Chebyshev Polynomial of the Second Kind

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 $\alpha=1$. The Chebyshev polynomials of the second kind $U_n(x)$ are illustrated above for $x\in[0,1]$ and $n=1$, 2, ..., 5.

The defining Generating Function of the Chebyshev polynomials of the second kind is

$$g_2(t,x)={1\over 1-2xt+t^2} = \sum_{n=0}^\infty U_n(x)t^n$$ (1)

for $\vert x\vert < 1$ and $\vert t\vert < 1$. To see the relationship to a Chebyshev Polynomial of the First Kind ($T$), take ${\partial g/\partial t}$,

$${\partial g\over \partial t} = -(1-2xt+t^2)^{-2}(-2x+2t)$$

$$= 2(t-x)(1-2xt+t^2)^{-2}$$

$$= \sum_{n=0}^\infty nU_n(x)t^{n-1}.$$ (2)

Multiply (2) by $t$,

$$(2t^2-2xt)(1-2xt+t^2)^{-2} = \sum_{n=0}^\infty nU_n(x)t^n$$ (3)

and take (3) minus (2),

$${(2t^2-2tx)-(1-2xt+t^2)\over(1-2xt+t^2)^2} = {t^2-1\over(1-2xt+t)^2} = \sum_{n=0}^\infty (n-1)U_n(x)t^n.$$ (4)

The Rodrigues representation is

$$U_n(x) = {(-1)^n(n+1)\sqrt{\pi}\over 2^{n+1}(n+{\textstyle{1\over 2}})!(1-x^2)^{1/2}} {d^n\over dx^n} [(1-x^2)^{n+1/2}].$$ (5)

The polynomials can also be written

$$U_n(x) = \sum_{r=0}^{\left\lfloor{n/2}\right\rfloor } (-1)^r {n-r\choose r} (2x)^{n-2r}$$

$$= \sum_{m=0}^{\left\lceil{n/2}\right\rceil } {n+1\choose 2m+1}x^{n-2m}(x^2-1)^m,$$ (6)

where $\left\lfloor{x}\right\rfloor$ is the Floor Function and $\left\lceil{x}\right\rceil$ is the Ceiling Function, or in terms of a Determinant

$$U_n=\left\vert\matrix{ 2x & 1 & 0 & 0 & \cdots & 0 & 0\cr 0 ... ...ts & \vdots\cr 0 & 0 & 0 & 0 & \cdots & 1 & 2x\cr}\right\vert.$$ (7)

The first few Polynomials are

$$\begin{eqnarray*} U_0(x) &=& 1\\ U_1(x) &=& 2x\\ U_2(x) &=& 4x^2-1\\ U_3(... ... U_5(x) &=& 32x^5-32x^3+6x\\ U_6(x) &=& 64x^6-80x^4+24x^2-1. \end{eqnarray*}$$

Letting $x\equiv\cos\theta$ allows the Chebyshev polynomials of the second kind to be written as

$$U_n(x)={\sin[(n+1)\theta]\over\sin\theta}.$$ (8)

The second linearly dependent solution to the transformed differential equation is then given by

$$W_n(x)={\cos[(n+1)\theta]\over\sin\theta},$$ (9)

which can also be written

$$W_n(x)=(1-x^2)^{-1/2} T_{n+1}(x),$$ (10)

where $T_n$ is a Chebyshev Polynomial of the First Kind. Note that $W_n(x)$ 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 $T_n(x)$ and $U_n(x)$.'' Ch. 22 in An Atlas of Functions. Washington, DC: Hemisphere, pp. 193-207, 1987.