Further generalizations of the first and the second irreducibility theorems of I. Schur

Let $g_{n}^{(\alpha)}(x)=a_{n}\frac{x^{n}}{(n+\alpha)!} +a_{n-1}\frac{x^{n-1}}{(n-1+\alpha)!}+\cdots + a_{1}\frac{x}{(1+\alpha)!} + \frac{a_{0}}{\alpha!}$ where $a_{j}$ are arbitrary integers with $\lvert a_{0}a_{n} \rvert =1$. I. Schur established the irreducibility of $g_{n}^{(0)}(x)$ and further showed that $g_{n}^{(1)}(x)$ is either irreducible or has a linear or a quadratic factor for every $n\ge 1$. M. Filaseta and T.Y. Lam extended Schur's results to a general non-negative integer $\alpha$.We extend these works to the case that $\lvert a_{0} \rvert \ge 1$.