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arXiv · 2008.05165

On the Galois Theory of Generalized Laguerre Polynomials and Trimmed Exponential

Lior Bary-Soroker·Or Ben-Porath
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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 $f_{n,n+k}=\sum_{i=0}^{k} \frac{x^{i}}{(n+i)!}$ 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 $S_k$.

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BibTeXRIS

Lior Bary-Soroker, Or Ben-Porath. 2020-08-12. On the Galois Theory of Generalized Laguerre Polynomials and Trimmed Exponential. https://arxiv.org/abs/2008.05165

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