Orthogonal Polynomials

Orthogonal polynomials are classes of polynomials defined over a range that obey an orthogonality relation

(1)

where is a nonnegative weighting function, has polynomial degree , , and is the Kronecker delta. If , then the polynomials are not only orthogonal, but orthonormal.

Orthogonal polynomials have very useful properties in the solution of mathematical and physical problems. Just as Fourier series provide a convenient method of expanding a periodic function in a series of linearly independent terms, orthogonal polynomials provide a natural way to solve, expand, and interpret solutions to many types of important differential equations. Orthogonal polynomials are especially easy to generate using Gram-Schmidt orthonormalization.

A table of common orthogonal polynomials is given below, where is the weighting function and

(2)

(Abramowitz and Stegun 1972, pp. 774-775).

polynomial	interval
Chebyshev polynomial of the first kind
Chebyshev polynomial of the second kind
Gegenbauer polynomial
Hermite polynomial
Jacobi polynomial
Laguerre polynomial			1
generalized Laguerre polynomial
Legendre polynomial		1

In the above table,

(3)

where is a gamma function.

More generally, orthogonality may be defined by a positive measure on the real line with finite moments of all orders, not concentrated on any finite set. The measure need not be given by a weighting function. The corresponding monic polynomials satisfy

(4)

where , , is real, and for , with . Favard's theorem supplies the converse. After normalization, this recurrence relation is represented by a Jacobi matrix. Its finite leading blocks have the zeros of the orthogonal polynomials as their eigenvalues, giving the nodes of Gaussian quadrature.

Orthogonality alone does not guarantee a complete orthogonal system in all square-integrable functions. For a finite positive measure on a bounded interval, polynomials are dense in , so the normalized sequence is complete. On unbounded intervals, additional hypotheses are needed. These distinctions matter when interpreting expansions in orthogonal polynomials as generalized Fourier series.

For the following properties, normalize the leading coefficients to be positive.

The roots of orthogonal polynomials possess many rather surprising and useful properties. For instance, let be the roots of the with and . Then each interval for , 1, ..., contains exactly one root of . Between two roots of there is at least one root of for .

Let be an arbitrary real constant, then the polynomial

(5)

has distinct real roots. If (), these roots lie in the interior of , with the exception of the greatest (least) root which lies in only for

(6)

The following decomposition into partial fractions holds

(7)

where are the roots of and

			(8)
			(9)

Another interesting property is obtained by letting be the orthonormal set of polynomials associated with the distribution on . Then the convergents of the continued fraction

(10)

are given by

			(11)
			(12)
			(13)

where , 1, ... and

(14)

Furthermore, the roots of the orthogonal polynomials associated with the distribution on the interval are real and distinct and are located in the interior of the interval .

The Jacobi polynomial, Laguerre polynomial, and Hermite polynomial families and their special cases, in their ranges of real orthogonality, also have a common differential structure. Write and . In monic normalization, they satisfy

(15)

for nonnegative integers (NIST, §18.8). For , let and be the coefficients of and in , respectively. Then

(16)