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Chebyshev Polynomial of the Second Kind

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ChebyshevU

The Chebyshev polynomials of the second kind are orthogonal polynomials denoted U_n(x) for nonnegative integers n, with the generating function defined below. These polynomials 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. They are 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

U_0(x)=1
(1)
U_1(x)=2x
(2)
U_2(x)=4x^2-1
(3)
U_3(x)=8x^3-4x
(4)
U_4(x)=16x^4-12x^2+1
(5)
U_5(x)=32x^5-32x^3+6x
(6)
U_6(x)=64x^6-80x^4+24x^2-1.
(7)

When the coefficients are ordered from smallest to largest powers, their nonzero entries are 1; 2; -1, 4; -4, 8; 1, -12, 16; 6, -32, 32; ... (OEIS A053117). The cited sequence includes the zero coefficients.

The defining generating function of the Chebyshev polynomials of the second kind is

g(t,x)=1/(1-2xt+t^2)
(8)
=sum_(n=0)^(infty)U_n(x)t^n
(9)

for real x with |x|<=1 and |t|<1. To see the relationship to a Chebyshev polynomial of the first kind T_n(x), take partial/partialt of equation (9) to obtain

(partialg)/(partialt)=2(x-t)(1-2xt+t^2)^(-2)
(10)
=sum_(n=1)^(infty)nU_n(x)t^(n-1).
(11)

Multiplying (11) by t then gives

(2xt-2t^2)(1-2xt+t^2)^(-2)=sum_(n=0)^inftynU_n(x)t^n
(12)

and adding (12) and (9) gives

((2xt-2t^2)+(1-2xt+t^2))/((1-2xt+t^2)^2)=(1-t^2)/((1-2xt+t^2)^2)
(13)
=sum_(n=0)^(infty)(n+1)U_n(x)t^n.
(14)

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.

For -1<x<1, the Rodrigues representation for U_n 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)].
(15)

The polynomials can also be defined in terms of the sums

U_n(x)=sum_(r=0)^(|_n/2_|)(-1)^r(n-r; r)(2x)^(n-2r)
(16)
=sum_(m=0)^(|_n/2_|)(n+1; 2m+1)x^(n-2m)(x^2-1)^m,
(17)

where |_x_| is the floor function and (n; k) is a binomial coefficient, or in terms of the product

U_n(x)=2^nproduct_(k=1)^n[x-cos((kpi)/(n+1))]
(18)

(Zwillinger 1995, p. 696).

For n>=1, U_n(x) also obeys the determinant identity for the n×n matrix

U_n(x)=|2x 1 0 0 ... 0 0; 1 2x 1 0 ... 0 0; 0 1 2x 1 ... 0 0; 0 0 1 2x ... 0 0; 0 0 0 1 ... 1 0; | ... ... ... ... ... 1; 0 0 0 0 ... 1 2x|.
(19)

The Chebyshev polynomials of the second kind are a special case of the Jacobi polynomials P_n^((alpha,beta)) with alpha=beta=1/2,

U_n(x)=(n+1)(P_n^((1/2,1/2))(x))/(P_n^((1/2,1/2))(1))
(20)
=(n+1)_2F_1(-n,n+2;3/2;1/2(1-x)),
(21)

where _2F_1(a,b;c;x) is a hypergeometric function (Koekoek and Swarttouw 1998).

Letting x=costheta with 0<theta<pi allows the Chebyshev polynomials of the second kind to be written as

U_n(x)=(sin[(n+1)theta])/(sintheta).
(22)

The composition identity

U_(m-1)(T_n(z))U_(n-1)(z)=U_(mn-1)(z)
(23)

holds for positive integers m and n, where T_n is the Chebyshev polynomial of the first kind (Benjamin and Walton 2010). It follows from the multiple-angle formulas above and extends to all complex numbers z by equality of polynomials.

On -1<x<1, a second linearly independent solution to the differential equation (1-x^2)y^('')-3xy^'+n(n+2)y=0 is given by

W_n(x)=(cos[(n+1)theta])/(sintheta),
(24)

which can also be written

W_n(x)=(1-x^2)^(-1/2)T_(n+1)(x),
(25)

where T_n(x) is a Chebyshev polynomial of the first kind. Note that W_n(x) is therefore not a polynomial.

The number triangle whose nth row consists of the resultants rho(U_n(x),U_k(x)) for k=1, 2, ..., n, starting at n=1, 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 Differential Equation, Chebyshev Polynomial of the First Kind, d-Chebyshev Polynomial, Function Composition, Gegenbauer Polynomial

Related Wolfram sites

https://functions.wolfram.com/Polynomials/ChebyshevU/, https://functions.wolfram.com/HypergeometricFunctions/ChebyshevUGeneral/

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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.Benjamin, A. T. and Walton, D. "Combinatorially Composing Chebyshev Polynomials." J. Statist. Plann. Inference 140, 2161-2167, 2010. https://doi.org/10.1016/j.jspi.2010.01.012.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. "MathPuzzle: ChebyshevU." Jun. 1, 2002. https://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.

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Chebyshev Polynomial of the Second Kind

Cite this as:

Weisstein, Eric W. "Chebyshev Polynomial of the Second Kind." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/ChebyshevPolynomialoftheSecondKind.html

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