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

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

No vote yet

1 vote

×

Problem Loading...

Note Loading...

Set Loading...

Easy Math Editor

`*italics*`

or`_italics_`

italics`**bold**`

or`__bold__`

boldNote: you must add a full line of space before and after lists for them to show up correctlyparagraph 1

paragraph 2

`[example link](https://brilliant.org)`

`> This is a quote`

Remember to wrap math in \( ... \) or \[ ... \] to ensure proper formatting.`2 \times 3`

`2^{34}`

`a_{i-1}`

`\frac{2}{3}`

`\sqrt{2}`

`\sum_{i=1}^3`

`\sin \theta`

`\boxed{123}`

## Comments

There are no comments in this discussion.