5 - Discrete orthogonal polynomials

Summary

In Chapter 4 we discussed the question of polynomials orthogonal with respect to a weight function, which was assumed to be a positive continuous function on a real interval. This is an instance of a measure. Another example is a discrete measure, for example, one supported on the integers with masses w_m, m = 0, ±1, ±2,… Most of the results of Section 4.1 carry over to this case, although if w_m is positive at only a finite number N + 1 of points, the associated function space has dimension N + 1 and will be spanned by orthogonal polynomials of degrees zero through N.

In this context the role of differential operators is played by difference operators. An analogue of the characterization in Theorem 3.4.1 is valid: up to normalization, the orthogonal polynomials that are eigenfunctions of a symmetric second-order difference operator are the “classical discrete polynomials,” associated with the names Charlier, Krawtchouk, Meixner, and Hahn.

The theory of the classical discrete polynomials can be developed in a way that parallels the treatment of the classical polynomials in Chapter 4, using a discrete analogue of the formula of Rodrigues.