Disturbing the Dyson conjecture in a \emph{generally} GOOD way

Andrew V. Sills

doi:10.1016/j.jcta.2005.12.005

paper

Disturbing the Dyson conjecture in a \emph{generally} GOOD way

arXiv:1812.05557 · doi:10.1016/j.jcta.2005.12.005

Abstract

Dyson's celebrated constant term conjecture ({\em J. Math. Phys.}, 3 (1962): 140--156) states that the constant term in the expansion of $\prod_{1 ≦ i \neq = j ≦ n} (1 - x_{i} / x_{j})^{a_{j}}$ is the multinomial coefficient $(a_{1} + a_{2} + \dots + a_{n})! / (a_{1}! a_{2}! \dots a_{n}!)$ . The definitive proof was given by I. J. Good ({\em J. Math. Phys.}, 11 (1970) 1884). Later, Andrews extended Dyson's conjecture to a $q$ -analog ({\em The Theory and Application of Special Functions}, (R. Askey, ed.), New York: Academic Press, 191--224, 1975.) In this paper, closed form expressions are given for the coefficients of several other terms in the Dyson product, and are proved using an extension of Good's idea. Also, conjectures for the corresponding $q$ -analogs are supplied. Finally, perturbed versions of the $q$ -Dixon summation formula are presented.

16 pages

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Disturbing the Dyson Conjecture (in a GOOD Way)

Disturbing the Dyson conjecture in a \emph{generally} GOOD way

Abstract

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