An extension of a second irreducibility theorem of I. Schur

Let $n\mathrm{\ne }8$ be a positive integer such that $n+1\mathrm{\ne }{2}^u$ for any integer $u\ge 2$. Let $\phi (x)$ belonging to $\mathbb{Z}[x]$ be a monic polynomial which is irreducible modulo all primes less than or equal to $n+1$. Let $a_j(x)$ with $0\le j\le n-1$ belonging to $\mathbb{Z}[x]$ be polynomials having degree less than $\mathrm{deg}\phi (x)$. Assume that the content of $(a_n{a}_0(x))$ is not divisible by any prime less than or equal to $n+1$. In this paper, we prove that the polynomial $f(x)=a_n\frac{\phi (x{)}^n}{(n+1)!}+{\sum }_{j=0}^{n-1}{a}_j(x)\frac{\phi (x{)}^j}{(j+1)!}$ is irreducible over the field $\mathbb{Q}$ of rational numbers. This generalises a well-known result of Schur which states that the polynomial ${\sum }_{j=0}^n{a}_j\frac{x^j}{(j+1)!}$ with $a_j\in \mathbb{Z}$ and $\mathrm{\mid }{a}_0\mathrm{\mid }=\mathrm{\mid }{a}_n\mathrm{\mid }=1$ is irreducible over $\mathbb{Q}$. We illustrate our result through examples.