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: the 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 w on an interval of the real line. The basic theory of general orthogonal polynomials of this type is covered in this chapter. This includes expressions as determinants, three-term recurrence relations, properties of the zeros, and basic asymptotics. It is shown that under a certain condition on the weight w(x), which is satisfied in each of the three classical cases, each element of the space Lw2 can be expanded in a series using the orthogonal polynomials, analogous to the Fourier series expansion. These results carry over to more general measures than those of the form w(x)dx.

Orthogonal polynomials occur naturally in connection with approximating the Stieltjes transform of the weight function or measure. This transform can also be viewed as a continued fraction.

The central role played by the three-term recurrence relations leads to the question: do such relations characterize orthogonal polynomials? The (positive) answer is known as Favard's theorem.

The chapter concludes with a brief discussion of the asymptotic distribution of zeros.

Weight functions and orthogonality

Let w(x) be a positive weight function on an open interval I =(a,b) and assume that the moments

are finite.

Let Δ−1 = 1 and let Δn, n ≥ 0, be the determinant

The associated quadratic form

is positive definite, so the determinant Δn is positive.

Consider the Hilbert space Lw2, with inner product

The polynomial

is orthogonal to xm, m < n, while (Qn, xn) = Δn. To see this, expand the determinant (4.1.2) along the last column. Computing the inner product of Qn with xm results in a determinant in which the last column of the determinant (4.1.1) has been replaced by column m+1 of the same determinant.