Irreducibility of generalized Hermite-Laguerre Polynomials III

TL;DR

This paper extends previous irreducibility results of Laguerre and Hermite polynomials, showing that certain generalized Laguerre polynomials are irreducible for almost all cases, with a complete factorization provided for a specific exception.

Contribution

The authors generalize Schur's irreducibility results to a broader family of Laguerre polynomials with specific fractional parameters, identifying all cases of irreducibility and one explicit factorization.

Findings

Most Laguerre polynomials with specified parameters are irreducible.

Complete factorization provided for the case q=1/4, n=2.

General result encompasses previous special cases.

Abstract

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 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.