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^{1}

Dedicated to Freeman Dyson (b. Dec. 15, 1923) on his 89\frac{2}{3}-th birthday

Very Important: As in all our joint papers, this document, the human-readable article, is not the main point, but its Maple implementation, qDYSON, written by DZ, available from

http://www.math.rutgers.edu/~zeilberg/tokhniot/qDYSON .

The ‘front’ of this article

http://www.math.rutgers.edu/~zeilberg/mamarim/mamarimhtml/qdyson.html ,

contains sample input and output files, chuckfull of rigorously-derived deep identities, computed by SBE.

In fact, since we believe in free open access, we supply the source code, that, in principle, is also humanly-readable, but only if one knows Maple, and since most people do not know it fluently enough, DZ kindly prepared the present article.

Let’s first do a redux of Gyula Károlyi and Zoltán Lóránt Nagy’s proof from the book[KN], of the Zeilberger-Bressoud[ZB] theorem (née Andrews’s q-Dyson Conjecture [An]), in a form that would be amenable for the extension promised in the title.

The Károlyi-Nagy Brilliant Proof of Zeilberger-Bressoud

Fact 1: If a polynomial of degree ≤ d vanishes at d + 1 different places it must be identically zero.

Proof: By induction on d. If d = 0 it is a constant, and since it happens to be zero somewhere, it must be zero everywhere, i.e. it must be the polynomial 0.

If a polynomial P (x) of degree ≤ d vanishes at d + 1 distinct places, let a be one of them. Then (thanks to Euclid), P (x) = (x−a)Q(x)+c for some polynomial Q(x) of degree ≤ d−1 and constant c, that must be 0, since P (a) = 0. Hence P (x) = (x − a)Q(x), and Q(x) is a polynomial of degree ≤ d − 1 that vanishes at d different places. By the induction hypothesis, it is the zero polynomial, and hence so is P (x).

1 Department of Mathematics, Rutgers University (New Brunswick), Hill Center-Busch Campus, 110 Frelinghuysen

Rd., Piscataway, NJ 08854-8019, USA. zeilberg at math dot rutgers dot edu , http://www.math.rutgers.edu/~zeilberg/ . Aug. 15, 2013. Accompanied by Maple package qDYSON downloadable from Zeilberger’s website. Supported in part by the NSF.

Fact 2: (Lagrange Interpolation Formula) If P (x) is a polynomial of degree ≤ d in x, and |A| = d+1, then

P (x) = ∑_{c∈A} ( ∏_{c′∈A\c} \frac{x − c′}{c − c′} ) P (c) .

Proof: The left-side minus the right side is a polynomial of degree ≤ d that vanishes at the d + 1 different members of A, and hence must be identically zero by Fact 1.

Fact 3: (Immediate consequence of the Lagrange Interpolation Formula) If P (x) is a polynomial of degree ≤ d in x, and |A| = d + 1, then

Coeff_{x^{d}}P (x) = ∑_{c∈A} \frac{P (c)}{∏_{c′∈A\c}(c − c′)} .

Proof: Extract the coefficient of x^{d} on both sides of Fact 2.

Fact 4: (‘Quantitative’ form ([L][KP]) of the Combinatorial Nullstellensatz ([A]), reproved in [KN], Lemma 2.1) Let F (x_{1}, …, x_{n}) be a polynomial of degree ≤ d_{1} + … + d_{n}. For arbitrary sets A_{1}, …, A_{n} with |A_{i}| = d_{i} + 1, the coefficient of ∏_{i=1}^{n} x_{i}^{d_{i}} in F (x_{1}, …, x_{n}) is

∑_{c_{1}∈A_{1}} ∑_{c_{2}∈A_{2}} … ∑_{c_{n}∈A_{n}} \frac{F (c_{1}, …, c_{n})}{ϕ′_{1}(c_{1})ϕ′_{2}(c_{2}) ⋯ ϕ′_{n}(c_{n})} ,

where ϕ_{i}(z) := ∏_{a∈A_{i}}(z − a), and ϕ′_{i}(c_{i}) is ϕ_{i}(z)/(z − c_{i}) evaluated at c_{i}.

