Written: Aug. 15, 2013

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

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, q^a1, q^a2, ..., q^an). 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, q^a1, ..., q^an).
In fact, more strongly, it is what Herb Wilf z"l and I called "proper q-hypergeometric".

Maple Package

• qDYSON

• vimeo: Part 1 Part 2.
• youtube: Part 1 Part 2.

Sample Output

• To see all expressions (all (automatically!) rigorously proved) for all coefficients of x1^a1x2^a2x3^a3 with a1+a2+a3=0 (of course) (and hence a3=-a1-a2) and a1 ≥ a2 > 0 and a1+a2 ≤ 5 etc. as well as a1 ≤ 5 and a2+a3=-a1 and a2 ≤ a3, of q-Dyson product with 3 variables
  the input file yields the output file.

• To see an extension of the above for a1 +a2 ≤ 8 (same conditions otherwise)
  the input file yields the output file.

• To see all expressions (all (automatically!) rigorously proved) for all coefficients of x1^a1x2^a2x3^a3x4^a4 with a1+a2+a3+a4=0 (of course) and a1 ≥ a2 ≥ a3 > 0 (and hence a4=-a1-a2-a3) and a1+a2+a3 ≤ 3 etc. of the q-Dyson product with 4 variables
  the input file yields the output file.

• To see an extension of the above file for all a1+a2+a3 ≤ 5 (same conditions otherwise)
  the input file yields the output file.

• To see all expressions (all (automatically!) rigorously proved) for all coefficients of x1^a1x2^a2x3^a3x4^a4 x5^a5 with a1+a2+a3+a4+a5=0 (of course), of "distance" 2 from (0,0,0,0,0), and the above convention of the q-Dyson product with 5 variables
  the input file yields the output file.

• To see an extension of the above file, for all coefficients of of "distance" 3 from (0,0,0,0,0),
  the input file yields the output file.

• To see all expressions (all (automatically!) rigorously proved) for all coefficients of x1^a1x2^a2x3^a3x4^a4 x5^a5 x6^a6 with a1+a2+a3+a4+a5+a6=0 (of course), of "distance" 2 from (0,0,0,0,0,0), of the q-Dyson product with 6 variables
  the input file yields the output file.

• To see the explicit expression of the coefficient of x1³x2^-2x3^-4x4x5 in the q-Dyson product with 6 variables (x1,x2,x3,x4,x5,x6), but due to its size, left un-simplified, the
  the input file yields the output file.

Doron Zeilberger's Home Page