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(in).

Then a couple days ago I concluded a blog post by noting that by taking the real part of this equation and replacing sin^2 with 1 - cos^2 one could express cos n as a polynomial in cos ,

and in fact this polynomial is the nth Chebyshev polynomial T_n 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 2+ 1.

Here we did a change of variables k = n - .

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