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Chebyshevs Polynomial

The $$n$$-th Chebyshev polynomial (of the first kind) is usually defined as the polynomial expressing $$\cos(nx)$$ in terms of $$\cos(x)$$.

Closely related is the polynomial $$P_n(x)$$ that expresses $$2\cos(nx)$$ in terms of $$2\cos(x)$$. This polynomial can be obtained by writing:

$$x^{n} +x^{−n}$$ in terms of $$x+x^{−1}$$.

Indeed, if $$x = \cos(t) +i \sin(t)$$, then $$x+x^{−1} = 2\cos(t)$$, while by the de Moivre formula $$x^{n}+x^{−n} = 2\cos(nt)$$.

Note that the sum-to-product formula $$\cos[(n+1)x]+\cos[(n−1)x] = 2\cos(x)\cos(nx)$$, allows us to prove by induction that $$P_n(x)$$ has integer coefficients, and we can easily compute

$\large P_0(x) = 2, P_1(x) = x, P_2(x) = x^2−2, P_3(x) = x^3−3x$

The fact that $$x^n +x^{-n}$$ can be written as a polynomial with integer coefficients in

$$x +x^{-1}$$ for all $$n$$ can also be proved inductively using the identity

$$x^n+x^{-n} = \left(x+x^{-1}\right)\left(x^n+x^{-n}\right) - x^{n-2} + x^{-(n-2)}$$.

Note by Sauditya Yo Yo
1 year, 11 months ago