#### 8.94 The Chebyshev polynomials Tn(x) and Un(x)

8.940 Definition

1. Chebyshev's polynomials of the first kind

NA 66, 71

2. Chebyshev's polynomials of the second kind:

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

##### Functional relations

8.941 Recursion formulas:

1.

NA 358

2.

3.

EH II 184(3) ...

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