New Proof: Since any polynomial of of degree ≤ d_{1}+…+d_{n} is a linear combination of monomials of degree ≤ d_{1} + … + d_{n}, it suffices, by linearity, to prove this for monomials F = x_{1}^{m_{1}} ⋯ x_{n}^{m_{n}} with m_{1} + … + m_{n} ≤ d_{1} + … + d_{n}. If there is an i such that m_{i} < d_{i} then the left side it 0, and the right side is 0 by Fact 3 with x = x_{i}, d = d_{i}, P (x_{i}) = x_{i}^{m_{i}}. If all m_{i} ≥ d_{i}, then, of course, m_{i} = d_{i} for all 1 ≤ i ≤ n, and the left side is 1 and the right side is 1^{n} = 1, by applying Fact 3 for each i, with P (x_{i}) = x_{i}^{d_{i}} and multiplying.

We are now ready for

Fact 5: (The Zeilberger-Bressoud Theorem([ZB])) Let, q and X be commuting indeterminates, and let n be a non-negative integer. Define first

(X)_{n} := ∏_{t=0}^{n−1}(1 − q^{t}X) .

Let a_{1}, …, a_{n} be non-negative integers, and let x_{1}, …, x_{n} be commuting indeterminates. The coefficient of x_{1}^{0} … x_{n}^{0} (i.e. the constant term) of

∏_{1≤i<j≤n}(x_{i}/x_{j})_{a_{i}}(qx_{j}/x_{i})_{a_{j}}  (qDyson)

equals the q-multinomial coefficient

\frac{(q)_{a_{1}+a_{2}+...+a_{n}}}{(q)_{a_{1}}(q)_{a_{2}} ⋯ (q)_{a_{n}}} .

Proof ([KN] with purely-routine stuff removed). If any of the a_{i} equals 0 then the theorem reduces to one with < n variables and would follow by induction on n, hence we can assume that all the a_{i} are strictly positive. Let σ = ∑_{i=1}^{n} a_{i}. We have to evaluate the coefficient of ∏_{i=1}^{n} x_{i}^{σ−a_{i}} of the polynomial (of degree (n − 1)σ)

F (x_{1}, …, x_{n}) := ∏_{1≤i<j≤n}(x_{i}/x_{j})_{a_{i}}(qx_{j}/x_{i})_{a_{j}} · x_{j}^{a_{i}}x_{i}^{a_{j}} .

Let’s Apply Fact 4 with F, d_{i} = σ − a_{i} and, for i = 1, …, n, A_{i} := {q^{α_{i}} ; 0 ≤ α_{i} ≤ σ − a_{i}} .

SubFact 5.1: If there exists a pair i, j with 1 ≤ i < j ≤ n such that

−(a_{i} − 1) ≤ α_{i} − α_{j} ≤ a_{j} ,

then F (q^{α_{1}}, …, q^{α_{n}}) = 0. Proof: Since (q^{−e})_{f} = 0 if 0 ≤ e < f, (x_{i}/x_{j})_{a_{i}}(qx_{j}/x_{i})_{a_{j}} with x_{i} = q^{α_{i}} and x_{j} = q^{α_{j}} vanishes, and hence so does F. SubFact 5.2: The set of lattice points (α_{1}, …, α_{n}) with 0 ≤ α_{i} ≤ σ − a_{i} such that for every pair 1 ≤ i < j ≤ n it is not the case that −(a_{i} − 1) ≤ α_{i} − α_{j} ≤ a_{j}, in other words the set

S(a_{1}, …, a_{n}) := {(α_{1}, …, α_{n}) ; 0 ≤ α_{i} ≤ σ − a_{i}} ⋂_{1≤i<j≤n} (α_{j} − α_{i} ≥ a_{i} OR α_{i} − α_{j} ≥ a_{j} + 1)

is the singleton set

{(0, a_{1}, a_{1} + a_{2}, …, a_{1} + … + a_{n−1})} .

