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Zbl 0565.33001
Zeilberger, Doron; Bressoud, David M.
A proof of Andrews' $q$-Dyson conjecture. (English)
[J] Discrete Math. 54, 201-224 (1985). ISSN 0012-365X

The $q$-shifted factorial is defined by $(x;q)\sb n=(1-x)(1-xq)\cdots(1- xq\sp{n-1})$. Andrews conjectured that the constant term in the Laurent polynomial $$ \prod\sb{1\le i\le j<n}(x\sb i/x\sb j;q)\sb{a\sb i}(qx\sb j/x\sb i;q)\sb{a\sb j}$$ is $$(q;q)\sb{a\sb 1+\dots+a\sb n}/(q;q)\sb{a\sb 1}\dots(q;q)\sb{a\sb n}, $$ as an extension of an earlier conjecture of {\it F. J. Dyson} [J. Math. Phys. 3, 140--156 (1962; Zbl 0105.41604)] that was proved by{\it J. Gunson} [ibid. 3, 752--753 (1962; Zbl 0111.43903)] and {\it 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.
[R.Askey]

MSC 2000:: *33D15 Basic hypergeometric functions of one variable
05A15 Combinatorial enumeration problems

Keywords: q-Dyson conjecture; identities; constant term

Citations: Zbl 0105.41604; Zbl 0111.43903; Zbl 0113.21403

Cited in: Zbl 1126.05013 Zbl 1104.05009 Zbl 1094.33003 Zbl 0942.05004 Zbl 0581.05004 Zbl 0569.33002

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