Chebyshev Polynomials

The $n$ th Chebyshev polynomial may be defined by

$$T_n(x) = \left\{\begin{array}{ll} \cos[n\cos^{-1}(x)], & \vert x\vert\le1 \\ [5pt] \cosh[n\cosh^{-1}(x)], & \vert x\vert>1 \\ \end{array} \right..$$

Clearly, $T_0(x)=1$ and $T_1(x)=x$ .
Using the double-angle trig formula $\cos(2\theta)=2\cos^2(\theta)-1$ , it can be verified that

$$\zbox{T_n(x) = 2x T_{n-1}(x) - T_{n-2}(x)} \quad (n\ge 2)$$