Disturbing the Dyson conjecture in a \emph{generally} GOOD way

TL;DR

This paper extends Dyson's conjecture by providing closed-form expressions for specific coefficients in the Dyson product, introduces conjectures for their q-analogs, and presents perturbed q-Dixon summation formulas, challenging the original conjecture in a constructive way.

Contribution

The paper offers new explicit formulas for coefficients in Dyson's product, extends these results to q-analogs, and introduces perturbed q-Dixon formulas, advancing understanding of Dyson's conjecture.

Findings

Closed form expressions for several coefficients in Dyson's product.

Conjectures for q-analogs of these coefficients.

Perturbed q-Dixon summation formulas.

Abstract

Dyson's celebrated constant term conjecture ({\em J. Math. Phys.}, 3 (1962): 140--156) states that the constant term in the expansion of $\prod_{1\leqq i\neq j\leqq n} (1-x_i/x_j)^{a_j}$ is the multinomial coefficient $(a_1 + a_2 + \cdots + a_n)!/ (a_1! a_2! \cdots a_n!)$. The definitive proof was given by I. J. Good ({\em J. Math. Phys.}, 11 (1970) 1884). Later, Andrews extended Dyson's conjecture to a $q$-analog ({\em The Theory and Application of Special Functions}, (R. Askey, ed.), New York: Academic Press, 191--224, 1975.) 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.

Equations

$$\mathbf{a}:=⟨a_1,a_2,\dots ,a_n⟩,$$

$$\mathbf{x}:=⟨x_1,x_2,\dots ,x_n⟩,$$

$$0:=⟨0,0,\dots ,0⟩,$$

$${\mathbf{e}}_k:=⟨0,0,\dots ,0,1,0,0,\dots ,0⟩,$$

$${\sigma }_n(\mathbf{a}):=a_1+a_2+\cdots +a_n,$$

$$(A;q{)}_n:=\prod _{i=0}^{n-1}(1-A{q}^i),$$

$$F_n(\mathbf{x};\mathbf{a}):=\prod _{1\leqq i<j\leqq n}{\left(1-\frac{x_i}{x_j}\right)}^{a_j}{\left(1-\frac{x_j}{x_i}\right)}^{a_i}$$

$${\mathcal{F}}_n(\mathbf{x};\mathbf{a};q):=\prod _{1\leqq i<j\leqq n}{\left(\frac{x_{i}q}{x_j};q\right)}_{a_j}{\left(\frac{x_j}{x_i};q\right)}_{a_i},$$

$$[x^3{y}^2](3+5x^3{y}^2-6xy)=5,$$

$$[1](3+5x^3{y}^2-6xy)=[x^0{y}^0](3+5x^3{y}^2-6xy)=3,$$

$$[x{y}^2](3+5x^3{y}^2-6xy)=0.$$

$$[1]F_n(\mathbf{x};\mathbf{a})=\frac{{\sigma }_n(\mathbf{a})!}{a_1!a_2!\cdots a_n!}.$$

$$[1]{\mathcal{F}}_n(\mathbf{x};\mathbf{a};q)=\frac{(q;q{)}_{{\sigma }_n(\mathbf{a})}}{(q;q{)}_{a_1}(q;q{)}_{a_2}\cdots (q;q{)}_{a_n}}.$$

$$[x_1^{b_1}{x}_2^{b_2}\cdots x_n^{b_n}]F_n(\mathbf{x};\mathbf{a}),$$

$$[x_r/x_s]F_n(\mathbf{x};\mathbf{a})=-\left(\frac{a_s}{1+{\sigma }_n(\mathbf{a})-a_s}\right)\frac{{\sigma }_n(\mathbf{a})!}{a_1!a_2!\cdots a_n!}.$$

