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

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

#Algebra #Polynomials #DeMoivre'sFormula

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Math	Appears as
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`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
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Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$