It was shown in Chapter 3 that there are three cases in which the eigenfunctions of a second-order ordinary differential operator that is symmetric with respect to a weight are polynomials. The polynomials in the three cases are the classical orthogonal polynomials: Hermite polynomials, Laguerre polynomials, and Jacobi polynomials.
Each of these sets of polynomials is an example of a family of polynomials that are orthogonal with respect to an inner product that is induced by a positive weight function on an interval of the real line. The basic theory of general orthogonal polynomials is covered in the first section: expressions as determinants, three-term recurrence relations, properties of the zeros, and so on. It is ...