LCM of sequences of polynomials
type: Item
instance of: scholarly article
title: LCM of sequences of polynomials
zbMATH Open document ID: 0786.11014
author: Edgard Bavencoffe
published in: Annales de la Faculté des Sciences de Toulouse. Mathématiques. Série VI
publication date: 1994-05-15T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/73298
review text: Let \({\mathbf P}= \{P_ k(T)\mid k\geq n_ 0\}\) be a sequence of monic polynomials with complex coefficients. Write \(\Pi(n,T,{\mathbf P})\) and \(M(n,T,{\mathbf P})\) for the product and least common multiple of \(\{P_ k(T)\mid n_ 0\leq k\leq n\}\), and let \(r(n,{\mathbf P})\) be the ratio of the degree of \(M(n,T,{\mathbf P})\) to the degree of \(\Pi(n,T,{\mathbf P})\). This paper is concerned with evaluating \(r({\mathbf P})= \lim_{n\to\infty} r(n,{\mathbf P})\) for certain sequences of polynomials, namely \(P_ k(T)= T^{kn}-w\), \(T^{km+q}-w\), and \((T^{k(nm+1)}- w)/(T^ k- w)\), where \(w\) is a primitive \(m\)-th root of unity. This has applications to second order linear recurrences. For example, for \(P_ k(T)=T^ k-w\), one has \[ r({\mathbf P})= {{6m\Omega_ 1(m) \Omega_ 2(m)} \over {\varphi(m)\pi^ 2}}, \] where \(\Omega_ 1(n)= \prod_{p\mid n}({p\over p+1})\), and \(\Omega_ 2(n)= \sum\{{1\over k^ 2}\mid 1\leq k\leq n\), \((k,n)=1\}\). (The English version of the abstract contains a number of misprints: one which is potentially misleading is the omission of ``the degree of'' before ``the LCM'' and ``the product of'').
Mathematics Subject Classification ID: 11C08
Mathematics Subject Classification ID: 11B37
zbMATH DE Number: 202534
zbMATH Keywords: linear recurrence sequence
zbMATH Keywords: degree of product
zbMATH Keywords: degree of least common multiple
zbMATH Keywords: monic polynomials with complex coefficients
reviewed by: David W. Boyd
MaRDI profile type: Publication
OpenAlex ID: W2329096135
cites work: A New Formula for &#960
cites work: Hilbert's Tenth Problem is Unsolvable
cites work: An asymptotic formula for \(\pi\)
cites work: Lehmer numbers and an asymptotic formula for \(\pi\)
cites work: Plus petit commun multiple des termes consécutifs d'une suite récurrente linéaire. (Least common multiple of consecutive terms of a linear recurrent sequence)
cites work: Q4293460
cites work: Q4101884
DOI: 10.5802/AFST.743
description: scientific article; zbMATH DE number 202534