Orthogonal polynomials are classes of polynomials {p_n(x)}" src="http://mathworld.wolfram.com/images/equations/OrthogonalPolynomials/Inline1.gif" style="height:15px; width:43px" /> defined over a range  that obey an orthogonalityrelation

(1)

where  is a weighting function 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			{pi for n=0; 1/2pi otherwise" src="http://mathworld.wolfram.com/images/equations/OrthogonalPolynomials/Inline11.gif" style="height:50px; width:106px" />
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.

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

(4)

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

(5)

The following decomposition into partial fractions holds

(6)

where {xi_nu}" src="http://mathworld.wolfram.com/images/equations/OrthogonalPolynomials/Inline49.gif" style="height:15px; width:23px" /> are the roots of  and

			(7)
			(8)

Another interesting property is obtained by letting {p_n(x)}" src="http://mathworld.wolfram.com/images/equations/OrthogonalPolynomials/Inline57.gif" style="height:15px; width:43px" /> 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 .

REFERENCES:

Abramowitz, M. and Stegun, I. A. (Eds.). "Orthogonal Polynomials." Ch. 22 in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 771-802, 1972.

Arfken, G. "Orthogonal Polynomials." Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 520-521, 1985.

Chihara, T. S. An Introduction to Orthogonal Polynomials. New York: Gordon and Breach, 1978.

Gautschi, W.; Golub, G. H.; and Opfer, G. (Eds.) Applications and Computation of Orthogonal Polynomials, Conference at the Mathematical Research Institute Oberwolfach, Germany, March 22-28, 1998. Basel, Switzerland: Birkhäuser, 1999.

Iyanaga, S. and Kawada, Y. (Eds.). "Systems of Orthogonal Functions." Appendix A, Table 20 in Encyclopedic Dictionary of Mathematics. Cambridge, MA: MIT Press, p. 1477, 1980.

Koekoek, R. and Swarttouw, R. F. The Askey-Scheme of Hypergeometric Orthogonal Polynomials and its -Analogue. Delft, Netherlands: Technische Universiteit Delft, Faculty of Technical Mathematics and Informatics Report 98-17, 1-168, 1998.

Nikiforov, A. F.; Uvarov, V. B.; and Suslov, S. S. Classical Orthogonal Polynomials of a Discrete Variable. New York: Springer-Verlag, 1992.

Sansone, G. Orthogonal Functions. New York: Dover, 1991.

Szegö, G. Orthogonal Polynomials, 4th ed. Providence, RI: Amer. Math. Soc., pp. 44-47 and 54-55, 1975.