More juice in the lemon

There’s more juice left in the lemon we’ve been squeezing lately.

A few days ago I first brought up the equation which holds because both sides equal exp(inθ).

Then a couple days ago I concluded a blog post by noting that by taking the real part of this equation and replacing sin²θ with 1 – cos²θ one could express cos nθ as a polynomial in cos θ, and in fact this polynomial is the nth Chebyshev polynomial Tn since these polynomials satisfy Now in this post I’d like to prove a relationship between Chebyshev polynomials and sines starting with the same raw material.

The relationship between Chebyshev polynomials and cosines is well known, even a matter of definition depending on where you start, but the connection to sines is less well known.

Let’s go back to the equation at the top of the post, replace n with 2n + 1, and take the imaginary part of both sides.

The odd terms of the sum contribute to the imaginary part, so we sum over 2ℓ+ 1.

Here we did a change of variables k = n – ℓ.

The final expression is the expression we began with, only evaluated at sin θ instead of cos θ.

That is, So for all n we have and for odd n we also have The sign is positive when n is congruent to 1 mod 4 and negative when n is congruent to 3 mod 4.

More Chebyshev posts Chebyshev interpolation Chebyshev approximation Chebyshev’s inequality.

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