Once in a time, I had to work with functions that have the following Taylor series expansion: $ t_m(x)=1-\frac{x^m}{m!}+\frac{x^{2m}}{(2m)!}+\cdots =\sum_{k=0}^\infty \frac{(-1)^k x^{km}}{(km)!}. $ Plugging in $m=2$ this is obviously the Taylor series expansion for $\cos(x)$. Now I found the following nice formula to get a closed formula for this functions, which is (proven here): $ t_m(x)=\frac{1}{m}\sum_{k=0}^{m-1} \exp( e^{i\frac{2k+1}{m}\pi}x ) $ and again it is obvious that for $m=2$ I'll get $\displaystyle\frac{e^{i\pi x}+e^{-i\pi x}}{2}=\cos(x)$. It is also easy to see that $\displaystyle\frac{d(t_m)^m}{dx^m}=-t_m.$
And now I have 2 questions:
Do these functions have a name and any application? One possible use would be in solving $m$th order differential equation over $\mathbb{R}$.
When I ask Wolfram for the roots, if $m=4$, I get $x_n=\frac{2\pi n + \pi}{\sqrt{2}}$ (and also $i\cdot x_n$). Asking for other $m\neq2,4$, I (so far) just get numerical values. Are there closed formulas for the roots in all case of $m$. Do they have a geometric interpretation?