Difference between revisions of "Chebyshev T"

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(Properties)
(Properties)
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=Properties=
 
=Properties=
 
[[T_(n+1)(x)-2xT_n(x)+T_(n-1)(x)=0]]<br />
 
[[T_(n+1)(x)-2xT_n(x)+T_(n-1)(x)=0]]<br />
 
+
[[Orthogonality of Chebyshev T on (-1,1)]]
<div class="toccolours mw-collapsible mw-collapsed" style="width:800px">
 
<strong>Theorem (Orthogonality):</strong> 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.$$
 
<div class="mw-collapsible-content">
 
<strong>Proof:</strong> █
 
</div>
 
</div>
 
  
 
{{:Relationship between Chebyshev T and hypergeometric 2F1}}
 
{{:Relationship between Chebyshev T and hypergeometric 2F1}}

Revision as of 22:15, 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
Orthogonality of Chebyshev T on (-1,1)

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.

Proof

References

Orthogonal polynomials