Difference between revisions of "Chebyshev U"
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[[Relationship between Chebyshev U and hypergeometric 2F1]]<br /> | [[Relationship between Chebyshev U and hypergeometric 2F1]]<br /> | ||
[[Relationship between Chebyshev U and Gegenbauer C]]<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 /> | [[U n(x)=Sum (-1)^k n!/((2k+1)!(n-2k-1)!)(1-x^2)^(k+1/2)x^(n-2k-1)]]<br /> | ||
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): $(7.2)$
