In Chapter 4 we discussed the question of polynomials that are 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 wm, m = 0,±1,±2, … Most of the results of Section 4.1 carry over to this case, although if wm 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 degree 0 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 5, using a discrete analogue of the formula of Rodrigues.

Working with functions of a complex variable and taking differences in the complex direction leads to two more characterizations: the Meixner–Pollaczek and continuous Hahn polynomials on the one hand, and the continuous dual Hahn and Wilson polynomials on the other. Except for the dual Hahn polynomials and the Racah polynomials, this completes the roster of polynomials in the “Askey scheme” – polynomials that can be expressed in terms of the generalized hypergeometric functions pFq, q+1 ≤ p = 1, 2, 3, 4.

Discrete weights and difference operators

Suppose that is a two-sided sequence of nonnegative numbers. The corresponding inner product

is well-defined for all real functions f and g for which the norms ∥f ∥w and ∥g∥w are finite, where

The norm and inner product depend only on the values taken by functions on the integers, although it is convenient to continue to regard polynomials, for example, as being defined on the line. Polynomials have a finite norm if and only if the even moments

are finite. If so, then orthogonal polynomials ψn can be constructed exactly as in the case of a continuous weight function.