A generalization of a fourth irreducibility theorem of I. Schur

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 $\mathrm{\mid }{a}_0\mathrm{\mid }=1$. In 1929, I. Schur established a general theorem about the factorization of $f(x)$ in the case that $\mathrm{\mid }{a}_n\mathrm{\mid }=1$. We establish a more general result in which $\mathrm{\mid }{a}_n\mathrm{\mid }$ is allowed to be larger, and show that the result is in some sense best possible.