I have tried integrating to find a relationship between $C_n^{(m)}$ and $C_n^{(m-1)}$, but I don't know if I'm right, as the integral is very confusing.
Also, even if I am wrong on the relationship, I attempted section b) and I did not know how to approach the change of variable.
Any help would be amazing. Thank you.
Observe that
$$\frac{d^2x^{2m}}{dx^2}=(2m)(2m-1)x^{2m-2}=(2m)(2m-1)x^{2(m-1)}.$$
Now use (or rederive, using integration by parts) the fact that
$$\hat{f''}(n)=in\hat{f'}(n)=-n^2\hat{f}(n).$$
Take the Fourier-series transform of the first equation, specifically the left and right sides, to get:
$$-n^2c^{(m)}_n=(2m)(2m-1)c^{(m-1)}_n.$$