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\bf
PROOF OF A q-ANALOG OF A 
CONSTANT TERM IDENTITY CONJECTURED BY FORRESTER\rm

{\it Doron Zeilberger\footnote{*}{
Department of Mathematics, Temple University,
Philadelphia, PA19122, USA; zeilberg@euclid.math.temple.edu.
Supported in part by NSF grant DMS8901610.
}}

{\bf Appeared in:} J. Comb. Theory Ser. A {\bf 66}(1994), 311-312.

As always in q-theory, $(X;Q)_n$ will stand for the product
$(1-X) (1-QX) ... (1- Q^{n-1} X)$, and when the "base" $Q$ is
q, we will abbreviate $(X;q)_n$ to $(X)_n$.
For any Laurent polynomial $f$ in $ x_1 , ... , x_n$, 
$CT (f)$  denotes
the coefficient of $x_1^0.. x_n^0$.
Throughout this paper $t := q^{a \,, \,s=} q^b  \,,  \,u= q^c $. 

{\bf Theorem( q-extension of (4.2) in [F2])}:

$$
 CT \prod_{i=1}^{n} (1+ y x_i ) \prod_{i=1}^{n} { {( x_i )}_b {( {{q}\over {x_i}} )}_c  } \prod_{ 1 <= i < j <= n } { {( {{x_i}\over{ x_j}} ) }_a  {( {{ q x_j}\over{ x_i}} ) }_a  } \,= \, 
\eqno(*)
$$ 

$$
  \prod_{j=0}^{n-1} 
{{(q)_{b+c+ja} (q)_{(j+1)a} } \over { (q)_{b+ja} (q)_{c+ja} (q)_a } } 
\sum_{r=0}^{n}  {{ (t;t)_n (u;t)_r (qs;t)_{n-r} } \over { (t;t)_r (t;t)_{n-r} 
(qs;t)_n }} {(-qy)}^r  \, \, \,, 
$$

{\bf Proof:} As in [Z], first use the Stembridge-Stanton
trick, to transform (*) to the equivalent "anti-symmetric" version,
let's call it (*'), in which
${( {{ q x_j}\over{ x_i}} ) }_a $ is replaced by
${( {{ q x_j}\over{ x_i}} ) }_{a-1}$, and the right side
of (*) gets multiplied by $(1-t)/ (t;t)_a$.
Next expand the very first product on the left of (*'), into
a sum of $2^n$ terms, and note that they are all
{\it bad guys} (see [Z], p. 314 ), except for the $n+1$
terms $x_1 ... x_r y^r $, $r= 0 , ... , n$, the corresponding
constant terms of which are evaluated by [Z]'s eq. (5.1). QED

{\bf Historical Notes}: The special
case $q=1$, $b=1$,$c=0$, of the above theorem was
conjectured in [F1]. Shaun Cooper[C] formulated a
conjecture for the general $q$-case, with still $b=1$, $c=0$.
After Peter Forrester received a preliminary version of
the present paper, that proved Cooper's conjectured
q-extension of his original conjecture, he
also received the preprint [K1], from which[F2] he was able to
derive the special case $q=1$ of the above theorem.
Forrester told us that the 
general theorem should follow from [K2] in an analogous way.

The present proof is shorter (even with [Z]) than Kaneko's proof,
and entirely elementary, 
but Kaneko proves an even more general theorem.

{\bf Acknowledgement}: Many thanks are due to Shaun Cooper 
and Peter Forrester for helpful remarks.

{\bf References:}

[C] S. Cooper, {\it private communication}.

[F1] P.J. Forrester, 
\it
Some multidimensional integrals related to many body
systems with the $1/ r^2$ potential\rm
, preprint, La Trobe
University, Bundoora, Victoria, Australia.

[F2] ", \it
Selberg correlation integrals and the
$1/ r^2$ quantum many body system\rm
,preprint, La Trobe
University, Bundoora, Victoria, Australia.

[K1] J. Kaneko, \it
Selberg integrals and hypergeometric functions
associated with Jack polynomials\rm , preprint.

[K2] ", \it
q-Selberg integrals and hypergeometric functions
associated with Macdonald symmetric polynomials\rm , in preparation.

[Z] D. Zeilberger, \it
A Stembridge-Stanton style elementary
proof of the Habsieger-Kadell q-Morris identity\rm
, Discrete
Math. {\bf 79}(1989/90) 313-322.

Jan 18, 1992.

Revised: Jan 31, 1992 ; June 9, 1992.
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