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Orthogonal Polynomials

Orthogonal polynomials are classes of polynomials ${p_n(x)}$ defined over a range that obey an orthogonality relation

(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			${(2^(1-2alpha)piGamma(n+2alpha))/(n!(n+alpha)[Gamma(alpha)]^2) for alpha!=0; (2pi)/(n^2) for alpha=0.$
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 ${xi_nu}$ are the roots of and

			(7)
			(8)

Another interesting property is obtained by letting ${p_n(x)}$ 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.

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Weisstein, Eric W. "Orthogonal Polynomials." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/OrthogonalPolynomials.html