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)),$$ |
| + | where $\sin$ denotes [[sine]] and $\mathrm{arcsin}$ denotes [[arcsin]]. | ||
=Properties= | =Properties= | ||
| − | < | + | [[Orthogonality of Chebyshev U on (-1,1)]]<br /> |
| − | < | + | [[Relationship between Chebyshev U and hypergeometric 2F1]]<br /> |
| − | + | [[Relationship between Chebyshev U and Gegenbauer C]]<br /> | |
| − | + | [[U n(x)=(-i/2)(x+i sqrt(1-x^2))^n+(-i/2)(x-i sqrt(1-x^2))^n]]<br /> | |
| − | + | [[U n(x)=Sum (-1)^k n!/((2k+1)!(n-2k-1)!)(1-x^2)^(k+1/2)x^(n-2k-1)]]<br /> | |
| − | + | ||
| − | + | =References= | |
| − | + | * {{BookReference|Special Functions for Scientists and Engineers|1968|W.W. Bell|prev=Chebyshev T|next=T n(x)=(1/2)(x+i sqrt(1-x^2))^n+(1/2)(x-i sqrt(1-x^2))^n}}: $(7.2)$ | |
| − | + | ||
| − | + | {{:Orthogonal polynomials footer}} | |
| − | + | ||
| + | [[Category:SpecialFunction]] | ||
Latest revision as of 19:33, 15 March 2018
The Chebyshev polynomials of the second kind are orthogonal polynomials defined by $$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
- 1968: W.W. Bell: Special Functions for Scientists and Engineers ... (previous) ... (next): $(7.2)$
