Chebyshev T

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

Properties[edit]

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 formula holds: $$\int_{-1}^1 \dfrac{T_m(x)T_n(x)}{\sqrt{1-x^2}} dx = \left\{ \begin{array}{ll} 0 &; m \neq n \\ \dfrac{\pi}{2} &; m=n\neq 0 \\ \pi &; m=n=0. \end{array} \right.$$

Proof:

Theorem: The following formula holds: $$T_n(x) = {}_2F_1 \left( -n,n ; \dfrac{1}{2}; \dfrac{1-x}{2} \right),$$ where $T_n$ denotes a Chebyshev T polynomial and ${}_2F_1$ denotes the hypergeometric pFq.

Proof:

Theorem: The following formula holds: $$T_n(x)=\dfrac{n}{2} \displaystyle\lim_{\lambda \rightarrow 0} \dfrac{C_n^{\lambda}(x)}{\lambda}; n\geq 1,$$ where $T_n$ denotes a Chebyshev T polynomial and $C_n^{\lambda}$ denotes a Gegenbauer C polynomial.

Proof:

Orthogonal polynomials