Proof: The condition, for each 1 ≤ i < j ≤ n

α_{j} − α_{i} ≥ a_{i} OR α_{i} − α_{j} ≥ a_{j} + 1

is equivalent to, for each 1 ≤ i ≠ j ≤ n, α_{j} − α_{i} ≥ a_{i} + [i > j] , where [statement] is 1 and 0 respectively, according to whether the statement is true or false. Obviously all the α_{i} are distinct, hence there exists a unique permutation π ∈ S_{n} such that

α_{π(1)} < α_{π(2)} < … < α_{π(n)} .

Because of the conditions, we have

α_{π(i+1)} − α_{π(i)} ≥ a_{π(i)} + [π(i) > π(i + 1)] .

Adding up from i = 1 to i = n − 1 we have

α_{π(n)} − α_{π(1)} ≥ ∑_{i=1}^{n−1} a_{π(i)} + des(π) ,

where des(π) is the number of descents of π (i.e. the number of i, 1 ≤ i < n, for which π(i) > π(i + 1)). Hence

α_{π(n)} − α_{π(1)} ≥ σ − a_{π(n)} + des(π) .

But, α_{π(n)} ≤ σ − a_{π(n)} and α_{π(1)} ≥ 0 so, α_{π(n)} − α_{π(1)} ≤ σ − a_{π(n)}, and hence

σ − a_{π(n)} + des(π) ≤ σ − α_{π(n)} .

Hence des(π) ≤ 0. Of course des(π) ≥ 0, hence des(π) = 0, and hence π must be the identity permutation: π(i) = i.

Going back to α_{π(i+1)} − α_{π(i)} ≥ a_{π(i)} + [π(i) > π(i + 1)] with π = Identity, we have

α_{1} ≥ 0 , α_{2} − α_{1} ≥ a_{1} , α_{3} − α_{3} ≥ a_{2} , …, α_{n} − α_{n−1} ≥ a_{n−1} .

None of these inequalities can be strict, or else, adding-them-up would imply that α_{n} > σ − a_{n}. Hence, the only solution to the above linear-diophantine system of inequalities is

α_{1} = 0 , α_{2} = a_{1} , α_{3} = a_{1} + a_{2} , …, α_{n} = a_{1} + … + a_{n−1} .

SubFact 5.3: If ϕ(z) = ∏_{i=0}^{d}(z − q^{i}), and 0 ≤ j ≤ d then ϕ′(q^{j}) = ∏_{i=0}^{j−1}(q^{j} − q^{i}) ∏_{i=j+1}^{d}(q^{j} − q^{i}) equals q^{EasyT oCompute}(−1)^{AlsoEasyT oCompute} times (q)_{j}(q)_{d−j}.

SubFact 5.4: Any evaluation of F (x_{1}, …, x_{n}) where x_{i} = q^{L_{i}} and L_{i} are affine-linear expressions in a_{1}, …, a_{n} can be written in terms of various (q)_{L} for some affine-linear expressions L times q^{EasyT oCompute}(−1)^{AlsoEasyT oCompute}

Note: This is better left to a computer, see procedure EvalFcBG(a,c,r,q,S) in qDYSON.

Plugging-in the unique non-zero point in S(a_{1}, …, a_{n}), namely (0, a_{1}, …, a_{1} + … + a_{n−1}), and doing purely-routine manipulations (better left to the computer), lo-and-behold, we get what we want, namely the q-multinomial coefficient \frac{(q)_{a_{1}+a_{2}+...+an}}{(q)_{a_{1}} (q)_{a_{2}} ···(q)_{an}}. .

(End of proof of Fact 5 (alias the Zeilberger-Bressoud q-Dyson theorem)).

How to Evaluate Any Other Coefficient

Fact 6:(The Generalized Zeilberger-Bressoud q-Dyson Theorem) Let δ = (δ_{1}, …, δ_{n}) be a fixed, numeric, vector of integers that add-up-to 0. The coefficient of ∏_{i=1}^{n} x_{i}^{δ_{i}} in the q-Dyson product (qDyson) given above is

R_{δ}(q, q^{a_{1}}, …, q^{a_{n}}) · \frac{(q)_{a_{1}+a_{2}+...+a_{n}}}{(q)_{a_{1}}(q)_{a_{2}} ⋯ (q)_{a_{n}}} ,

for some easily-computable (using qDYSON) rational function R_{δ}. Furthermore, the denominator of R_{δ} is ‘nice’ (a product of terms of the form 1 − q^{L}, where L are affine-linear-combinations in the a_{i}’s).

