Irreducibility of generalized Hermite-Laguerre Polynomials II

TL;DR

This paper proves that for all positive integers n, the generalized Hermite-Laguerre polynomials G_{1/4} and G_{3/4} are either irreducible or factor into a linear polynomial times an irreducible polynomial of degree n-1.

Contribution

It establishes the irreducibility or near-irreducibility of specific generalized Hermite-Laguerre polynomials for all positive integers n.

Findings

G_{1/4} and G_{3/4} are either irreducible or factor into a linear and an irreducible polynomial

The result holds for all n ≥ 1

Extends previous work on polynomial irreducibility

Abstract

In this paper, we show that for each $n\geq 1$, the generalised Hermite-Laguerre Polynomials $G_{\frac{1}{4}}$ and $G_{\frac{3}{4}}$ are either irreducible or linear polynomial times an irreducible polynomial of degree $n-1$.