$$[x_r/x_s]{\mathcal{F}}_n(\mathbf{x};\mathbf{a};q)=-q^{L(r,s)}\left(\frac{1-q^{a_s}}{1-q^{1+{\sigma }_n(\mathbf{a})-a_s}}\right)\frac{(q;q{)}_{{\sigma }_n(\mathbf{a})}}{(q;q{)}_{a_1}(q;q{)}_{a_2}\cdots (q;q{)}_{a_n}},$$

$$L(r,s)=\left\{\begin{array}[]{ll}1+\sigma_{n}(\mathbf{a})-\sum_{k=r}^{s}a_{k},&\mbox{if $r<s$}\\ \sum_{k=s+1}^{r-1}a_{k},&\mbox{if $r>s$.}\end{array}\right.$$

$$\underset{i\ne k}{\prod _{i=1}^n}{\left(1-\frac{x_i}{x_s}\right)}^{a_s},$$

$$\left[\frac{x_{r}^{2}}{x_{s}x_{t}}\right]F_{n}(\mathbf{x};\mathbf{a})=\left(\frac{a_{s}a_{t}\Big{(}(1+\sigma_{n}(\mathbf{a}))+(1+\sigma_{n}(\mathbf{a})-a_{s}-a_{t})\Big{)}}{(1+\sigma_{n}(\mathbf{a})-a_{s}-a_{t})(1+\sigma_{n}(\mathbf{a})-a_{s})(1+\sigma_{n}(\mathbf{a})-a_{t})}\right)\frac{\sigma_{n}(\mathbf{a})!}{a_{1}!a_{2}!\cdots a_{n}!},$$

$$\displaystyle\left[\frac{x_{r}^{2}}{x_{s}x_{t}}\right]\mathcal{F}_{n}(\mathbf{x};\mathbf{a};q)=q^{L(r,s,t)}\left(\frac{(1-q^{a_{s}})(1-q^{a_{t}})\Big{(}(1-q^{1+\sigma_{n}(\mathbf{a})})+q^{M(r,s,t)}(1-q^{1+\sigma_{n}(\mathbf{a})-a_{s}-a_{t}})\Big{)}}{(1-q^{1+\sigma_{n}(\mathbf{a})-a_{s}-a_{t}})(1-q^{1+\sigma_{n}(\mathbf{a})-a_{s}})(1-q^{1+\sigma_{n}(\mathbf{a})-a_{t}})}\right)$$

$$\times \frac{(q;q{)}_{{\sigma }_n(\mathbf{a})}}{(q;q{)}_{a_1}(q;q{)}_{a_2}\cdots (q;q{)}_{a_n}},$$

$$L(r,s,t)=\left\{\begin{array}[]{ll}2+2\sigma_{n}(\mathbf{a})-2\sum_{k=r}^{t}a_{k}+\sum_{k=s+1}^{t-1}a_{k},&\mbox{if $r<s<t$,}\\ 1+\sigma_{n}(\mathbf{a})-\sum_{k=s}^{t}a_{k}+2\sum_{k=s+1}^{r-1}a_{k},&\mbox{if $s<r<t$,}\\ 2\sum_{k=t+1}^{r-1}a_{k}+\sum_{k=s+1}^{t-1}a_{k},&\mbox{if $s<t<r$,}\end{array}\right.$$

$$M(r,s,t)=\left\{\begin{array}[]{ll}a_{t},&\mbox{if $r<s<t$ or $s<t<r$,}\\ a_{s},&\mbox{if $s<r<t$.}\end{array}\right.$$

$$\left[\frac{x_{r}x_{s}}{x_{t}x_{u}}\right]F_{n}(\mathbf{x};\mathbf{a})=\left(\frac{a_{t}a_{u}\Big{(}(1+\sigma_{n}(\mathbf{a}))+(1+\sigma_{n}(\mathbf{a})-a_{t}-a_{u})\Big{)}}{(1+\sigma_{n}(\mathbf{a})-a_{t}-a_{u})(1+\sigma_{n}(\mathbf{a})-a_{t})(1+\sigma_{n}(\mathbf{a})-a_{u})}\right)\frac{\sigma_{n}(\mathbf{a})!}{a_{1}!a_{2}!\cdots a_{n}!}.$$

