A generalization of a fourth irreducibility theorem of I. Schur

Martha Allen; Michael Filaseta

paper

A generalization of a fourth irreducibility theorem of I. Schur

arXiv:2307.05877

Abstract

Let $u_{2 j}$ be the product of the odd positive integers $< 2 j$ . For $n$ an integer $\geq 1$ , define \[ f(x)=\sum_{j=0}^{n}a_j\frac{x^{2j}}{u_{2j+2}}, \] where the $a_{j}$ 's are arbitrary integers with $∣ a_{0} ∣ = 1$ . In 1929, I. Schur established a general theorem about the factorization of $f (x)$ in the case that $∣ a_{n} ∣ = 1$ . We establish a more general result in which $∣ a_{n} ∣$ is allowed to be larger, and show that the result is in some sense best possible.