An extension of a second irreducibility theorem of I. Schur

Let $n \neq 8$ be a positive integer such that $n+1 \neq 2^u$ for any integer $u\geq 2$. 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 $0\leq j\leq n-1$ belonging to $\mathbb{Z}[x]$ be polynomials having degree less than $\deg\phi(x)$. Assume that the content of $(a_na_0(x))$ is not divisible by any prime less than or equal to $n+1$. In this paper, we prove that the polynomial $f(x) = a_n\frac{\phi(x)^n}{(n+1)!}+ \sum\limits_{j=0}^{n-1}a_j(x)\frac{\phi(x)^{j}}{(j+1)!}$ is irreducible over the field $\mathbb{Q}$ of rational numbers. This generalises a well-known result of Schur which states that the polynomial $\sum\limits_{j=0}^{n}a_j\frac{x^{j}}{(j+1)!}$ with $a_j \in \mathbb{Z}$ and $|a_0| = |a_n| = 1$ is irreducible over $\mathbb{Q}$. We illustrate our result through examples.