I'm searching for the formula of the series $ \sum_{n=0}^{\infty}a^{n^l} $ I'm searching for the sum-formula (if exists) of the following power series:
$$
\sum_{n=0}^{\infty}a^{n^l}
$$
where $l=2,3,....$, and $|a|<1$.
 A: For $l\geq 3$, I believe the sum cannot be expressed in terms of standard special functions.
For $l=2$, the answer is (almost by definition)
$$\frac{1+\vartheta_3(a)}{2},$$
where $\vartheta_3(a)=\vartheta_3(0,a)$ denotes the Jacobi theta function of zero argument (theta constant).
A: The version of this question for $l = 3$ was asked here.
I was putting together a more useful answer than any that appear here or there when it was closed.  Here is that answer for the $l = 3$ case since it can't be provided there.

By the Fabry gap theorem, since $n^3/n \rightarrow \infty$ as $n \rightarrow \infty$, this function has the unit disk as a natural boundary in the complex plane.  This already makes is unusual in the basket of elementary, and "usual" transcendental functions.  However, as suggested in comments, this is a typical behaviour for a lacunary function.
Here's a plot of (the magnitude of) your function for complex inputs (necessarily from the open unit disk).

As is common for lacunary functions, the behaviour is calm near $0$ and becomes quite vigorous near the boundary of the disk.  (The vertical flailing is much larger in amplitude than shown -- the magnitudes are cut off at $0$ and $3$ in order to make the less violent parts of the plot clear.)
(Full disclosure:  I don't have a magic machine which can directly evaluate this function.  What is plotted is partial sums of the series.  Near $0$, fewer terms are used and near the boundary more terms are used.  About $100$ terms are used for points "at" the boundary -- this is probably excessive.)
If we restrict to the reals, we obtain (summing $300$ terms)

There seems to be a corner at the left edge of that, let's check that more closely.

So not a real corner.
(For the real plots, I have attempted to estimate error bounds for the truncation used. When plotted, the intervals guaranteed to contain the values of the function vanish behind the trace shown except for one or two pixels near $\pm 1$, so I don't bother showing them here.)
