The irreducibility of all but finitely many Bessel polynomials
type: Item
instance of: scholarly article
title: The irreducibility of all but finitely many Bessel polynomials
zbMATH Open document ID: 0845.11037
DOI: 10.1007/BF02392470
author: Michael Filaseta
published in: Acta Mathematica
publication date: 1996-09-11T00:00:00Z
review text: Grosswald conjectured that the Bessel polynomials \(y_n(x)= \sum_{j= 0}^n x^j(n+ j)!/2^j(n- j)! j!\) are all irreducible over the rationals. The author proves that this is indeed the case for all but finitely many (possibly 0) nonnegative integers \(n\). In theory, the method used in this paper (which relies on results about Newton polygons of polynomials) would yield an effective bound on the number of reducible \(y_n(x)\). However, such a bound would be too large to be able to check Grosswald's conjecture for the remaining cases. More generally, the author shows that for any integers \(a_0, a_1,\dots, a_n\) with \(|a_0|= |a_n|= 1\) the polynomials \(\sum_{j= 0}^n a_j x^j(n+ j)!/2^j(n- j)! j!\) are irreducible for all but finitely many \(n\). He conjectures that this is even true for all \(n\).
Mathematics Subject Classification ID: 11R09
Mathematics Subject Classification ID: 12E05
Mathematics Subject Classification ID: 11B83
zbMATH DE Number: 811810
zbMATH Keywords: irreducibility
zbMATH Keywords: Bessel polynomials
zbMATH Keywords: Newton polygons of polynomials
reviewed by: C. Krattenthaler
MaRDI profile type: Publication
cites work: Irreducibility of Polynomials
cites work: The irreducibility of almost all Bessel polynomials
cites work: An estimate for the number of reducible Bessel polynomials of bounded degree
cites work: On the Maximal Distance of Numbers with a Large Prime Factor
description: scientific article; zbMATH DE number 811810