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 alpha=1. They are also intimately connected with trigonometric multiple-angle formulas. The Chebyshev polynomials of the second kind are denoted U_n(x), and implemented in the Wolfram Language as ChebyshevU [n, x]. The polynomials U_n(x) are illustrated above for x in [-1,1] and n=1, 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; -1, 4; -4, 8; 1, -12, 16; 6, -32, 32; ... (OEIS A053117).
The defining generating function of the Chebyshev polynomials of the second kind is
for |x|<1 and |t|<1. To see the relationship to a Chebyshev polynomial of the first kind T(x), take partial/partialt of equation (9) to obtain
Multiplying (◇) by t then gives
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 1-2xt+t^2 in the denominator.
The Rodrigues representation for U_n is
| [画像: 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)]. ] |
(15)
|
![画像: 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)].](/index.cgi/contrast/https://mathworld.wolfram.com/images/equations/ChebyshevPolynomialoftheSecondKind/NumberedEquation2.svg){kind=link}
The polynomials can also be defined in terms of the sums
where |_x_| is the floor function and [x] is the ceiling function, or in terms of the product
![画像: U_n(x)=2^nproduct_(k=1)^n[x-cos((kpi)/(n+1))]](/index.cgi/contrast/https://mathworld.wolfram.com/images/equations/ChebyshevPolynomialoftheSecondKind/NumberedEquation3.svg){kind=link}
(Zwillinger 1995, p. 696).
U_n(x) also obey the interesting determinant identity
The Chebyshev polynomials of the second kind are a special case of the Jacobi polynomials P_n^((alpha,beta)) with alpha=beta=1/2,
where _2F_1(a,b;c;x) is a hypergeometric function (Koekoek and Swarttouw 1998).
Letting x=costheta allows the Chebyshev polynomials of the second kind to be written as
![画像: U_n(x)=(sin[(n+1)theta])/(sintheta).](/index.cgi/contrast/https://mathworld.wolfram.com/images/equations/ChebyshevPolynomialoftheSecondKind/NumberedEquation5.svg){kind=link}
The second linearly dependent solution to the transformed differential equation is then given by
![画像: W_n(x)=(cos[(n+1)theta])/(sintheta),](/index.cgi/contrast/https://mathworld.wolfram.com/images/equations/ChebyshevPolynomialoftheSecondKind/NumberedEquation6.svg){kind=link}
which can also be written
| W_n(x)=(1-x^2)^(-1/2)T_(n+1)(x), |
(24)
|
where T_n(x) is a Chebyshev polynomial of the first kind. Note that W_n(x) is therefore not a polynomial.
The triangle of resultants rho(U_n(x),U_k(x)) is given by {0}, {-4,0}, {0,-64,0}, {16,256,4096,0}, {0,0,0,1048576,0}, ... (OEIS A054376).
See also
Chebyshev Approximation Formula, Chebyshev Polynomial of the First Kind, Gegenbauer PolynomialRelated Wolfram sites
http://functions.wolfram.com/Polynomials/ChebyshevU/, http://functions.wolfram.com/HypergeometricFunctions/ChebyshevUGeneral/Explore with Wolfram|Alpha
More things to try:
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 q-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 T_n(x) and U_n(x)." 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.Referenced on Wolfram|Alpha
Chebyshev Polynomial of the Second KindCite this as:
Weisstein, Eric W. "Chebyshev Polynomial of the Second Kind." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/ChebyshevPolynomialoftheSecondKind.html