  For a rational $q=u+\frac{\alpha}{d}$ with $u, \alpha, d\in \ACOBZ$ with
$u\ge 0, 1\le \alpha<d$, $\gcd(\alpha, d)=1$, the \emph{generalized
Hermite-Laguerre polynomials $G_q(x)$} are defined by \begin{align*}
G_q(x)&=a_nx^n+a_{n-1}(\alpha +(n-1+u)d)x^{n-1}+\cdots\\
&\quad+a_1\left(\prod^{n-1}_{i=1}(\alpha +(i+u)d)\right)x+a_0
\left(\prod^{n-1}_{i=0}(\alpha +(i+u)d)\right) \end{align*} where $a_0, a_1,
\cdots, a_n$ are arbitrary integers. We prove some irreducibility results of
$G_q(x)$ when $q\in \{\frac{1}{3}, \frac{2}{3}\}$ and extend some of the
earlier irreducibility results when $q$ of the form $u+\frac{1}{2}$. We also
prove a new improved lower bound for greatest prime factor of product of
consecutive terms of an arithmetic progression whose common difference is 2 and
3.
