Orthogonal Polynomials

Orthogonal polynomials are classes of polynomials  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
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  are the roots of  and

			(7)
			(8)

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 .

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.