Expressing non solvable roots?

Question

So i was playing around with $x^{n}-x-1$, for $n>4$ for this is irreducible over Q and the Galois group of its splitting field over Q corresponds to $\mathbf{S_{n}}$, thus is non solvable, and can not be expressed by radicals.
But on wolframe, if you click "exact" form you can apparently express them in something called hypergeometric generalized functions, and i am struggling to find resources or a theory behind them and how they are used to acheieve this.
So my question is, if there is some extended "galois theory" where you learn which roots over for example Q are expressibale using Hypergeometic functions or any other method , and when this happens? And if yoou can maybe express every single root theoreticly using something that resembles a closed form expression?
Thanks and happy holiday (:

Answer 1

I doubt that there is anything that works in all cases, but one particular "exact" form can be derived from a series expansion for a root of $y - c y^{1/n} - 1$. See e.g. my answer here.

Somewhat more generally, if you want a series solution, you'll want to expand around a known solution in powers of some parameter. Thus suppose you write your equation as $F(w)− \epsilon G(w)=0$, where $F$ and $G$ are analytic, and $F(0)=0$ so that $w=0$ is a solution when $\epsilon = 0$, while $G(0)\ne 0$ and $F′(0)\ne 0$. Then the Lagrange inversion theorem gives a series expansion for $w$ in powers of $\epsilon$, convergent for sufficiently small $\epsilon$.