# §18.2 General Orthogonal Polynomials

## §18.2(i) Definition

### ¶ Orthogonality on Intervals

Let be a finite or infinite open interval in . A system (or set) of polynomials , is said to be orthogonal on with respect to the weight function () if

Here is continuous or piecewise continuous or integrable, and such that for all .

It is assumed throughout this chapter that for each polynomial that is orthogonal on an open interval the variable is confined to the closure of unless indicated otherwise. (However, under appropriate conditions almost all equations given in the chapter can be continued analytically to various complex values of the variables.)

### ¶ Orthogonality on Finite Point Sets

Let be a finite set of distinct points on , or a countable infinite set of distinct points on , and , , be a set of positive constants. Then a system of polynomials , is said to be orthogonal on with respect to the weights if

when is infinite, or

when is a finite set of distinct points. In the former case we also require

whereas in the latter case the system is finite: .

More generally than (18.2.1)–(18.2.3), may be replaced in (18.2.1) by a positive measure , where is a bounded nondecreasing function on the closure of with an infinite number of points of increase, and such that for all . See McDonald and Weiss (1999, Chapters 3, 4) and Szegö (1975, §1.4).

## §18.2(ii) -Difference Operators

If the orthogonality discrete set is or , then the role of the differentiation operator in the case of classical OP’s (§18.3) is played by , the forward-difference operator, or by , the backward-difference operator; compare §18.1(i). This happens, for example, with the Hahn class OP’s (§18.20(i)).

If the orthogonality interval is or , then the role of can be played by , the central-difference operator in the imaginary direction (§18.1(i)). This happens, for example, with the continuous Hahn polynomials and Meixner–Pollaczek polynomials (§18.20(i)).

## §18.2(iii) Normalization

The orthogonality relations (18.2.1)–(18.2.3) each determine the polynomials uniquely up to constant factors, which may be fixed by suitable normalization.

If we define

and

18.2.7

then two special normalizations are: (i) orthonormal OP’s: , ; (ii) monic OP’s: .

## §18.2(iv) Recurrence Relations

As in §18.1(i) we assume that .

### ¶ First Form

Here , (), and () are real constants, and for . Then

18.2.9

### ¶ Second Form

Here , (), () are real constants, and (). Then

18.2.11

If the OP’s are orthonormal, then (). If the OP’s are monic, then ().

Conversely, if a system of polynomials satisfies (18.2.10) with (), then is orthogonal with respect to some positive measure on (Favard’s theorem). The measure is not necessarily of the form nor is it necessarily unique.

## §18.2(vi) Zeros

All zeros of an OP are simple, and they are located in the interval of orthogonality . The zeros of and separate each other, and if then between any two zeros of there is at least one zero of .

For illustrations of these properties see Figures 18.4.118.4.7.