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

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 oneroot
of .
Between two roots of there is at least one root of
for .

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

(9)

are given by

(10)

(11)

(12)

where ,
1, ... and

(13)

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 .