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

## TL;DR

This paper extends Dyson's conjecture by providing closed-form expressions for specific coefficients in the Dyson product, introduces conjectures for their q-analogs, and presents perturbed q-Dixon summation formulas, challenging the original conjecture in a constructive way.

## Contribution

The paper offers new explicit formulas for coefficients in Dyson's product, extends these results to q-analogs, and introduces perturbed q-Dixon formulas, advancing understanding of Dyson's conjecture.

## Key findings

- Closed form expressions for several coefficients in Dyson's product.
- Conjectures for q-analogs of these coefficients.
- Perturbed q-Dixon summation formulas.

## Abstract

Dyson's celebrated constant term conjecture ({\em J. Math. Phys.}, 3 (1962): 140--156) states that the constant term in the expansion of $\prod_{1\leqq i\neq j\leqq n} (1-x_i/x_j)^{a_j}$ is the multinomial coefficient $(a_1 + a_2 + \cdots + a_n)!/ (a_1! a_2! \cdots a_n!)$. 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.