Extension of Laguerre polynomials with negative arguments II

TL;DR

This paper investigates the factorization properties of extended Laguerre polynomials with negative arguments, providing explicit classifications for certain parameter ranges and offering new proofs for known irreducibility results.

Contribution

It explicitly determines the pairs (n, s) for which the polynomial is reducible when s ≤ 30, and offers a new proof of irreducibility for s ≤ 22.

Findings

Explicit classification of reducible pairs (n, s) for s ≤ 30.

New proof of irreducibility for s ≤ 22.

Extension of previous results on Laguerre polynomial irreducibility.

Abstract

For $n \geq 3$ and $s \leq 92$, it is proved in \cite{ShSi} that, except for finitely many pairs $(n, s), G_1(x) = G_1(x, n, s) $ is either irreducible or linear factor times an irreducible polynomial. If $s \leq 30$, we determine here explicitely the set of pairs $(n, s)$ in the above assertion. This implies a new proof of the result of Nair and Shorey \cite{NaSh1} that $G_1(x)$ is irreducible for $s \leq 22$.