Orthogonal Polynomials and Gaussian Quadrature with Nonclassical Weight Functions

The subject of orthogonal polynomials crops up in a variety of different applications in both analytic and numerical work. Every different weight function (defined below) implies a different set of orthogonal polynomials. For the few most common, "classical" weight functions, the resulting polynomials are tabulated in standard references. In this column we will show you how to construct your own orthogonal polynomials when the weight functiol1 in your problem is not one of the classical forms. As an application, we will show how you can use a set of such orthogonal polynomials to carry out an otherwise intractable Gaussian quadrature.