Disturbing the Dyson conjecture, in a generally GOOD way

Abstract

Dyson's celebrated constant term conjecture [F.J. Dyson, Statistical theory of the energy levels of complex systems I, J. Math. Phys. 3 (1962) 140–156] states that the constant term in the expansion of ∏ 1 ≦ i ≠ j ≦ n ( 1 − x i / x j ) a j is the multinomial coefficient ( a 1 + a 2 + ⋯ + a n ) ! / ( a 1 ! a 2 ! ⋯ a n ! ) . The definitive proof was given by I.J. Good [I.J. Good, Short proof of a conjecture of Dyson, J. Math. Phys. 11 (1970) 1884]. Later, Andrews extended Dyson's conjecture to a q-analog [G.E. Andrews, Problems and prospects for basic hypergeometric functions, in: R. Askey (Ed.), The Theory and Application of Special Functions, Academic Press, New York, 1975, pp. 191–224]. 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.