A proof of Andrews' \(q\)-Dyson conjecture
type: Item
instance of: scholarly article
title: A proof of Andrews' \(q\)-Dyson conjecture
zbMATH Open document ID: 0565.33001
DOI: 10.1016/0012-365X(85)90081-0
author: Doron Zeilberger
author: David M. Bressoud
published in: Discrete Mathematics
publication date: 1985-01-01T00:00:00Z
review text: The \(q\)-shifted factorial is defined by \((x;q)_ n=(1-x)(1-xq)\cdots(1- xq^{n-1})\). Andrews conjectured that the constant term in the Laurent polynomial \[ \prod_{1\leq i\leq j<n}(x_ i/x_ j;q)_{a_ i}(qx_ j/x_ i;q)_{a_ j} \] is \[ (q;q)_{a_ 1+\dots+a_ n}/(q;q)_{a_ 1}\dots(q;q)_{a_ n}, \] as an extension of an earlier conjecture of \textit{F. J. Dyson} [J. Math. Phys. 3, 140--156 (1962; Zbl 0105.41604)] that was proved by\textit{J. Gunson} [ibid. 3, 752--753 (1962; Zbl 0111.43903)] and \textit{K. G. Wilson} [ibid. 3, 1040--1043 (1962; Zbl 0113.21403)] (the case \(q=1)\). A combinatorial proof is given in the present paper. This is a major advance, and is one more indication that enumerative combinatorics has come of age, and should be learned by many people who could use it, as well as those who are developing it.
Mathematics Subject Classification ID: 33D15
Mathematics Subject Classification ID: 05A15
zbMATH DE Number: 3901160
zbMATH Keywords: q-Dyson conjecture
zbMATH Keywords: identities
zbMATH Keywords: constant term
Wikidata QID: Q60305308
reviewed by: R. Askey
MaRDI profile type: Publication
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full work available at URL: 0012-365x(85)90081-0
OpenAlex ID: W4205310059
DBLP publication ID: journals/dm/ZeilbergerB85
description: scientific article; zbMATH DE number 3901160