Table of Integrals, Series, and Products, 8th Edition

8.94 The Chebyshev polynomials T_n(x) and U_n(x)

8.940 Definition

1. Chebyshev's polynomials of the first kind

$\begin{array}{ccc}{T}_n\left(x\right)& =& \cos \left(n\text{ arccos }x\right)=\frac{1}{2}\left[{\left(x+i\sqrt{1-x^2}\right)}^n+{\left(x-i\sqrt{1-x^2}\right)}^n\right]\\ =& x^n-\left|\left(2_n^{}\right)x^{n-2}\left(1-x^2\right)+\left(4_n^{}\right)x{\left(1-x^2\right)}^2-\left(6_n^{}\right)x^{n-6}{\left(1-x^2\right)}^3+\mathrm{\dots }\right|\end{array}$

  NA 66, 71

2. Chebyshev's polynomials of the second kind:

$\begin{array}{ccc}{U}_n\left(x\right)& =& \frac{\sin \left[\left(n+1\right)\arccos x\right]}{\sin \left[\arccos x\right]}=\frac{1}{2i\sqrt{1-x^2}}\left[{\left(x+i\sqrt{1-x^2}\right)}^{n+1}-{\left(x-i\sqrt{1-x^2}\right)}^{n+1}\right]\\ =& \left(1_{n+1}^{}\right)x^n-\left(3_{n+1}^{}\right)x^{n-2}\left(1-x^2\right)+\left(5_{n+1}^{}\right)x^{n-4}{\left(1-x^2\right)}^2-\mathrm{\dots }\end{array}$

Functional relations

8.941 Recursion formulas:

1. 

$T_{n+1}\left(x\right)-2x{T}_n\left(x\right)+T_{n-1}\left(x\right)=0$

  NA 358

2. 

$U_{n+1}\left(x\right)-2x{U}_n\left(x\right)+U_{n-1}\left(x\right)=0$

3. 

$T_n\left(x\right)=U_n\left(x\right)-x{U}_{n-1}\left(x\right)$

  EH II 184(3) ...