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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:

  1. Do these functions have a name and any application? One possible use would be in solving $m$th order differential equation over $\mathbb{R}$.

  2. 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?

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up vote 9 down vote accepted

What you have at hand is known as Mittag-Leffer function $t_m(x) = E_{m}(-x^m)$.

There is a very nice article on ArXiv, "Mittag-Leffler functions and their applications", arXiv:0909.0230

To the best of my knowledge roots are hard to come by in closed form. For instance, for $m=3$, $$ t_3(x) = \frac{1}{3} \mathrm{e}^{-x} + \frac{2}{3} \mathrm{e}^{x/2} \cos\left(\frac{\sqrt{3}}{2} x\right) $$ Its roots are solutions of $$ \exp\left(\frac{3}{2} x\right) \cos\left(\frac{\sqrt{3}}{2} x\right) = -\frac{1}{2} $$ which is a transcendental equation. It has infinitely many real solutions, all of which are positive: $$ x_{n, n \geqslant 0} = \frac{\pi}{\sqrt{3}} \left(2n+1\right) + \frac{(-1)^n}{\sqrt{3}} \exp\left(-\frac{\sqrt{3}}{2} \pi (2n+1) \right) + \mathcal{o}\left(\mathrm{e}^{-\pi \sqrt{3} n}\right) $$

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+1 Thx, very cool reference! – draks ... Jan 25 '12 at 22:25
Mittag-Leffler's function features prominently, among other places, in the solution of fractional differential equations. A very interesting function, it is... – J. M. Jan 26 '12 at 1:04

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