the coefficients Bk+1 and Ck+1. Such evaluations are made completely formally through a Mathematica program to obtain great precision. A comparison between various methods, starting from moments and modified moments, is shown. Numerical results are also presented.