Orthogonal Polynomials

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


where w(x) is a weighting function and delta_(mn) is the Kronecker delta. If c_n=1, 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 w(x) is the weighting function and


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

In the above table,


where Gamma(z) is a gamma function.

The roots of orthogonal polynomials possess many rather surprising and useful properties. For instance, let x_1<x_2<...<x_n be the roots of the p_n(x) with x_0=a and x_(n+1)=b. Then each interval [x_nu,x_(nu+1)] for nu=0, 1, ..., n contains exactly one root of p_(n+1)(x). Between two roots of p_n(x) there is at least one root of p_m(x) for m>n.

Let c be an arbitrary real constant, then the polynomial


has n+1 distinct real roots. If c>0 (c<0), these roots lie in the interior of [a,b], with the exception of the greatest (least) root which lies in [a,b] only for

 c<=(p_(n+1)(b))/(p_n(b))    (c>=(p_(n+1)(a))/(p_n(a))).

The following decomposition into partial fractions holds


where {xi_nu} are the roots of p_(n+1)(x) and


Another interesting property is obtained by letting {p_n(x)} be the orthonormal set of polynomials associated with the distribution dalpha(x) on [a,b]. Then the convergents R_n/S_n of the continued fraction


are given by


where n=0, 1, ... and


Furthermore, the roots of the orthogonal polynomials p_n(x) associated with the distribution dalpha(x) on the interval [a,b] are real and distinct and are located in the interior of the interval [a,b].

See also

Appell Polynomial, Charlier Polynomial, Chebyshev Polynomial of the First Kind, Chebyshev Polynomial of the Second Kind, Christoffel-Darboux Identity, Complete Biorthogonal System, Complete Orthogonal System, Ferrers' Function, Gegenbauer Polynomial, Gram-Schmidt Orthonormalization, Hahn Polynomial, Hermite Polynomial, Jack Polynomial, Jacobi Polynomial, Krawtchouk Polynomial, Laguerre Polynomial, Legendre Polynomial, Meixner-Pollaczek Polynomial, Multivariate Orthogonal Polynomials, Orthogonal Functions, Pollaczek Polynomial, Spherical Harmonic, Stieltjes-Wigert Polynomial, Zernike Polynomial

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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 q-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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Orthogonal Polynomials

Cite this as:

Weisstein, Eric W. "Orthogonal Polynomials." From MathWorld--A Wolfram Web Resource.

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