Proof: Now we apply Fact 4 with

A_{i} := {q^{α_{i}} ; 0 ≤ α_{i} ≤ σ − a_{i} + δ_{i}} .

SubFact 6.1: If there exists a pair i, j with 1 ≤ i < j ≤ n such that

−(a_{i} − 1) ≤ α_{i} − α_{j} ≤ a_{j} ,

then F (q^{α_{1}}, …, q^{α_{n}}) = 0.

Proof: SubFact 6.1 is the same as SubFact 5.1, see the above proof.

SubFact 6.2: The set of lattice points (α_{1}, …, α_{n}) with 0 ≤ α_{i} ≤ σ − a_{i} + δ_{i} such that for every pair 1 ≤ i < j ≤ n it is not the case that −(a_{i} − 1) ≤ α_{i} − α_{j} ≤ a_{j}, in other words the set

S_{δ}(a_{1}, …, a_{n}) := {(α_{1}, …, α_{n}) ; 0 ≤ α_{i} ≤ σ−a_{i}+δ_{i}} ⋂_{1≤i<j≤n} (α_{j} − α_{i} ≥ a_{i} OR α_{i} − α_{j} ≥ a_{j} + 1)

is a finite set, easily constructed by the Maple package qDYSON, whose cardinality does not depend on (a_{1}, …, a_{n}), only on δ.

Proof: The condition, for each 1 ≤ i < j ≤ n

α_{j} − α_{i} ≥ a_{i} OR α_{i} − α_{j} ≥ a_{j} + 1

is equivalent to, for each 1 ≤ i ≠ j ≤ n,

α_{j} − α_{i} ≥ a_{i} + [i > j] ,

where [statement] is 1 and 0 respectively, according to whether the statement is true or false. Obviously all the α_{i} are distinct, hence there exists a unique permutation π ∈ S_{n} such that

α_{π(1)} < α_{π(2)} < … < α_{π(n)} .

Because of the conditions, we have

α_{π(i+1)} − α_{π(i)} ≥ a_{π(i)} + [π(i) > π(i + 1)] .

Adding up from i = 1 to i = n − 1, we have

α_{π(n)} − α_{π(1)} ≥ ∑_{i=1}^{n−1} a_{π(i)} + des(π) ,

where des(π) is the number of descents of π (i.e. the number of i, 1 ≤ i < n, for which π(i) > π(i + 1)). Hence

α_{π(n)} − α_{π(1)} ≥ σ − a_{π(n)} + des(π) .

But, α_{π(n)} ≤ σ − a_{π(n)} + δ_{π(n)} and α_{π(1)} ≥ 0 so, α_{π(n)} − α_{π(1)} ≤ σ − a_{π(n)} + δ_{π(n)}, and hence

σ − a_{π(n)} + des(π) ≤ σ − α_{π(n)} + δ_{π(n)} .

Hence des(π) ≤ δ_{π(n)}. For a given π ∈ S_{n} that satisfies this condition, we have

σ − a_{π(n)} + des(π) ≤ ∑_{i=1}^{n−1}(α_{π(i+1)} − α_{π(i)}) ≤ σ − a_{π(n)} + δ_{π(n)} .

Since

α_{π(i+1)} − α_{π(i)} ≥ a_{π(i)} + [π(i) > π(i + 1)] .

we can write, for i = 1, …, n − 1, and for some integers m_{i} ≥ 0 (1 ≤ i ≤ n): α_{π(1)} = m_{1}, and for 1 ≤ i ≤ n − 1:

α_{π(i+1)} − α_{π(i)} = a_{π(i)} + [π(i) > π(i + 1)] + m_{i+1} .

