On the arithmetic of Pad\'e approximants to the exponential function

TL;DR

This paper investigates the algebraic properties of Padé approximants to exponential functions, focusing on their Galois groups and irreducibility by connecting them to generalized Laguerre polynomials.

Contribution

It establishes the Galois groups and irreducibility properties of diagonal Padé approximants for exponential functions, linking them to generalized Laguerre polynomials.

Findings

Determined Galois groups of certain Padé approximants.

Proved irreducibility in specific cases.

Connected Padé approximants to generalized Laguerre polynomials.

Abstract

The $(u,v)$-Pad\'e 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 Pad\'e 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.