$$\displaystyle\left[\frac{x_{r}x_{s}}{x_{t}x_{u}}\right]\mathcal{F}_{n}(\mathbf{x};\mathbf{a};q)=q^{L(r,s,t,u)}\left(\frac{(1-q^{a_{t}})(1-q^{a_{u}})\Big{(}(1-q^{1+\sigma_{n}(\mathbf{a})})+q^{M(r,s,t,u)}(1-q^{1+\sigma_{n}(\mathbf{a})-a_{t}-a_{u}})\Big{)}}{(1-q^{1+\sigma_{n}(\mathbf{a})-a_{t}-a_{u}})(1-q^{1+\sigma_{n}(\mathbf{a})-a_{t}})(1-q^{1+\sigma_{n}(\mathbf{a})-a_{u}})}\right)$$

$$\times \frac{(q;q{)}_{{\sigma }_n(\mathbf{a})}}{(q;q{)}_{a_1}(q;q{)}_{a_2}\cdots (q;q{)}_{a_n}},$$

$$L(r,s,t,u)=\left\{\begin{array}[]{ll}2+2\sigma_{n}(\mathbf{a})-2\sum_{k=r}^{u}a_{k}+\sum_{k=r}^{s-1}a_{k}+\sum_{k=t+1}^{u-1}a_{k},&\mbox{if $r<s<t<u$,}\\ 1+\sigma_{n}(\mathbf{a})-\sum_{k=r}^{u}a_{k}+\sum_{k=t+1}^{s-1}a_{k},&\mbox{if $r<t<s<u$,}\\ 1+\sigma_{n}(\mathbf{a})-\sum_{k=r}^{s-1}a_{k}+2\sum_{k=t+1}^{r-1}a_{k}+\sum_{k=t+1}^{u-1}a_{k}+2\sum_{k=u+1}^{s-1}a_{k},&\mbox{if $r<t<u<s$,}\\ 1+\sigma_{n}(\mathbf{a})-\sum_{k=t}^{u}a_{k}+\sum_{k=r}^{s-1}a_{k}+2\sum_{k=t+1}^{r-1}a_{k},&\mbox{if $t<r<s<u$,}\\ \sum_{k=t+1}^{r-1}a_{k}+\sum_{k=u+1}^{s-1}a_{k},&\mbox{if $t<r<u<s$,}\\ \sum_{k=r}^{s-1}a_{k}+\sum_{k=t+1}^{u-1}a_{k}+2\sum_{k=u+1}^{r-1}a_{k},&\mbox{if $t<u<r<s$,}\end{array}\right.$$

$$M(r,s,t,u)=\left\{\begin{array}[]{ll}a_{u},&\mbox{if $r<s<t<u$ or $r<t<u<s$ or $t<u<r<s$,}\\ 1+\sigma_{n}(\mathbf{a})&\mbox{if $r<t<s<u$ or $t<r<u<s$,}\\ a_{t},&\mbox{if $t<r<s<u$.}\\ \end{array}\right.$$

$$[x_1^{b_1}{x}_2^{b_2}\cdots x_n^{b_n}]{\mathcal{F}}_n(\mathbf{x};\mathbf{a};q)=(-1{)}^{\tau }{q}^{b_1+b_2+\cdots +b_{\rho }+am}\frac{(q;q{)}_{an}(q^a;q^a{)}_{\tau }(q;q^a{)}_{\rho +\sigma }}{(q;q{)}_a^n(q;q^a{)}_n},$$

$$c_n^{\mathbf{b}}(\mathbf{a}):=[x_1^{b_1}{x}_2^{b_2}\cdots x_n^{b_n}]F_n(\mathbf{x};\mathbf{a}).$$