Summing from i = 1 to i = n − 1 gives

0 ≤ ∑_{i=1}^{n} m_{i} ≤ δ_{π(n)} − des(π) .

Of course, there are only finitely-many such {(m_{1}, …, m_{n})}. So, for each permutation π obeying des(π) ≤ δ_{π(n)} and for each vector (m_{1}, m_{2}, …, m_{n}) of non-negative integers whose sum is ≤ δ_{π(n)} − des(π), we have a member of S_{δ}(a_{1}, …, a_{n}),

α_{π(i)} = ∑_{r=1}^{i} m_{r} + ∑_{r=1}^{i−1}(a_{π(r)} + [π(r) > π(r + 1)]) .

Note that a_{i} are symbolic, and the sets of feasible π and (m_{1}, …, m_{n}) only depend on δ not on (a_{1}, …, a_{n}). It is immediately seen that these points satisfy the condition for membership in S_{δ}.

SubFact 6.3: For a given feasible permutation π and feasible vector m = (m_{1} …, m_{n}), and with A_{i} as above, the summand of Fact 4 is a simple (factored) rational function of (q, q^{a_{1}}, …, q^{a_{n}}) times the q-multinomial coefficient \frac{(q)_{a_{1}+a_{2}+...+an}}{(q)_{a_{1}} (q)_{a_{2}} ···(q)_{an}}.

Proof: Routine (and programmed into qDYSON).

Adding up these finitely many contributions concludes the proof of Fact 6.

Remark: One can get much smaller sets of evaluation-points S_{δ}, by shifting the A_{i}’s by (positive or negative) c_{i}, in other words consider

A_{i} := {q^{α_{i}} ; c_{i} ≤ α_{i} ≤ σ − a_{i} + δ_{i} + c_{i}} .

Of course, one should get the same output, regardless of the c’s, but for the sake of efficiency it would be nice to make S_{δ} as small as possible. The Maple package qDYSON has a procedure BestShift(d), that finds the optimal shift.

Another Remark: Fact 6 is only valid for numeric (specific) δ, and each numeric, specific, number of variables n. There is no closed form formula for the general coefficient of ∏_{i=1}^{n} x_{i}^{δ_{i}} of the q-Dyson product where the δ, as well as a = (a_{1}, …, a_{n}), are symbolic. For any specific n, it follows from WZ theory[WZ], that this quantity is holonomic, i.e. there exist ‘pure’ linear recurrences (in each of a_{i}) with coefficients that are polynomials in (q^{a_{1}}, …, q^{a_{n}}, q^{δ_{1}}, …, q^{δ_{n}}), but these are already fairly complicated for n = 3, and the orders get larger and larger with larger n.

The miracle of q-Dyson is that for the constant term, these recurrences are always first-order (that is what it means to be closed-form), for any number of variables, n, and the generalized Zeilberger-Bressoud (Fact 6) extends this miracle to any, specific, other coefficient.

Yet another remark: Even though Fact 6 is only valid, in general, for specific n and specific δ, it sometimes happens that if you take a specific, numeric, δ = (δ_{1}, …, δ_{n}) then the coefficient of δ_{r} := (δ_{1}, …, δ_{n}, 0(r times)) in the q-Dyson product in n+r variables can be expressed ‘uniformly’, since the size of S_{δ_{r}} (shifted by a judicious shift) remains the same. Of course, this is the case with the original q-Dyson, with δ = (0, …, 0), where S_{δ} is always a singleton, leading to a general statement valid for all n (i.e. symbolic n). This is also the case with the conjectures of Drew Sills[S2] proved by Lun Lv, Guoce Xin and Yue Zhou [LXZ], by extending the method of [GX]. In [LXZ] there is also a beautiful, much more general theorem. We are sure that their result can be reproved, quicker, using the Károly-Nagy approach, as extended in our present article, but we leave this to the interested reader.

The Maple package qDYSON

Everything (and more) is implemented in the Maple package qDYSON. Its only limitation is that the number of variables, n, is numeric, not symbolic, but, as noted above, often, by running it for n ≤ 5, one can deduce, and easily translate for general n, the proofs given by the package for specific n.

