Irreducibility of generalized Hermite-Laguerre polynomials

Abstract

For a rational $q=u+\frac{\alpha }{d}$ with $u, α, d\in \ACOBZ$ 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}(α+(n-1+u)d)x^{n-1}+\cdots\\ &\quad+a_1\left(\prod^{n-1}_{i=1}(α+(i+u)d)\right)x+a_0 \left(\prod^{n-1}_{i=0}(α+(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.

Cited by in corpus (2)

• Irreducibility of generalized Hermite-Laguerre Polynomials II
• Irreducibility of generalized Hermite-Laguerre Polynomials III