Difference between revisions of "Chebyshev T"
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| − | Chebyshev polynomials of the first kind are [[orthogonal polynomials]] defined by | + | Chebyshev polynomials of the first kind are [[orthogonal polynomials]] defined for $n=0,1,2,\ldots$ by |
| − | $$T_n(x) = \cos(n \mathrm{arccos}(x)) | + | $$T_n(x) = \cos(n \mathrm{arccos}(x)),$$ |
| + | where $\cos$ denotes [[cosine]] and $\mathrm{arccos}$ denotes [[arccos]]. | ||
=Properties= | =Properties= | ||
| − | + | [[T_{n+1}(x)-2xT_n(x)+T_{n-1}(x)=0]]<br /> | |
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| − | </ | ||
<div class="toccolours mw-collapsible mw-collapsed" style="width:800px"> | <div class="toccolours mw-collapsible mw-collapsed" style="width:800px"> | ||
Revision as of 22:12, 19 December 2017
Chebyshev polynomials of the first kind are orthogonal polynomials defined for $n=0,1,2,\ldots$ by $$T_n(x) = \cos(n \mathrm{arccos}(x)),$$ where $\cos$ denotes cosine and $\mathrm{arccos}$ denotes arccos.
Properties
[[T_{n+1}(x)-2xT_n(x)+T_{n-1}(x)=0]]
Theorem (Orthogonality): The following formula holds: $$\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.$$
Proof: █
Theorem
The following formula holds for $n \in \{0,1,2,\ldots\}$: $$T_n(x) = {}_2F_1 \left( -n,n ; \dfrac{1}{2}; \dfrac{1-x}{2} \right),$$ where $T_n$ denotes a Chebyshev polynomial of the first kind and ${}_2F_1$ denotes the hypergeometric pFq.
Proof
References
Theorem
The following formula holds for $n \in \{1,2,3,\ldots\}$: $$T_n(x)=\dfrac{n}{2} \displaystyle\lim_{\lambda \rightarrow 0} \dfrac{C_n^{\lambda}(x)}{\lambda},$$ where $T_n$ denotes a Chebyshev polynomial of the first kind and $C_n^{\lambda}$ denotes a Gegenbauer C polynomial.
