Journal of the Ramanujan Mathematical Society

Volume 37, Issue 3, September 2022  pp. 207–219.

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

Authors:  John Cullinan and Nick Scheel
Author institution:Department of Mathematics, Bard College, Annandale-On-Hudson, NY 12504

Summary:  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.