Irreducibility of generalized Hermite-Laguerre Polynomials III

For a positive integer $n$ and a real number $\alpha$, the generalized Laguerre polynomials are defined by \begin{align*} L^{(\alpha)}_n(x)=\sum^n_{j=0}\frac{(n+\alpha)(n-1+\al)\cdots (j+1+\alpha)(-x)^j}{j!(n-j)!}. \end{align*} These orthogonal polynomials are solutions to \emph{Laguerre's Differential Equation} which arises in the treatment of the harmonic oscillator in quantum mechanics. Schur studied these Laguerre polynomials for its interesting algebraic properties. He obtained irreducibility results of $L^{(\pm \frac{1}{2})}_n(x)$ and $L^{(\pm \frac{1}{2})}_n(x^2)$ and derived that the Hermite polynomials $H_2n(x)$ and $\frac{H_{2n+1}(x)}{x}$ are irreducible for each $n$. In this article, we extend Schur's result by showing that the family of Laguerre polynomials $L^{(q)}_n(x)$ and $L^{(q)}_n(x^d)$ with $q\in \{\pm \frac{1}{3}, \pm \frac{2}{3}, \pm \frac{1}{4}, \pm \frac{3}{4}\}$, where $d$ is the denominator of $q$, are irreducible for every $n$. In fact, we derive it from a more general result.