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Topic: Closed expression for coefficients Chebychev polynomial
Replies: 1   Last Post: Jun 14, 1996 8:10 PM

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robert bristow-johnson

Posts: 19
Registered: 12/7/04
Re: Closed expression for coefficients Chebychev polynomial
Posted: Jun 14, 1996 8:10 PM
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In article <4prj8q$2pq@rzsun02.rrz.uni-hamburg.de>, Hauke Reddmann (fc3a501@GEOMAT.math.uni-hamburg.de) writes:
>I could err (must look up my math dictionary),
>but wasn't simply Tn(x)=sin(n*arccos(x)),Un(x)=cos(thesame)?


i think you switched the Tn and Un.

T[n](x) = cos( n*arccos(x) ) for |x| <= 1
= cosh( n*arccosh(x) ) for |x| >= 1

it is what you say but it is also a polynomial (because T[0](x) = 1, T[1](x) =
x, and there is this cool recursion formula, T[n+1](x) = 2*x*T[n](x) - T[n-1])
and the original poster (and myself) want to know if there is a closed form
expression for the kth coefficient of T[n](x) that is a simple function of n
and k. i suppose there is (that can be inductively proven using the recursion
formula and boundary conditions above) but i have never seen it.

r b-j
wave mechanics, inc.
robert@audioheads.com or robert@wavemechanics.com








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