Evaluate infinite s, series, similar to $\cos(z)$ 
Evaluate the sum
$$\sum_{k=0}^\infty\frac{(-1)^kz^{ak}}{\Gamma(1+ak)}$$
where $a\in \mathbb{R}_{>0}$ and $z\in\mathbb{C}$.

I know if $a=2$ then this is the series expansion for $\cos(z)$. But for arbitrary positive real $a$ I have no idea what this is.
 A: This is a Mittag-Leﬄer function :
$$E_{\alpha, \beta} (z) = \sum_{k=0}^\infty \frac{z^k}{\Gamma(\alpha k + \beta)},\quad \alpha,\beta\in\mathbb{C},\;\Re(\alpha)>0,\,\Re(\beta)>0, z\in \mathbb{C}$$
or in your case ($\beta=1$ is implicit) :
$$E_{\,a} (-z^a) = \sum_{k=0}^\infty \frac{(-z^a)^k}{\Gamma(1+a k)}$$
with a table for the first values of $a$ :
\begin{array} {c|c}
a&E_a(x)\\
\hline
0&\frac 1{1-x}\\
1&e^x\\
2&\cosh\sqrt{x}\\
3&\frac 13\left(e^{x^{1/3}}+2\,e^{-\frac 12 x^{1/3}}\cos\left(\frac{\sqrt{3}}2x^{\frac 13}\right)\right)\\
4&\frac 12\left(\cos(x^{1/4})+\cosh(x^{1/4})\right)\\
\end{array}
Of course the corresponding expressions for $E_a(z^a)$ for $a$ a positive integer would be somewhat simpler and this for a good reason :
we are simply using the Taylor series of the exponential $\;\displaystyle \exp(z):=\sum_{k=0}^\infty \frac {z^k}{k!}\;$ and keeping only the terms with $k$ multiple of $\,a\,$ which may be rewritten as :
$$E_a(z^a)=\frac 1a\sum_{k=0}^{a-1}\exp\left(z\;e^{\dfrac{2\pi i k}a}\right)$$
(for more details see this answer )
For $a=\frac n2$ you should get the hypergeometric function proposed by Claude for $x=-z^a$
$$E_{\frac n2}(x)=  _0F_{n-1}\left(;\frac{1}{n},\frac{2}{n},\cdots,\frac{n-1}n;\frac{x^2}{n^n}\right)+\frac {2^{(n+1)/2}}{n!\sqrt{\pi}} {}_1F_{2n-1}\left(1;\frac{n+2}{2n},\frac{n+3}{2n},\cdots,\frac{3n}{2n};\frac{x^2}{n^n}\right)$$
The name of the function should help you to find more (for example the previous results) :
Haubold, Mathai and Saxena's "Mittag-Leffler Functions and Their Applications".
A: $$f_a=\sum_{k=0}^\infty\frac{(-1)^kz^{ak}}{\Gamma(1+ak)}$$
If $a$ is a positive integer, asking Wolfram Alpha, you will get the expressions for $a=1,2,3,4$.
For larger values, only hypergeometric functions
$$f_a=\,
   _0F_{a-1}\left(;\frac{1}{a},\frac{2}{a},\cdots,\frac{a-1}
   {a};-\left(\frac{z}{a}\right)^a\right)$$
