Theoretical Statistics and Mathematics Unit, ISI Delhi

Irreducibility of generalized Hermite-Laguerre Polynomials III

by Shanta Laishram and T. N. Shorey

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+\alpha)\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 $n$ except when
$q=\frac{1}{4}, n=2$ where we give the complete factorization. In fact, we derive it from a more general result.

isid/ms/2015/17 [fulltext]

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