On the Galois Theory of Generalized Laguerre Polynomials and Trimmed Exponential

Inspired by the work of Schur on the Taylor series of the exponential and Laguerre polynomials, we study the Galois theory of trimmed exponentials $f_{n,n+k}={\sum }_{i=0}^k\frac{x^i}{(n+i)!}$ and of the generalized Laguerre polynomials $L_k^{(n)}$ of degree $k$. We show that if $n$ is chosen uniformly from $\{1,\dots ,x\}$, then, asymptotically almost surely, for all $k\leq x^{o(1)}$theGaloisgroupsof$f_{n,n+k}$andof$L_{k}^{(n)}$ are the full symmetric group $S_k$.