How to Extend Károlyi and Nagy's BRILLIANT Proof of the Zeilberger-Bressoud q-Dyson Theorem in order to Evaluate ANY Coefficient of the q-Dyson Product

Shalosh B. Ekhad and Doron Zeilberger

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Written: Aug. 15, 2013

Dedicated to Freeman Dyson on the occasion of his eighty-nine-and-two-thirds -th birthday

(Exclusively published in the Personal Journal of Shalosh B. Ekhad and Doron Zeilberger, and

One of the results that I was most proud of was the proof of, way back in my innocent, pre-computer days (1984) (even before I used TeX, and even before I used troff, I got a professional typist to convert the hand-written manuscript to a typed one (on paper)) was the Zeilberger-Bressoud q-Dyson Theorem conjectured, in 1975, by George Andrews, who very generously commented on p. 38 of his insightful lecture notes (based on a ten-lecture series that I was fortunate to attend, at Arizona State, 1985) that "the proof is an astounding tour de force".

Alas, that "astounding tour-de-force" took quite a few pages to prove. A shorter, lovely, formal Laurent series proof was given, in 2004, by Ira Gessel, and his brilliant former student, Guoce Xin, that took ten pages.

But like all human-proved results, Andrews' q-Dyson conjecture (that became the Zeilberger-Bressoud theorem) also turned out (in hindsight!) to be almost trivial. Last year, the amazing Hungarians (alias "aliens"), Gyula Károlyi and Zoltan Lóránt Nagy found the proof from the book, using a quantitative form, due to Karasev and Petrov, and independently to Lason, of Noga Alon's Combinatorial Nullstellensatz, that has already "trivialized" (or more politely, simplified) many "deep" results in combinatorics.

But a minor extension of their brilliant idea also implies a way to find not just the "constant term", but a closed-form expression for ANY coefficient, as the q-multinomial coefficient TIMES a certain rational function in (q, qa1, qa2, ..., qan). Now we can start getting non-trivial results, since the further you move from the constant-term, the more complicated that rational function becomes, but thanks to the Maple package qDYSON, Shalosh B. Ekhad found many such coefficients, and anyone who wishes can find many more!

The main "theoretical" novelty of this semi-expository, semi-implementation, article is the observation that ANY coefficient of the q-Dyson product is equal to the q-multinomial coefficient
(q)a1+...+an/((q)a1 ... (q)an)
times SOME (possibly and, usually, very complicated) RATIONAL function of
(q, qa1, ..., qan).
In fact, more strongly, it is what Herb Wilf z"l and I called "proper q-hypergeometric".

Maple Package

Added Sept. 23, 2013: watch the lecture:

Sample Output

Personal Journal of Shalosh B. Ekhad and Doron Zeilberger

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