An irreducibility question concerning modifications of Laguerre polynomials

Abstract

This paper addresses a question recently posed by Hajir concerning the irreducibility of certain modifications [Formula: see text] of generalized Laguerre polynomials [Formula: see text] where [Formula: see text] is an integer. For a fixed [Formula: see text], we obtain lower bounds [Formula: see text] on [Formula: see text] in terms of [Formula: see text] such that [Formula: see text] is irreducible over the rationals for all [Formula: see text]. Furthermore, for [Formula: see text], it is shown that [Formula: see text] is either irreducible or is a product of a linear polynomial and a polynomial of degree [Formula: see text]. The set of circumstances in which [Formula: see text] has a linear factor for [Formula: see text], is completely described.