Difference between revisions of "Chebyshev T"

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Chebyshev polynomials of the first kind are [[orthogonal polynomials]] defined by
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Chebyshev polynomials of the first kind are [[orthogonal polynomials]] defined for $n=0,1,2,\ldots$ and $-1 \leq x \leq 1$ by
$$T_n(x) = \cos(n \mathrm{arccos}(x)).$$
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$$T_n(x) = \cos(n \mathrm{arccos}(x)),$$
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where $\cos$ denotes [[cosine]] and $\mathrm{arccos}$ denotes [[arccos]].
  
 
=Properties=
 
=Properties=
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[[T_(n+1)(x)-2xT_n(x)+T_(n-1)(x)=0]]<br />
<strong>Theorem:</strong> The polynomials $T_n(x)$ and $U_n(x)$ are two independent solutions of the following equation, called Chebyshev's equation:
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[[Orthogonality of Chebyshev T on (-1,1)]]<br />
$$(1-x^2)\dfrac{d^2y}{dx^2}-x\dfrac{dy}{dx}+n^2y=0.$$
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[[Relationship between Chebyshev T and hypergeometric 2F1]]<br />
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[[Relationship between Chebyshev T and Gegenbauer C]]<br />
<strong>Proof:</strong>
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[[T n(x)=(1/2)(x+i sqrt(1-x^2))^n+(1/2)(x-i sqrt(1-x^2))^n]]<br />
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[[T n(x)=Sum (-1)^k n!/((2k)! (n-2k)!) (1-x^2)^k x^(n-2k)]]<br />
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=References=
<strong>Theorem:</strong> The following formula holds:
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* {{BookReference|Special Functions for Scientists and Engineers|1968|W.W. Bell|prev=findme|next=Chebyshev U}}: $(7.1)$
$$T_{n+1}(x)-2xT_n(x)+T_{n-1}(x)=0.$$
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* {{BookReference|An Introduction to Orthogonal Polynomials|1978|T.S. Chihara|prev=Orthogonality of Chebyshev T on (-1,1)|next=findme}} $(1.4)$ (<i>note: calls them Tchebichef polynomials of the first kind</i>)
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<strong>Proof:</strong> █
 
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{{:Orthogonal polynomials footer}}
<strong>Theorem (Orthogonality):</strong> 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.$$
 
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<strong>Proof:</strong> █
 
</div>
 
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{{:Orthogonal polynomials footer}}
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[[Category:SpecialFunction]]

Latest revision as of 19:33, 15 March 2018

Chebyshev polynomials of the first kind are orthogonal polynomials defined for $n=0,1,2,\ldots$ and $-1 \leq x \leq 1$ 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)
Relationship between Chebyshev T and hypergeometric 2F1
Relationship between Chebyshev T and Gegenbauer C
T n(x)=(1/2)(x+i sqrt(1-x^2))^n+(1/2)(x-i sqrt(1-x^2))^n
T n(x)=Sum (-1)^k n!/((2k)! (n-2k)!) (1-x^2)^k x^(n-2k)

References

Orthogonal polynomials