A proof of Andrews' q-Dyson conjecture

Doron Zeilberger; David M. Bressoud

doi:10.1016/j.disc.2006.03.023

A proof of Andrews' q-Dyson conjecture

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Abstract

Let (y)_a = (1 - y) (1 - qy) ⋯ (1 - q^{a - 1} y). We prove that the constant term of the Laurent polynomial ∏_{1 ≤ i < j ≤ n} (x_i / x_j)_ai (qx_j / x_i)_aj, where x₁, ..., x_n, q are commuting indeterminates and a₁, ..., a_n are non-negative integers, equals (q)_a1 _{+ ⋯ + an} / (q)_a1 ... (q)_an. This settles in the affirmative a conjecture of George Andrews (in: R.A. Askey, ed., Theory and Applications of Special Functions, Academic Press, New York, 1975, 191-224].

Original language	English (US)
Pages (from-to)	1039-1059
Number of pages	21
Journal	Discrete Mathematics
Volume	306
Issue number	10-11
DOIs	https://doi.org/10.1016/j.disc.2006.03.023
State	Published - May 28 2006
Externally published	Yes

All Science Journal Classification (ASJC) codes

Theoretical Computer Science
Discrete Mathematics and Combinatorics

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10.1016/j.disc.2006.03.023

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abstract = "Let (y)a = (1 - y) (1 - qy) ⋯ (1 - qa - 1 y). We prove that the constant term of the Laurent polynomial ∏1 ≤ i < j ≤ n (xi / xj)ai (qxj / xi)aj, where x1, ..., xn, q are commuting indeterminates and a1, ..., an are non-negative integers, equals (q)a1 + ⋯ + an / (q)a1 ... (q)an. This settles in the affirmative a conjecture of George Andrews (in: R.A. Askey, ed., Theory and Applications of Special Functions, Academic Press, New York, 1975, 191-224].",

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AU - Bressoud, David M.

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N2 - Let (y)a = (1 - y) (1 - qy) ⋯ (1 - qa - 1 y). We prove that the constant term of the Laurent polynomial ∏1 ≤ i < j ≤ n (xi / xj)ai (qxj / xi)aj, where x1, ..., xn, q are commuting indeterminates and a1, ..., an are non-negative integers, equals (q)a1 + ⋯ + an / (q)a1 ... (q)an. This settles in the affirmative a conjecture of George Andrews (in: R.A. Askey, ed., Theory and Applications of Special Functions, Academic Press, New York, 1975, 191-224].

AB - Let (y)a = (1 - y) (1 - qy) ⋯ (1 - qa - 1 y). We prove that the constant term of the Laurent polynomial ∏1 ≤ i < j ≤ n (xi / xj)ai (qxj / xi)aj, where x1, ..., xn, q are commuting indeterminates and a1, ..., an are non-negative integers, equals (q)a1 + ⋯ + an / (q)a1 ... (q)an. This settles in the affirmative a conjecture of George Andrews (in: R.A. Askey, ed., Theory and Applications of Special Functions, Academic Press, New York, 1975, 191-224].

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A proof of Andrews' q-Dyson conjecture

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