How to Extend Károlyi and Nagy's BRILLIANT Proof of the
ZeilbergerBressoud qDyson Theorem in order to Evaluate ANY Coefficient of the qDyson Product
By
Shalosh B. Ekhad and Doron Zeilberger
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Written: Aug. 15, 2013
Dedicated to Freeman Dyson on the occasion of his eightynineandtwothirds th birthday
(Exclusively published in the Personal Journal of Shalosh B. Ekhad and Doron Zeilberger, and arxiv.org)
One of the results that I was most proud of was the proof of, way back in my innocent, precomputer days (1984)
(even before I used TeX, and even before I used troff, I got a professional typist to convert
the handwritten manuscript to a typed one (on paper)) was the
ZeilbergerBressoud qDyson Theorem conjectured, in 1975, by George Andrews, who very
generously commented on p. 38 of his
insightful lecture notes (based on a tenlecture series that I was fortunate to attend, at Arizona State, 1985)
that "the proof is an astounding tour de force".
Alas, that "astounding tourdeforce" 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 humanproved results, Andrews' qDyson conjecture (that became the ZeilbergerBressoud 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 closedform expression for ANY coefficient, as the qmultinomial coefficient TIMES a certain rational function in
(q, q^{a1}, q^{a2}, ..., q^{an}).
Now we can start getting nontrivial results, since the further you move from the constantterm, 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 semiexpository, semiimplementation, article is the
observation that ANY coefficient of the qDyson product is equal to the qmultinomial 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 qhypergeometric".
Maple Package
Added Sept. 23, 2013: watch the lecture:
Sample Output

To see all expressions (all (automatically!) rigorously proved) for
all coefficients of x1^{a1}x2^{a2}x3^{a3} with a1+a2+a3=0 (of course)
(and hence a3=a1a2)
and a1 ≥ a2 > 0 and a1+a2 ≤ 5 etc.
as well as a1 ≤ 5 and a2+a3=a1 and a2 ≤ a3,
of qDyson 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^{a1}x2^{a2}x3^{a3}x4^{a4} with a1+a2+a3+a4=0 (of course)
and a1 ≥ a2 ≥ a3 > 0 (and hence a4=a1a2a3) and a1+a2+a3 ≤ 3 etc.
of the qDyson 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^{a1}x2^{a2}x3^{a3}x4^{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 qDyson 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^{a1}x2^{a2}x3^{a3}x4^{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 qDyson product with 6 variables
the input file
yields the
output file.

To see the explicit expression of the coefficient of
x1^{3}x2^{2}x3^{4}x4x5 in the qDyson product with 6 variables
(x1,x2,x3,x4,x5,x6), but due to its size, left unsimplified, the
the input file
yields the
output file.
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