Orthogonal polynomials are classes of Polynomials over a range which obey an
Orthogonality relation

(1) |

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. Abramowitz and Stegun (1972, pp. 774-775) give a table of common orthogonal polynomials.

In the above table, the normalization constant is the value of

(2) |

(3) |

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) |

(5) |

The following decomposition into partial fractions holds

(6) |

(7) |

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

(8) |

(9) | |||

(10) |

where , 1, ...and

(11) |

**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.

© 1996-9

1999-05-26