$$c_n^0(\mathbf{a})=\frac{{\sigma }_n(\mathbf{a})!}{a_1!a_2!\cdots a_n!}.$$

$$F_n(\mathbf{x};\mathbf{a})=\sum _{k=1}^n{F}_n(\mathbf{x};\mathbf{a}-{\mathbf{e}}_k).$$

$$c_n^0(\mathbf{a})=\sum _{k=1}^n{c}_n^0(\mathbf{a}-{\mathbf{e}}_k).$$

$$c_n^0(0)=1.$$

$$F_n(\mathbf{x};⟨a_1,a_2,\dots ,a_{k-1},0,a_{k+1},\dots a_n⟩)$$

$$=F_{n-1}(⟨x_1,x_2,\dots ,x_{k-1},x_{k+1},\dots ,x_n⟩;⟨a_1,\dots ,a_{k-1},a_{k+1},\dots a_n⟩)\left\{\underset{i\ne k}{\prod _{i=1}^n}\frac{(x_i-x_k{)}^{a_i}}{x_i^{a_i}}\right\}$$

Full text

††journal: J. Combin. Theory Ser. A

url]www.math.rutgers.edu/~asills

1 Introduction

1.1 Notation

For $n$ a nonnegative integer, we define the following symbols:

[TABLE]

and let $[Y]Z$ denote the coefficient of $Y$ in the expression $Z$, thus e.g.

[TABLE]

1.2 Background

F. J. Dyson [5, p. 152, Conjecture C] conjectured that the constant term in the Laurent polynomial $\prod_{1\leqq i<j\leqq n}\left(1-\frac{x_{i}}{x_{j}}\right)^{a_{j}}\left(1-\frac{x_{j}}{x_{i}}\right)^{a_{i}}$ is the multinomial coefficient; i.e.

Dyson’s conjecture * For $n\in\Zset_{+}$,*

[TABLE]

Dyson’s conjecture (1.1) was first proved independently by J. Gunson [9] and K. Wilson [17]. Later I. J. Good [8] supplied the most compact and elegant proof.

G. E. Andrews [1, p. 216] extended (1.1) to a $q$-analog:

Andrews’ $q$-Dyson conjecture * For $n\in\Zset_{+}$,*

[TABLE]

The first proof of (1.2) was given by D. Zeilberger and D. M. Bressoud [20]. Recently, another proof was given by I. M. Gessel and G. Xin [7].

In [14], together with Zeilberger, I showed that with the aid of our Maple/Mathematica packages GoodDyson, the computer can, subject only to limitations of time and memory capacity, conjecture a closed form expression for

[TABLE]

and automatically supply a proof for any fixed positive integer $n$ and fixed vector $\mathbf{b}=\langle b_{1},b_{2},\dots,b_{n}\rangle$.

1.3 Theorems and Conjectures

The results of [14] are extended here to generic $n$ for certain vectors $\mathbf{b}$, and a corresponding $q$-analog is conjectured for each. I made heavy use of Maple in forming these conjectures. I will prove

Theorem 1.1

Let $r$ and $s$ be fixed integers with $1\leqq r\neq s\leqq n$ and $n\geqq 2$. Then

[TABLE]

and provide a conjecture for its $q$-analog:

Conjecture 1.2** ($q$-analog of Theorem 1.1)**

Let $r$ and $s$ be fixed integers with $1\leqq r\neq s\leqq n$ and $n\geqq 2$. Then

[TABLE]

where

[TABLE]

Remark 1.3

Notice that the right hand side of Eq. (1.3) is independent of $r$, the subscript of the variable which appears to a positive power. In other words, $[x_{k}/x_{s}]F_{n}(\mathbf{x};\mathbf{a})$ is the same for all $k\neq s$. This can be explained by the fact that the only factors contributing to the $x_{k}/x_{s}$ term in the expansion of $F_{n}(\mathbf{x};\mathbf{a})$ are

[TABLE]

which is clearly invariant under any permutation of the subscripts of the $x_{i}$. The analogous phenomenon occurs in Theorems 1.4 and 1.6 as well.

