Chebyshev T

Chebyshev polynomials of the first kind are orthogonal polynomials defined by $$T_n(x) = \cos(n \mathrm{arccos}(x)).$$

Properties

Theorem: The polynomials $T_n(x)$ and $U_n(x)$ are two independent solutions of the following equation, called Chebyshev's equation: $$(1-x^2)\dfrac{d^2y}{dx^2}-x\dfrac{dy}{dx}+n^2y=0.$$

Proof: █

Theorem: The following formula holds: $$T_{n+1}(x)-2xT_n(x)+T_{n-1}(x)=0.$$

Proof: █

Theorem (Orthogonality): The following formulas hold: $$\int_{-1}^1 \dfrac{T_m(x)T_n(x)}{\sqrt{1-x^2}} dx = \left\{ \begin{array}{ll} 0 &; m \neq n \\ \dfrac{\pi}{2} &; m=n\neq 0 \\ \pi &; m=n=0 \end{array} \right.$$ and $$\int_{-1}^1 \dfrac{U_m(x)U_n(x)}{\sqrt{1-x^2}} dx = \left\{ \begin{array}{ll} 0 &; m \neq n \\ \dfrac{\pi}{2} &; m=n\neq 0\\ 0 &; m=n=0. \end{array} \right.$$

Proof: █