A generalization of a fourth irreducibility theorem of I. Schur

TL;DR

This paper extends Schur's 1929 irreducibility theorem for a class of polynomials, allowing larger leading coefficients and demonstrating the optimality of the generalized result.

Contribution

It generalizes Schur's irreducibility theorem by permitting larger coefficients and proves that this generalization is essentially optimal.

Findings

Extended the irreducibility conditions to larger coefficients

Proved the generalized theorem is in some sense best possible

Built upon and generalized a classical result from 1929

Abstract

Let $u_{2j}$ be the product of the odd positive integers $< 2j$. For $n$ an integer $\ge 1$, 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.