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Chebyshev polynomial∗ drini† 2013-03-21 13:34:56 The Chebyshev polynomials of first kind are defined by the simple formula Tn (x) = cos(nt), where x = cos t. It is an example of a trigonometric polynomial. This can be seen to be a polynomial by expressing cos(kt) as a polynomial of cos(t), by using the formula for cosine of angle-sum: cos(1t) = cos(t) cos(2t) = cos(t) cos(t) − sin(t) sin(t) = 2(cos(t))2 − 1 cos(3t) = .. . 4(cos(t))3 − 3 cos(t) So we have T0 (x) = 1 T1 (x) = x T2 (x) = 2x2 − 1 T3 (x) = .. . 4x3 − 3x These polynomials obey the recurrence relation: Tn+1 (x) = 2xTn (x) − Tn−1 (x) for n = 1, 2, . . . ∗ hChebyshevPolynomiali created: h2013-03-21i by: hdrinii version: h32161i Privacy setting: h1i hDefinitioni h42C05i h42A05i h33C45i † This text is available under the Creative Commons Attribution/Share-Alike License 3.0. You can reuse this document or portions thereof only if you do so under terms that are compatible with the CC-BY-SA license. 1 Related are the Chebyshev polynomials of the second kind that are defined as Un−1 (cos t) = sin(nt) , sin(t) which can similarly be seen to be polynomials through either a similar process as the above or by the relation Un−1 (t) = nTn0 (t). The first few are: U0 (x) = 1 U1 (x) = 2x U2 (x) = 4x2 − 1 U3 (x) = .. . 8x3 − 4x The same recurrence relation also holds for U : Un+1 (x) = 2xUn (x) − Un−1 (x) for n = 1, 2, . . .. 2