On the factorization of Eulerian polynomials
type: Item
instance of: scholarly article
title: On the factorization of Eulerian polynomials
zbMATH Open document ID: 0535.10010
DOI: 10.1016/0022-314X(84)90050-7
author: Arnošt J. J. Heidrich
published in: Journal of Number Theory
publication date: 1984-01-01T00:00:00Z
review text: The Eulerian polynomials \(P_ k(x)\) can be defined by setting \(R_ k(x)=\sum^{k}_{s=0}A_{ks}x^ s\) where \(A_{ks}=\sum^{s}_{j=0}(-1)^ j\left( \begin{matrix} k+1\\ j\end{matrix} \right)(s-j)^ k,\) and then \(P_ k(x)=R_ k(x)\) if k is odd, \(P_ k(x)=R_ k(x)/(x+1)\) if k is even. It has been conjectured that \(P_ k(x)\) is irreducible over the rationals. The author obtains partial results in this direction by showing that \(P_ k(x)\) must have an irreducible factor of degree at least 11 k/14 and improving this lower bound, even proving the conjecture, for certain special values of k.
Mathematics Subject Classification ID: 11B39
Mathematics Subject Classification ID: 05A15
Mathematics Subject Classification ID: 11R09
zbMATH DE Number: 3849313
zbMATH Keywords: Eulerian polynomials
zbMATH Keywords: irreducible factor
zbMATH Keywords: lower bound
reviewed by: David M. Bressoud
MaRDI profile type: Publication
cites work: Eulerian Numbers and Polynomials
cites work: Q4927338
cites work: Q3259107
cites work: Q5800952
cites work: On the interval containing at least one prime number
cites work: Q4333657
cites work: Q5798024
cites work: An Arithmetical Theory of the Bernoulli Numbers
full work available at URL: 0022-314x(84)90050-7
OpenAlex ID: W2113349151
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description: scientific article; zbMATH DE number 3849313