Chebyshev polynomials

Important

Chebyshev polynomials are a special sequence of polynomials that are useful in algebra and trigonometry. The most common are the Chebyshev polynomials of the first kind, denoted $T_n(x)$ . They are defined by the recurrence relation:

T_0(x) = 1,\quad T_1(x) = x,\quad T_{n+1}(x) = 2x T_n(x) - T_{n-1}(x)

Alternatively, $T_n(x)$ can be written as $T_n(x) = \cos(n \arccos x)$ for $x$ in $[-1, 1]$ .

Chebyshev polynomials have many applications, especially in approximating functions and solving equations.

Important properties

Each $T_n(x)$ is a polynomial of degree $n$ .
They satisfy $T_n(\cos \theta) = \cos(n\theta)$ .
The roots of $T_n(x)$ are $x_k = \cos\left(\frac{\pi (2k-1)}{2n}\right)$ for $k = 1, 2, ..., n$ .
They are orthogonal with respect to the weight $\frac{1}{\sqrt{1-x^2}}$ on $[-1,1]$ .