Next, we have

Theorem 1.4

Let $r$, $s$, and $t$ be distinct fixed integers with $1\leqq r,s,t\leqq n$ and $n\geqq 3$. Then

[TABLE]

and the following conjecture for its $q$-analog:

Conjecture 1.5** ($q$-analog of Theorem 1.4)**

Let $r$, $s$, and $t$ be distinct fixed integers with $1\leqq r,s,t\leqq n$ and $n\geqq 3$. Without loss of generality we may assume that $s<t$. Then

[TABLE]

where

[TABLE]

and

[TABLE]

Finally, we have

Theorem 1.6

Let $r$, $s$, $t$, and $u$ be distinct fixed integers with $1\leqq r,s,t,u\leqq n$ and $n\geqq 4$. Then

[TABLE]

Conjecture 1.7** ($q$-analog of Theorem 1.6)**

Let $r$, $s$, $t$ and $u$ be distinct fixed integers with $1\leqq r,s,t,u\leqq n$ and $n\geqq 4$. Without loss of generality we may assume that $r<s$ and $t<u$. Then

[TABLE]

where

[TABLE]

and

[TABLE]

Remark 1.8

Certain special cases of Conjectures 1.2, 1.5, and 1.7 have been proved by John Stembridge [15, p. 347, Cor. 7.4]. Stembridge proved that in the case where $\mathbf{a}=\langle a,a,\dots,a\rangle$, and $b_{\rho+1}=b_{\rho+2}=\cdots=b_{\rho+\tau}=-1$, for $\rho$ and $\tau$ satisfying $0\leqq\rho\leqq n$ and $1\leqq\tau\leqq n-\rho$,

[TABLE]

where $m=\sigma\tau+\sum_{i=1}^{\rho}(i-1)b_{i}-\sum_{i=1}^{n-\rho-\tau}i\ b_{n-i+1}$. Conjectures 1.2, 1.5, and 1.7 do indeed agree with (1.4) where they overlap, which, of course, provides some evidence in favor of the conjectures.

The theorems will be proved in §2. Special cases of the conjectured $q$-analogs will be discussed in some detail in §3, followed by some concluding remarks in §4.

2 Generalized Good Proofs

2.1 Good’s proof of Dyson’s conjecture

It will be instructive to review the proof of (1.1) due to Good [8] presented in a way that will make it easy to see how it naturally generalizes to the variations of Dyson’s conjecture under consideration here. The proof divides neatly into three parts: recurrence, initial condition, and boundary conditions. Let

[TABLE]

Thus Dyson’s conjecture is the assertion that

[TABLE]

2.1.1 Recurrence

For $a_{1},a_{2},\dots,a_{n}>0$, we have, by Lagrange interpolation,

[TABLE]

Thus the same recurrence must hold term by term when (2.1) is expanded, and in particular the recurrence must hold for the constant term, so we have

[TABLE]

2.1.2 Initial Condition

It is easily verified that

[TABLE]

2.1.3 Boundary Conditions

For $k$ fixed and $1\leqq k\leqq n$,

[TABLE]

Notice that we have segregated the factors involving $x_{k}$ (those in braces) from those which are independent of $x_{k}$. Find the Taylor expansion of $\underset{i\neq k}{\prod_{i=1}^{n}}\frac{(x_{i}-x_{k})^{a_{i}}}{x_{i}^{a_{i}}}$ about $x_{k}=0$. Extract the coefficient of $x_{k}^{0}$ from both sides of (2.2) to obtain

[TABLE]

where

[TABLE]

In the case of Dyson’s original conjecture, we have $P_{k}^{\mathbf{0}}=1$ for all $k$ and $n$.

