A modified set of Chebyshev Polynomials defined by a slightly different Generating Function. Used to develop four-dimensional Spherical Harmonics in angular momentum theory. They are also a special case of the Ultraspherical Polynomial with . The Chebyshev polynomials of the second kind are illustrated above for and , 2, ..., 5.

The defining Generating Function of the Chebyshev polynomials of the second kind is

(1) |

(2) |

Multiply (2) by ,

(3) |

(4) |

(5) |

(6) |

where is the Floor Function and is the Ceiling Function, or in terms of a Determinant

(7) |

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

(8) |

(9) |

(10) |

**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.

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