Consider the function $f(x) = e^{-x^2}$, notice that $f$ convoluted with itself is of the form $a\cdot f(bx+c)$ for reals $a, b$ and $c$. Another way of saying this is that the shape of the function $f(x)$ does not change when convoluted with itself. It only gets stretch and shifted.
Here is another such function: $f(x) = \frac1{1+x^2}$
I am interested in a categorization of all such functions. Is such a categorization known? I am interested in any resources discussing this topic. If you are unable to do either, I would still be interested in any example of such a function other than the two I listed.
The classification is known. You're looking for alpha-stable distributions.