Extension of Laguerre polynomials with negative arguments

TL;DR

This paper investigates the irreducibility of certain Laguerre polynomials with negative arguments, extending previous results and providing new bounds on their factorization properties.

Contribution

It proves that, except for finitely many cases, these polynomials are either irreducible or have a linear factor times an irreducible polynomial, improving earlier estimates.

Findings

Proves irreducibility for most cases of the polynomial family.

Establishes a new inequality s > 1.9k for polynomial factors.

Sharpens previous bounds by Shorey, Tijdeman, and Nair.

Abstract

We consider the irreducibility of polynomial $L_n^{(\alpha)} (x) $ where $\alpha$ is a negative integer. We observe that the constant term of $L_n^{(\alpha)} (x) $ vanishes if and only if $n \geq |\alpha| = -\alpha$. Therefore we assume that $\alpha = -n-s-1$ where $s$ is a non-negative integer. Let $$ g(x) = (-1)^n L_n^{(-n-s-1)}(x) = \sum\limits_{j=0}^{n} a_j \frac{x^j}{j!} $$ and more general polynomial, let $$ G(x) = \sum\limits_{j=0}^{n} a_j b_j \frac{x^j}{j!} $$ where $b_j$ with $0 \leq j \leq n$ are integers such that $|b_0| = |b_n| = 1$. Schur was the first to prove the irreducibility of $g(x)$ for $s=0$. It has been proved that $g(x)$ is irreducibile for $0 \leq s \leq 60$. In this paper, by a different method, we prove : Apart from finitely many explicitely given posibilities, either $G(x)$ is irreducible or $G(x)$ is linear factor times irreducible polynomial. This is a consequence of the estimate $s > 1.9 k$ whenever $G(x)$ has a factor of degree $k \geq 2$ and $(n,k,s) \neq (10,5,4)$. This sharpens earlier estimates of Shorey and Tijdeman and Nair and Shorey.