Difference between revisions of "Chebyshev U"

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The Chebyshev polynomials of the second kind are orthogonal polynomials defined by
 
The Chebyshev polynomials of the second kind are orthogonal polynomials defined by
$$U_n(x) = \sin(n \mathrm{arcsin}(x)).$$
+
$$U_n(x) = \sin(n \mathrm{arcsin}(x)),$$
 +
where $\sin$ denotes [[sine]] and $\mathrm{arcsin}$ denotes [[arcsin]].
  
 
=Properties=
 
=Properties=
{{:Chebychev differential equation}}
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[[Orthogonality of Chebyshev U on (-1,1)]]<br />
 +
[[Relationship between Chebyshev U and hypergeometric 2F1]]<br />
 +
[[Relationship between Chebyshev U and Gegenbauer C]]<br />
 +
[[U n(x)=(-i/2)(x+i sqrt(1-x^2))^n+(-i/2)(x-i sqrt(1-x^2))^n]]<br />
 +
[[U n(x)=Sum (-1)^k n!/((2k+1)!(n-2k-1)!)(1-x^2)^(k+1/2)x^(n-2k-1)]]<br />
  
<div class="toccolours mw-collapsible mw-collapsed" style="width:800px">
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=References=
<strong>Theorem:</strong> The following formula holds:
+
* {{BookReference|Special Functions for Scientists and Engineers|1968|W.W. Bell|prev=Chebyshev T|next=T n(x)=(1/2)(x+i sqrt(1-x^2))^n+(1/2)(x-i sqrt(1-x^2))^n}}: $(7.2)$
$$\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.$$
 
<div class="mw-collapsible-content">
 
<strong>Proof:</strong> █
 
</div>
 
</div>
 
 
 
{{:Relationship between Chebyshev U and hypergeometric 2F1}}
 
 
 
{{:Relationship between Chebyshev U and Gegenbauer C}}
 
  
 
{{:Orthogonal polynomials footer}}
 
{{:Orthogonal polynomials footer}}
  
 
[[Category:SpecialFunction]]
 
[[Category:SpecialFunction]]

Latest revision as of 19:33, 15 March 2018

The Chebyshev polynomials of the second kind are orthogonal polynomials defined by $$U_n(x) = \sin(n \mathrm{arcsin}(x)),$$ where $\sin$ denotes sine and $\mathrm{arcsin}$ denotes arcsin.

Properties

Orthogonality of Chebyshev U on (-1,1)
Relationship between Chebyshev U and hypergeometric 2F1
Relationship between Chebyshev U and Gegenbauer C
U n(x)=(-i/2)(x+i sqrt(1-x^2))^n+(-i/2)(x-i sqrt(1-x^2))^n
U n(x)=Sum (-1)^k n!/((2k+1)!(n-2k-1)!)(1-x^2)^(k+1/2)x^(n-2k-1)

References

Orthogonal polynomials