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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
2 years, 5 months ago

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