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) |
(3) |
(4) |
(5) |
(6) |
(7) |
Letting
allows the Chebyshev polynomials of the second kind to be written as
(8) |
(9) |
(10) |
See also Chebyshev Approximation Formula, Chebyshev Polynomial of the First Kind, Ultraspherical 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. ``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 Eric W. Weisstein