Apply the constant term operator to both sides of (2.3) to obtain

[TABLE]

for $k=1,2,\dots,n$.

Finally, since ($R$), ($I$), and ($B$) uniquely determine $c_{n}^{\mathbf{0}}(\mathbf{a}),$ and the multinomial coefficient $\sigma_{n}(\mathbf{a})!/a_{1}!\cdots a_{n}!$ also satisfies  ($R$), ($I$), and ($B$), the result follows. ∎

2.2 Proof of Theorem 1.1

Theorem 1.1 asserts that if $\mathbf{b}=\mathbf{e}_{r}-\mathbf{e}_{s}$,

[TABLE]

2.2.1 Recurrence

It was already noted that by Lagrange interpolation, for $a_{1},a_{2},\dots,a_{n}>0$, we have

[TABLE]

Thus the same recurrence must hold term by term when (2.6) is expanded, and in particular the recurrence must hold for the $x_{r}/x_{s}$ term, and so

[TABLE]

2.2.2 Initial Condition

[TABLE]

2.2.3 Boundary Conditions

For $k$ fixed and $1\leqq k\leqq n$,

[TABLE]

Once again, we have segregated the factors involving $x_{k}$ (those in braces) from those which are independent of $x_{k}$. Next, find the Taylor expansion of $\underset{i\neq k}{\prod_{i=1}^{n}}\frac{(x_{i}-x_{k})^{a_{i}}}{x_{i}^{a_{i}}}$ about $x_{k}=0$. Extract the coefficient of $x_{k}^{b_{k}}$ from both sides of (2.7) to obtain

[TABLE]

where

[TABLE]

and thus by extracting the coefficient of $x_{r}x_{s}^{-1}x_{k}^{b_{k}}$ from both sides of (2.8), we obtain

[TABLE]

where

[TABLE]

with $\delta_{i,j}$ denoting the Kronecker delta function.

2.2.4 The RHS of (2.5) also satisfies (R), (I), and (B)

Since  ($R^{\prime}$), ($I^{\prime}$), and (2.9) uniquely determine $c_{n}^{\mathbf{e}_{r}-\mathbf{e}_{s}}(\mathbf{a}),$ once we establish that $d^{\mathbf{e}_{r}-\mathbf{e}_{s}}_{n}(\mathbf{a}):=-\left(\frac{a_{s}}{1+\sigma_{n}(\mathbf{a})-a_{s}}\right)\left(\frac{\sigma_{n}(\mathbf{a})!}{a_{1}!a_{2}!\cdots a_{n}!}\right)$ also satisfies  ($R^{\prime}$), ($I^{\prime}$), and (2.9), the result will follow. While this fact may not be obious a priori, we shall soon see that nothing beyond elementary algebra is required to establish its truth.

Without loss of generality, we may assume that $r=1$ and $s=n$, for if not, the indeterminants in $F_{n}(\mathbf{x};\mathbf{a})$ may be relabeled accordingly. We note that

[TABLE]

[TABLE]

and thus ($R^{\prime}$) is satisfied.

Clearly,

[TABLE]

so ($I^{\prime}$) is satisfied.

Also,

[TABLE]

and thus $d_{n}^{\mathbf{e}_{r}-\mathbf{e}_{s}}(\mathbf{a})$ satisfies $(B^{\prime})$ when $a_{r}=0$.

Clearly,

[TABLE]

and so $d_{n}^{\mathbf{e}_{r}-\mathbf{e}_{s}}(\mathbf{a})$ satisfies $(B^{\prime})$ when $a_{s}=0$.

