Chebyshev Polynomial of the Second Kind
المؤلف:
Abramowitz, M. and Stegun, I. A.
المصدر:
"Orthogonal Polynomials." Ch. 22 in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover
الجزء والصفحة:
...
3-8-2019
4533
Chebyshev Polynomial of the Second Kind
A modified set of Chebyshev polynomials defined by a slightly different generating function. They arise in the development of four-dimensional spherical harmonics in angular momentum theory. They are a special case of the Gegenbauer polynomial with 
. They are also intimately connected with trigonometric multiple-angle formulas. The Chebyshev polynomials of the second kind are denoted 
, and implemented in the Wolfram Language as ChebyshevU[n, x]. The polynomials 
are illustrated above for ![x in [-1,1]](http://mathworld.wolfram.com/images/equations/ChebyshevPolynomialoftheSecondKind/Inline4.gif)
and 
, 2, ..., 5.
The first few Chebyshev polynomials of the second kind are
When ordered from smallest to largest powers, the triangle of nonzero coefficients is 1; 2; 
, 4; 
, 8; 1, 
, 16; 6, 
, 32; ... (OEIS A053117).
The defining generating function of the Chebyshev polynomials of the second kind is
for 
and 
. To see the relationship to a Chebyshev polynomial of the first kind 
, take 
of equation (9) to obtain
Multiplying (◇) by 
then gives
 |
(12)
|
and adding (12) and (◇) gives
This is the same generating function as for the Chebyshev polynomial of the first kind except for an additional factor of 
in the denominator.
The Rodrigues representation for 
is
![U_n(x)=((-1)^n(n+1)sqrt(pi))/(2^(n+1)(n+1/2)!(1-x^2)^(1/2))(d^n)/(dx^n)[(1-x^2)^(n+1/2)].](http://mathworld.wolfram.com/images/equations/ChebyshevPolynomialoftheSecondKind/NumberedEquation2.gif) |
(15)
|
The polynomials can also be defined in terms of the sums
where 
is the floor function and ![[x]](http://mathworld.wolfram.com/images/equations/ChebyshevPolynomialoftheSecondKind/Inline63.gif)
is the ceiling function, or in terms of the product
![U_n(x)=2^nproduct_(k=1)^n[x-cos((kpi)/(n+1))]](http://mathworld.wolfram.com/images/equations/ChebyshevPolynomialoftheSecondKind/NumberedEquation3.gif) |
(18)
|
(Zwillinger 1995, p. 696).
also obey the interesting determinant identity
 |
(19)
|
The Chebyshev polynomials of the second kind are a special case of the Jacobi polynomials 
with 
,
where 
is a hypergeometric function (Koekoek and Swarttouw 1998).
Letting 
allows the Chebyshev polynomials of the second kind to be written as
![U_n(x)=(sin[(n+1)theta])/(sintheta).](http://mathworld.wolfram.com/images/equations/ChebyshevPolynomialoftheSecondKind/NumberedEquation5.gif) |
(22)
|
The second linearly dependent solution to the transformed differential equation is then given by
![W_n(x)=(cos[(n+1)theta])/(sintheta),](http://mathworld.wolfram.com/images/equations/ChebyshevPolynomialoftheSecondKind/NumberedEquation6.gif) |
(23)
|
which can also be written
 |
(24)
|
where 
is a Chebyshev polynomial of the first kind. Note that 
is therefore not a polynomial.
The triangle of resultants 
is given by ![<span style=]()
{0}" src="http://mathworld.wolfram.com/images/equations/ChebyshevPolynomialoftheSecondKind/Inline78.gif" style="height:14px; width:17px" />, ![<span style=]()
{-4,0}" src="http://mathworld.wolfram.com/images/equations/ChebyshevPolynomialoftheSecondKind/Inline79.gif" style="height:14px; width:42px" />, ![<span style=]()
{0,-64,0}" src="http://mathworld.wolfram.com/images/equations/ChebyshevPolynomialoftheSecondKind/Inline80.gif" style="height:14px; width:64px" />, ![<span style=]()
{16,256,4096,0}" src="http://mathworld.wolfram.com/images/equations/ChebyshevPolynomialoftheSecondKind/Inline81.gif" style="height:14px; width:104px" />, ![<span style=]()
{0,0,0,1048576,0}" src="http://mathworld.wolfram.com/images/equations/ChebyshevPolynomialoftheSecondKind/Inline82.gif" style="height:14px; width:123px" />, ... (OEIS A054376).
REFERENCES:
Abramowitz, M. and Stegun, I. 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.
Koekoek, R. and Swarttouw, R. F. "Chebyshev." §1.8.2 in The Askey-Scheme of Hypergeometric Orthogonal Polynomials and its 
-Analogue. Delft, Netherlands: Technische Universiteit Delft, Faculty of Technical Mathematics and Informatics Report 98-17, pp. 41-43, 1998.
Koepf, W. "Efficient Computation of Chebyshev Polynomials." In Computer Algebra Systems: A Practical Guide (Ed. M. J. Wester). New York: Wiley, pp. 79-99, 1999.
Pegg, E. Jr. "ChebyshevU." http://www.mathpuzzle.com/ChebyshevU.html.
Rivlin, T. J. Chebyshev Polynomials. New York: Wiley, 1990.
Sloane, N. J. A. Sequences A053117 and A054376 in "The On-Line Encyclopedia of Integer Sequences."
Spanier, J. and Oldham, K. B. "The Chebyshev Polynomials 
and 
." Ch. 22 in An Atlas of Functions. Washington, DC: Hemisphere, pp. 193-207, 1987.
Vasilyev, N. and Zelevinsky, A. "A Chebyshev Polyplayground: Recurrence Relations Applied to a Famous Set of Formulas." Quantum 10, 20-26, Sept./Oct. 1999.
Zwillinger, D. (Ed.). CRC Standard Mathematical Tables and Formulae. Boca Raton, FL: CRC Press, 1995.
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