The main procedure is: ‘Gyula(z,d,q);’. It inputs a variable-name, z, a numeric list of intgers (of length n, say), d = [d_{1}, …, d_{n}] whose sum is 0, as well as a variable-name, q. It outputs the rational function R_{d}(q, q^{a_{1}}, …, q^{a_{n}}) promised by Fact 6, where, for the sake of clarity, q^{a_{i}} is replaced by z_{i}. For example,

‘Gyula(z,[2,-2,0,0],q);’ yields

(−z_{4}^{2}z_{2}^{2}q^{3}z_{3}z_{1} + z_{4}^{3}z_{2}^{2}q^{3}z_{3}z_{1} + z_{3}z_{2}^{2}z_{4}q^{3} − z_{2}^{2}q^{3}z_{3}^{2}z_{4}^{2}z_{1} + z_{2}^{2}q^{3}z_{3}^{3}z_{4}^{2}z_{1} − z_{2}^{2}z_{4}q^{2}z_{1}z_{3}−

z_{2}q^{2}z_{3} −z_{4}^{2}z_{2}q^{2} +z_{2}z_{4}q^{2}z_{1}z_{3} −z_{2}z_{3}^{2}z_{4}q^{2} +z_{3}z_{2}z_{4}q^{2} −z_{3}^{2}z_{4}^{2}z_{2}q^{2}z_{1} +z_{2}z_{4}qz_{1} +z_{1}z_{3}z_{2}q +z_{3}z_{4}q −z_{1})·

\frac{(1 − z_{1})}{(z_{4}z_{2}q − 1) (z_{3}z_{2}q − 1) (z_{2}z_{3}z_{4}q − 1) (z_{3}z_{2}z_{4}q^{2} − 1)} .

As d and/or n gets larger, the size of S_{d} gets larger, and Maple takes a long time to bring everything under a common denominator (using the command normal), so for these, a much faster alternative is the unsimplified version, ‘Zoltan (z,d,q);’ , that in some sense is better, since it displays the output as a sum of simple rational functions. For example, ‘Zoltan(z,[2,-2,0,0],q);‘ yields

−\begin{matrix}(−1 + z_{1}) & (−z_{1} + q) \\ (qz_{2}z_{3}z_{4} − 1) & (z_{3}z_{2}z_{4}q^{2} − 1) \end{matrix} + \frac{z_{2}q^{2} (−1 + z_{1}) (z_{4}z_{1} − 1) (−1 + z_{3}) (z_{2}qz_{3}z_{4}z_{1} − 1)}{(−1 + z_{2}q) (z_{4}z_{2}q − 1) (z_{2}z_{3}q − 1) (z_{3}z_{2}z_{4}q^{2} − 1)}

+\frac{z_{4}z_{2}q^{2} (−1 + z_{4}) (−1 + z_{1}) (z_{3}z_{1} − 1) (z_{2}qz_{3}z_{4}z_{1} − 1)}{(z_{4}z_{2}q − 1) (z_{2}z_{3}q − 1) (qz_{2}z_{3}z_{4} − 1) (z_{3}z_{2}z_{4}q^{2} − 1)}−\frac{q (−1 + z_{3}z_{4}) (−1 + z_{1}) (z_{2}z_{4}qz_{1} − 1) (z_{2}qz_{3}z_{4}z_{1} − 1)}{(−1 + z_{2}q) (z_{4}z_{2}q − 1) (qz_{2}z_{3}z_{4} − 1) (z_{3}z_{2}z_{4}q^{2} − 1)} .

Other important procedures are GyulaTh , ZoltanTh, that are verbose forms of the above, and Sefer, that outputs full articles. Some examples are given in the front of this article

http://www.math.rutgers.edu/~zeilberg/mamarim/mamarimhtml/qdyson.html .

Also of interest is procedure ‘BestShift(d);’, already mentioned above, that finds the best shift to make S_{d} as small as possible. There are also plenty of checking procedures, that make sure that everything is correct! Please consult the on-line help (gotten by typing ezra();, ezra1();,

ezraS(); , and ezraC();).

