On the Galois theory of generalized Laguerre polynomials and trimmed exponential
Volume 200 / 2021
Acta Arithmetica 200 (2021), 183-196
MSC: Primary 11R09; Secondary 11R32, 12E05, 33C45.
DOI: 10.4064/aa200825-7-3
Published online: 12 July 2021
Abstract
Inspired by the work of Schur on the Taylor series of the exponential and Laguerre polynomials, we study the Galois theory of trimmed exponentials $f_{n,n+k}=\sum _{i=0}^{k} {x^{i}/(n+i)!}$ 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 $S_k$.
