Orthogonal Polynomials
Orthogonal polynomials are classes of Polynomials 
over a range 
which obey anOrthogonality relation
 | (1) |
is a Weighting Function and 
is the Kronecker Delta. If 
, then the Polynomialsare not only orthogonal, but orthonormal.Orthogonal polynomials have very useful properties in the solution of mathematical and physical problems. Just asFourier 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 importantDifferential Equations. Orthogonal polynomials are especially easy to generate usingGram-Schmidt Orthonormalization. Abramowitz and Stegun (1972, pp. 774-775) give a table of common orthogonalpolynomials.
| Type | Interval | |  |
| Chebyshev Polynomial of the First Kind |  |  |  |
| Chebyshev Polynomial of the Second Kind | |  |  |
| Hermite Polynomial |  |  |  |
| Jacobi Polynomial |  |  |  |
| Laguerre Polynomial |  |  | 1 |
| Laguerre Polynomial (Associated) | |  |  |
| Legendre Polynomial | | 1 |  |
| Ultraspherical Polynomial | |  |  |
In the above table, the normalization constant is the value of
 | (2) |
 | (3) |
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) |
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) |
are the Roots of and |  |  | |
|  | (7) |
Another interesting property is obtained by letting be the orthonormal set of Polynomialsassociated with the distribution 
on . Then the Convergents 
of theContinued Fraction
 | (8) |
 | |  | |
|  | (9) | |
 | |  | (10) |
where
, 1, ...and | (11) |
associated with the distribution onthe interval are Real and distinct and are located in the interior of the interval .See also Chebyshev Polynomial of the First Kind, Chebyshev Polynomial of the Second Kind, Gram-SchmidtOrthonormalization, Hermite Polynomial, Jacobi Polynomial, Krawtchouk Polynomial, LaguerrePolynomial, Legendre Polynomial, Orthogonal Functions, Spherical Harmonic, UltrasphericalPolynomial, Zernike Polynomial
References
Abramowitz, M. and Stegun, C. 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.
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.
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.
- z-Distribution
- Zeckendorf Representation
- Zeckendorf's Theorem
- Zeeman's Paradox
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- Zeisel Number
- Zeno's Paradoxes
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- Zermelo-Fraenkel Set Theory
- Zermelo's Axiom of Choice
- Zermelo Set Theory
- Zernike Polynomial
- Zero
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- Zero (Root)
- Zero-Sum Game
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- Zeta Function
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