Proof of a conjecture about unimodal polynomials
type: Item
instance of: scholarly article
title: Proof of a conjecture about unimodal polynomials
zbMATH Open document ID: 0678.05002
DOI: 10.1016/0022-314X(89)90096-6
author: Gert Almkvist
published in: Journal of Number Theory
publication date: 1989-01-01T00:00:00Z
review text: The author investigates the well-known conjecture that the polynomial \(\prod^{r}_{\nu =1}(1-t^{n\nu})/(1-t^{\nu})\) is symmetric unimodal if n is even and \(r\geq 1\), or n is odd and \(r\geq 11\). It is proved that conjecture is true for \(3\leq n\leq 20\) and for \(n=100\) and \(n=101\). The method of proof makes use of a clever induction for ``large'' r, say \(r\geq r_ 0(n)\), where \(r_ 0(n)\approx 10\) for large n. As by-product in the course of proof, two elegant inequalities for Chebyshev polynomials are obtained, namely \(| U_{n-1}(x)| \leq n^{x^ 2}\) and \(| T_ n'(x)| \leq n^{1+x^ 2}(| x| \leq 1).\)
Mathematics Subject Classification ID: 05A10
Mathematics Subject Classification ID: 05A17
Mathematics Subject Classification ID: 33C45
zbMATH DE Number: 4110695
zbMATH Keywords: unimodal polynomial
zbMATH Keywords: number of restrictive partitions
zbMATH Keywords: Chebyshev polynomials
Wikidata QID: Q123196650
reviewed by: Leetsch C. Hsu
MaRDI profile type: Publication
cites work: Representations of \(\text{SL}(2,\mathbb{C})\) and unimodal polynomials.
cites work: Q3947818
cites work: On the unimodality of some partition polynomials
cites work: Some aspects of groups acting on finite posets
full work available at URL: 0022-314x(89)90096-6
OpenAlex ID: W2002862278
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description: scientific article; zbMATH DE number 4110695