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 shown that under a certain condition on the weight, which is satisfied in each of the three classical cases, each element of the L2 space can be expanded in a series using the orthogonal polynomials, analogous to the Fourier series expansion.
We then examine some features common to the three classical cases, including Rodrigues formulas and representations as integrals. In succeeding sections each of the three classical cases is considered in more detail, as well as some special cases of Jacobi polynomials (Legendre and Chebyshev polynomials). The question of pointwise convergence of the expansion in orthogonal polynomials is addressed.
Finally we return to integral representations and the construction of a second solution of each of the differential equations.