A set of Orthogonal Polynomials defined as the solutions to the Chebyshev Differential Equation and denoted . They are used as an approximation to a Least Squares Fit, and are a special case of the Ultraspherical Polynomial with . The Chebyshev polynomials of the first kind are illustrated above for and , 2, ..., 5.

The Chebyshev polynomials of the first kind can be obtained from the generating functions

(1) |

(2) |

(3) |

(4) |

(5) |

(6) |

(7) |

(8) |

(9) |

(10) |

(11) |

(12) |

(13) |

(14) |

(15) |

Using Gram-Schmidt Orthonormalization in the range (,1) with Weighting Function
gives

(16) | |||

(17) | |||

(18) |

etc. Normalizing such that gives

The Chebyshev polynomial of the first kind is related to the Bessel Function of the First Kind and
Modified Bessel Function of the First Kind by the relations

(19) |

(20) |

Letting
allows the Chebyshev polynomials of the first kind to be written as

(21) |

(22) |

(23) |

(24) |

The Polynomial

(25) |

(26) |

**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. ``Chebyshev (Tschebyscheff) Polynomials'' and ``Chebyshev Polynomials--Numerical Applications.''
§13.3 and 13.4 in
*Mathematical Methods for Physicists, 3rd ed.* Orlando, FL: Academic Press, pp. 731-748, 1985.

Beeler, M.; Gosper, R. W.; and Schroeppel, R. *HAKMEM.* Cambridge, MA: MIT Artificial Intelligence Laboratory, Memo AIM-239, Feb. 1972.

Iyanaga, S. and Kawada, Y. (Eds.). ``Cebysev (Tschebyscheff) Polynomials.'' Appendix A, Table 20.II in
*Encyclopedic Dictionary of Mathematics.* Cambridge, MA: MIT Press, pp. 1478-1479, 1980.

Rivlin, T. J. *Chebyshev Polynomials*. New York: Wiley, 1990.

Spanier, J. and Oldham, K. B. ``The Chebyshev Polynomials and .''
Ch. 22 in *An Atlas of Functions.* Washington, DC: Hemisphere, pp. 193-207, 1987.

© 1996-9

1999-05-26