| 释义 |
Chebyshev Polynomial of the Second KindA 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) |
 and  . To see the relationship to a Chebyshev Polynomial of the First Kind ( ), take ,
Multiply (2) by  ,
 | (3) |
 | (4) |
 | (5) |
where  is the Floor Function and  is the Ceiling Function, or in terms of aDeterminant
 | (7) |
Letting  allows the Chebyshev polynomials of the second kind to be written as
 | (8) |
 | (9) |
 | (10) |
 is a Chebyshev Polynomial of the First Kind. Note that  is therefore not a Polynomial.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.
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