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arXiv · 2007.01329

On the arithmetic of Padé approximants to the exponential function

John Cullinan·Nick Scheel
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Abstract

The $(u,v)$-Padé approximation to a function $f$ is the (unique, up to scaling) rational approximation $f(x) = P(x)/Q(x) + O(x^{u+v+1})$, where $P$ has degree $u$ and $Q$ has degree $v$. Motivated by recent work of Molin, Pazuki, and Rabarison, we study the arithmetic of the Padé approximants of the exponential polynomials. By viewing the approximants as certain Generalized Laguerre Polynomials, we determine the Galois groups of the diagonal approximants and prove some special cases of irreducibility.

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John Cullinan, Nick Scheel. 2020-07-02. On the arithmetic of Padé approximants to the exponential function. https://arxiv.org/abs/2007.01329

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