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 $L^{ (n)}_k$ of degree $k$. We show that if $n$ is chosen uniformly from $\{ 1,\ldots, x\}$, then, asymptotically almost surely, for all $k\leq x^{ o(1)}$ the Galois groups of $f_{ n,n+k}$ and of $L_{ k}^{ (n)}$ are the full symmetric group $S_k$.
Categories: math.NT math.AC