[Math] Binomial expansion with 2n power

Question

I'm trying to expand $$(z+z^{-1})^{2n}$$.

I've written the first 3 terms out but I can't figure out a way to work out the last 3 terms.
I'm trying to produce an expression for $$\cos^{2n}x$$ using the complex numbers.

Answer 1

Just apply the binomial formula $$ (z+z^{-1})^{2n}=\sum_{k=0}^{2n}\binom{2n}kz^k(z^{-1})^{2n-k} =\sum_{k=0}^{2n}\binom{2n}kz^{2(k-n)} =\sum_{k=-n}^n\binom{2n}{n+k}z^{2k}. $$ Now putting $\def\i{\mathbf i}z=\exp(\i x)$ and dividing by $2^{2n}$ you get $$ \cos(x)^{2n}= 2^{-2n}\sum_{k=-n}^n\binom{2n}{n+k}\exp(2\i kx) = 2^{-2n}\sum_{k=-n}^n\binom{2n}{n+k}\cos(2kx). $$