>Neither the Mathieu nor the Hill equation is easily understood in terms of a Fourier series.

I am not sure what you want to say with that.
The Mathieu equation is a special case of the Hill equation in terms of its Fourier expansion, as written in the :
Hill equation (or almost the same [here](http://mathworld.wolfram.com/HillsDifferentialEquation.html))

>They are closer to quasiperiodic.

Periodicity is (as I know it) a special case of Quasiperiodicity, you probably meant aperiodic. ?? By the way it seems it is actually rather the constant term, which dictates the
periodicity, like according to Wikipedia the "a" here in the Wikipedia article about the [Mathieu function](http://en.wikipedia.org/wiki/Mathieu_function) enters the Mathieu sine/cosine.
[Weisstein](http://mathworld.wolfram.com/MathieuFunction.html) says the same. Apart from the special cos/sin case in order to find periodic solutions one would need to find characteristic values.