Difference between revisions of "Chebyshev T"
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| − | Chebyshev polynomials of the first kind are [[orthogonal polynomials]] defined for $n=0,1,2,\ldots$ by | + | Chebyshev polynomials of the first kind are [[orthogonal polynomials]] defined for $n=0,1,2,\ldots$ and $-1 \leq x \leq 1$ by |
$$T_n(x) = \cos(n \mathrm{arccos}(x)),$$ | $$T_n(x) = \cos(n \mathrm{arccos}(x)),$$ | ||
where $\cos$ denotes [[cosine]] and $\mathrm{arccos}$ denotes [[arccos]]. | where $\cos$ denotes [[cosine]] and $\mathrm{arccos}$ denotes [[arccos]]. | ||
| Line 5: | Line 5: | ||
=Properties= | =Properties= | ||
[[T_(n+1)(x)-2xT_n(x)+T_(n-1)(x)=0]]<br /> | [[T_(n+1)(x)-2xT_n(x)+T_(n-1)(x)=0]]<br /> | ||
| − | [[Orthogonality of Chebyshev T on (-1,1)]] | + | [[Orthogonality of Chebyshev T on (-1,1)]]<br /> |
| + | [[Relationship between Chebyshev T and hypergeometric 2F1]]<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 /> | ||
| − | {{ | + | =References= |
| − | + | * {{BookReference|Special Functions for Scientists and Engineers|1968|W.W. Bell|prev=findme|next=Chebyshev U}}: $(7.1)$ | |
| − | {{ | + | * {{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>) |
{{:Orthogonal polynomials footer}} | {{:Orthogonal polynomials footer}} | ||
[[Category:SpecialFunction]] | [[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
- 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)
