Irreducibility of extensions of Laguerre Polynomials

TL;DR

This paper proves the irreducibility of certain Laguerre polynomial extensions for most parameter values, identifying specific exceptions and extending previous results to broader parameter ranges.

Contribution

It establishes new irreducibility results for Laguerre polynomial extensions with explicit exceptions, generalizing prior work to wider parameter ranges.

Findings

Most Laguerre polynomial extensions are irreducible for specified parameters.

Explicit finite set of exceptions identified, with only one non-trivial exception.

Extends previous irreducibility results to larger parameter ranges.

Abstract

For integers $a_0,a_1,\ldots,a_n$ with $|a_0a_n|=1$ and either $\alpha =u$ with $1\leq u \leq 50$ or $\alpha=u+ \frac{1}{2}$ with $1 \leq u \leq 45$, we prove that $\psi_n^{(\alpha)}(x;a_0,a_1,\cdots,a_n)$ is irreducible except for an explicit finite set of pairs $(u,n)$. Furthermore all the exceptions other than $n=2^{12},\alpha=89/2$ are necessary. The above result with $0\leq\alpha \leq 10$ is due to Filaseta, Finch and Leidy and with $\alpha \in \{-1/2,1/2\}$ due to Schur.