A very brief history

The original Dyson conjecture appeared in 1962, in a very important paper[Dy] (that according to google scholar (viewed Aug. 13, 2013) was cited 1483 times), and (whose sequel) lead to, inter alia, intriguing connections to the Riemann Zeta function discovered by Hugh Montgomery, and extended by Andrew Odlyzko, Jonathan Keating, Nina Snaith and others. The Dyson conjecture was proved shortly after by Jack Gunson[Gu] and Kenneth G. Wilson(1936-2013, Physics Nobel 1982)[W], who later on went on to revolutionize physics by creating renormalization group theory. The proof from the book of the original Dyson conjecture (using Fact 3 with d = n−1) was given[Go] in 1970 by Bayesian pioneer (and collaborator of Alan Turing at Bletchley Park) Jack Good(1916-2009). In 1982, one of us (Zeilberger) found a longer, but equally nice, combinatorial proof[Z]. A couple of years later, Dave Bressoud and Doron Zeilberger succeeded in q-ifing this proof, thereby giving the first proof of the q-analog, conjectured, in 1975, by George Andrews[An]. A shorter proof of the Zeilberger-Bressoud q-Dyson theorem was given by Ira Gessel and Guoce Xin [GX], and as already noted, the proof from the book was given by Károlyi and Nagy[KN], that formed the inspiration for the present article. Other far-reaching applications of their method are given by Gyula Károlyi in collaboration with Alain Lascoux and Ole Warnaar [KLW].

The problem of computing other coefficients, besides the constant term, for the original Dyson product, was launched by Sills and Zeilberger[SZ], followed by Sills’ more general article [S1], that was q-analogized in [S2], where he conjectured interesting ‘uniform’ expressions for a few other coefficients of the q-Dyson, proved, and vastly generalized, in the article [LXZ], already mentioned above.

Yet another approach

While we love the Károlyi-Nagy proof, as extended in this article, let us end by remarking that the original approach of [ZB] could also be used to prove Fact 6, and even implemented on a computer. Now the multi-tournaments of [Z] and [ZB] have different score vectors, and unlike the original case, where the ‘good guys’ can be completely translated into words, with no ‘left-overs’, now, we do get ‘minimal left-overs’. It is easy to see that, for specific δ, and specific n, there are only finitely many of them, and each of them could be q-counted. But, since the Károlyi-Nagy approach is so efficient, there is little motivation to extend the original Zeilberger-Bressoud approach.

References

[Al] N. Alon, Combinatorial Nullstellensatz, Combin. Probab. Comput. 8(1999) 7-29.

http://www.tau.ac.il/~nogaa/PDFS/null2.pdf

[An] G. E. Andrews, Problems and prospects for basic hypergeometric functions, in: “Theory and Application of Special Functions”, (R. A. Askey, ed.), Academic Press, New York (1975), 191-224.

[Dy] F. Dyson, Statistical theory of energy levels of complex systems, J. Math. Phys. 3(1962), 140-156.

[GX] I. M. Gessel and G. Xin, A short proof of the ZeilbergerBressoud q-Dyson theorem, Proc. Amer. Math. Soc. 134(2006), 2179-2187.

http://arxiv.org/abs/math/0412339

[Go] I. J. Good, Short proof of a conjecture by Dyson, J. Math. Phys. 11(1970) 1884.

[Gu] J. Gunson, Proof of a conjecture of Dyson in the statistical theory of energy levels, J. Math. Phys. 3(1962) 752-753.

[KLW] Gy. Károlyi, A. Lascoux, and S. O. Warnaar, Constant term identities and Poincaré polynomials, http://arxiv.org/abs/1209.0855 (also to appear in Trans. Amer. Math. Soc.).

[KN] Gy. Károlyi and Z.L. Nagy, A simple proof of the Zeilberger-Bressoud q-Dyson theorem, http://arxiv.org/abs/1211.6484 (also to appear in Proc. Amer. Math. Soc.) .

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#N32, 6 pages. http://arxiv.org/abs/1302.4647

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