Difference between revisions of "Chebyshev T"
From specialfunctionswiki
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[[Relationship between Chebyshev T and hypergeometric 2F1]]<br /> | [[Relationship between Chebyshev T and hypergeometric 2F1]]<br /> | ||
[[Relationship between Chebyshev T and Gegenbauer C]]<br /> | [[Relationship between Chebyshev T and Gegenbauer C]]<br /> | ||
| + | [[T n(x)=(1/2)(x+i sqrt(1-x^2))^n+(1/2)(x-i sqrt(1-x^2))^n]]<br /> | ||
[[T n(x)=Sum (-1)^k n!/((2k)! (n-2k)!) (1-x^2)^k x^(n-2k)]]<br /> | [[T n(x)=Sum (-1)^k n!/((2k)! (n-2k)!) (1-x^2)^k x^(n-2k)]]<br /> | ||
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
- 1968: W.W. Bell: Special Functions for Scientists and Engineers ... (previous) ... (next): $(7.1)$
- 1978: T.S. Chihara: An Introduction to Orthogonal Polynomials ... (previous) ... (next) $(1.4)$ (note: calls them Tchebichef polynomials of the first kind)
