Irreducibility of generalized Hermite-Laguerre polynomials

TL;DR

This paper investigates the irreducibility of generalized Hermite-Laguerre polynomials for specific rational parameters and extends previous results, also providing new bounds on prime factors of certain arithmetic progressions.

Contribution

It establishes irreducibility results for $G_q(x)$ when $q$ is 1/3, 2/3, or of the form $u+1/2$, and introduces improved bounds for prime factors in specific arithmetic progressions.

Findings

Proved irreducibility of $G_q(x)$ for $q=1/3$ and $2/3$.

Extended irreducibility results to $q=u+1/2$ cases.

Derived new lower bounds for prime factors in arithmetic progressions with difference 2 and 3.

Abstract

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.