Disturbing the Dyson conjecture, in a generally GOOD way. (English) Zbl 1114.33025

Sills, Andrew V.

J. Comb. Theory, Ser. A 113, No. 7, 1368-1380 (2006).

Dyson’s celebrated constant term conjecture [F. J. Dyson, “Statistical theory of the energy levels of complex systems I–III”, J. Math. Phys. 3, 140–156, 157–165, 166–175 (1962; Zbl 0105.41604)] states that the constant term in the expansion of \[ \prod _{1\leq i\neq j\leq n}\Big(1 - {{x_{i}}\over{x_{j}}}\Big)^{a j} \] is the multinomial coefficient \[ {{(a_1 + a_2 + \cdots + a_n)!}\over{(a_1!a_2! \cdots a_n!)}}. \] The definitive proof was given by I. J. Good [“Short proof of a conjecture of Dyson”, J. Math. Phys. 11, 1884 (1970)]. Later, Andrews extended Dyson’s conjecture to a \(q\)-analog [G. E. Andrews, “Problems and prospects for basic hypergeometric functions”, in: The Theory and Application of Special Functions, Academic Press, New York, 1975, 191–224 (1975; Zbl 0342.33001)].
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.

Reviewer: Wolfram Koepf (Kassel)

MSC:

33D45	Basic orthogonal polynomials and functions (Askey-Wilson polynomials, etc.)
05A15	Exact enumeration problems, generating functions
05A30	\(q\)-calculus and related topics

Keywords:

Dyson conjecture; \(q\)-Dyson conjecture; Zeilberger-Bressoud theorem; \(q\)-Dixon sum

Citations:

Zbl 0105.41604; Zbl 0342.33001

Software:

qZeil; qMultiSum; qMultiZeilberger; GoodDyson; NewZeil.m; GuessRat; qZEILBERGER; ZEILBERGER

References:

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