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]]. | ||
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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 /> | ||
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| + | =References= | ||
| + | * {{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]] | ||
Revision as of 22:38, 19 December 2017
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
References
- 1978: T.S. Chihara: An Introduction to Orthogonal Polynomials ... (previous) ... (next) $(1.4)$ (note: calls them Tchebichef polynomials of the first kind)