Finally, for $1<k<n$, we have

[TABLE]

where $d_{n}^{\mathbf{0}}(\mathbf{a})=\sigma_{n}(\mathbf{a})!/a_{1}!\cdots a_{n}!$ by (1.1), and thus $d_{n}^{\mathbf{e}_{r}-\mathbf{e}_{s}}(\mathbf{a})$ satisfies $(B^{\prime})$ when $k$ is different from both $r$ and $s$. ∎

Remark 2.1

Clearly, the only nontrivial difference between the proof of (1.1) and that of Theorem 1.1 lies in the observation that $P_{k}^{\mathbf{b}}$ (see (2.4)) varies with $\mathbf{b}$. Once $P_{k}^{\mathbf{b}}$ is known for a given $\mathbf{b}$, the boundary condition (($B$) and (2.9) in the two previous cases) follows immediately.

2.3 Proof of Theorem 1.4

In light of Remark 2.1, we need only supply $P_{k}^{\mathbf{b}}$, for $\mathbf{b}=2\mathbf{e}_{r}-\mathbf{e}_{s}-\mathbf{e}_{t}$.

[TABLE]

which implies

[TABLE]

2.4 Proof of Theorem 1.6

Similarly,

[TABLE]

which implies

[TABLE]

3 Perturbed versions of $q$-Dixon

It is well known (see [1]) that the $n=3$ case of the $q$-Dyson conjecture is equivalent to a $q$-analog of a hypergeometric summation formula of A. C. Dixon [4].

This is because

[TABLE]

where the last equality follows from a triple application of a corollary of the $q$-binomial theorem due to Rothe (see [3, p. 490, Cor. 10.2.2 (c)]), and

[TABLE]

It is then a straightforward exercise in linear algebra combined with the change of variable $k=j+c$ to obtain

[TABLE]

For $\alpha=\beta=0$, combined with the $n=3$ case of the $q$-Dyson theorem, we obtain the $q$-Dixon sum of Andrews [1, p. 216, equation (5.6)], which he proved using the $q$-Pfaff-Saalschütz summation (see [6, equation (II.12)].)

Similarly, the following six identities follow from the $n=3$ case of Conjecture 1.2:

[TABLE]

where

[TABLE]

The corresponding identities arising from the $n=3$ case of Conjecture 1.5 are

[TABLE]

[TABLE]

[TABLE]

Remark 3.1

Each of the identities (3.1) through (3.9) is a ${}_{3}\phi_{2}$ summation formula, and as such is automatically verifiable by the $q$-WZ algorithm of Wilf and Zeilberger [16]. It is well known that Zeilberger’s algorithm and its $q$-analog does not always find the minimal order recurrence satisfied by a given summand (see, e.g.  [2] or [12, p. 116 ff.]). In each case considered here, the $q$-Zeilberger algorithm, as implemented in Maple by Zeilberger’s package qEKHAD and in Mathematica by A. Riese’s package qZeil.m (see [11]), a recurrence of order at least three was found for the sum side, even though there must be a first order recurrence since the right hand side is a sum of a fixed number of finite products. Even Paule’s creative symmetrization technique (see [11, section 5.2]) does not improve the order of the recurrence in these examples.

Remark 3.2

The same technique could be used to produce $q$-hypergeometric summation formulas corresponding to the case $n=4$. Here the resulting sum sides would be triple sums, and one could attempt to obain automated proofs of these in Mathematica using Riese’sqMultiSum.m package of [13], or in Maple using Zeilberger’s qMultiZeilberger package  [19].

Due to computer memory and time limitations, it is highly doubtful that the identities corresponding to $n>4$ could be successfully handled on today’s computers.

4 Conclusion

The obvious next step is to try to find proofs for the conjectured $q$-analogs. A combinatorial proof would be particularly nice, since would potentially explain the role played by the factors $q^{L}$ and $q^{M}$ in the conjectures, a feature that disappears in the ordinary $q=1$ case.

I thank Doron Zeilberger for getting me interested in the $q$-Dyson conjecture, and for the many stimulating conversations which occured as a result. I also with to express my gratitude to the referees, whose thoughtful suggestions helped to improve the paper.