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\define\GaRaAA{1}
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\topmatter 
\title Determinant evaluations and $U(n)$ extensions of Heine's
$_2\phi_1$-transformations  
\endtitle 
\author
R. A. Gustafson\\
\address 
Department of Mathematics,
Texas A\&M University,
College Station, TX 77843, USA.
\it e-mail: \tt robert.gustafson\@math.tamu.edu.
\endaddress
C.~Krattenthaler\\
\address Institut f\"ur Mathematik der Universit\"at Wien,
Strudlhofgasse 4, A-1090 Wien, Austria.
\it e-mail: \tt KRATT\@Pap.Univie.Ac.At.
\endaddress
\endauthor 
\affil 
Department of Mathematics,
Texas A\&M University,\\
College Station, TX 77843, USA\\
e-mail: robert.gustafson\@math.tamu.edu\\\vskip6pt
Institut f\"ur Mathematik der Universit\"at Wien,\\
Strudlhofgasse 4, A-1090 Wien, Austria.\\
e-mail: KRATT\@Pap.Univie.Ac.At
\endaffil
%\email KRATT\@Pap.Univie.Ac.At\endemail
%\dedicatory \enddedicatory
%\date \enddate
\thanks The first author was supported in part by NSF
Grant DMS-9304290\endthanks
\thanks The second author was supported in
part by the 
Austrian Science Foundation FWF, grant P10191-PHY\endthanks

\cvol{00}
\cvolyear{0000}
\cyear{0000}

\subjclass Primary 33D20;
 Secondary 33C20, 33C70
\endsubjclass
\keywords 
Basic hypergeometric series for $U(n)$, $A_n$ basic hypergeometric
series, Heine transformation
\endkeywords
\abstract 
We give new proofs for $U(n)$ extensions of Heine's three
$_2\phi_1$-transformation formulas that were recently discovered in
another paper by the authors. These new proofs proceed by combining
Heine's original transformation formulas with certain determinant
evaluations. As a by-product we are able to generalize two of the
$U(n)$ transformation formulas.

\endabstract
\endtopmatter
\document
\leftheadtext{R. A. Gustafson and C. Krattenthaler}
\rightheadtext{$U(n)$ Heine transformations}
\subhead 1. Introduction\endsubhead
In the recent paper \cite{Gustafson
and Krattenthaler [1994]} we discovered $U(n)$ extensions of
Heine's $_2\phi_1$-transformation formulas (see (2.1)--(2.3) for
Heine's formulas) which
involve $U(n)$ basic hypergeometric series of a new type. With the
$q$-notation $(a)_n:=\prod _{i=1} ^{n}(1-aq^{i-1})$ (note that this
differs from the standard notation $(a;q)_n$ in Gasper and Rahman's
book (\cite{Gasper and Rahman [1990]}) in that we drop
the base $q$ for the sake of being able to write 
our formulas more compactly), these
transformation formulas are the following. Our $U(n)$ extension
(\cite{Gustafson and Krattenthaler [1994], (1.7)}) of the
first Heine transformation reads
$$\multline \sum _{k_1,\dots,k_n\ge0} ^{}\bigg(\prod _{1\le i<j\le n}
^{}\frac {1-\frac {X_j} {X_i} q^{k_j-k_i}} {1-\frac {X_j} {X_i}} \prod
_{i=1} ^{n} \frac {(A)_{k_i}\,(BX_i)_{k_i}}
{(q)_{k_i}\,(CX_i)_{k_i}}q^{k_i (1-i)}Z^{k_i}\bigg)\\
=\prod _{i=1} ^{n}\frac {(Aq^{1-i})_\infty\,(BZX_i)_\infty}
{(Zq^{1-i})_\infty \,(CX_i)_\infty}\sum _{k_1,\dots,k_n\ge0} ^{}
\bigg(\prod _{1\le i<j\le n}
^{}\frac {1-\frac {X_j} {X_i} q^{k_j-k_i}} {1-\frac {X_j}
{X_i}}\\
\times \prod _{i=1} ^{n} \frac {(Z)_{k_i}\,(CX_i/A)_{k_i}}
{(q)_{k_i}\,(BZX_i)_{k_i}}q^{k_i (1-i)}A^{k_i}\bigg),
\endmultline\tag1.1$$
provided the series terminate or $\vert q\vert<1$, $\vert Zq^{1-n}\vert<1$,
$\vert Aq^{1-n}\vert<1$. Our $U(n)$ extension
(\cite{Gustafson and Krattenthaler [1994], (1.8)}) of the
second Heine transformation reads
$$\multline \sum _{k_1,\dots,k_n\ge0} ^{}\bigg(\prod _{1\le i<j\le n}
^{}\frac {1-\frac {X_j} {X_i} q^{k_j-k_i}} {1-\frac {X_j} {X_i}} \prod
_{i=1} ^{n} \frac {(A)_{k_i}\,(BX_i)_{k_i}}
{(q)_{k_i}\,(CX_i)_{k_i}}q^{k_i (1-i)}Z^{k_i}\bigg)\\
=\prod _{i=1} ^{n}\frac {(Cq^{1-i}/B)_\infty\,(BZX_i)_\infty}
{(Zq^{1-i})_\infty \,(CX_i)_\infty}\sum _{k_1,\dots,k_n\ge0} ^{}
\bigg(\prod _{1\le i<j\le n}
^{}\frac {1-\frac {X_j} {X_i} q^{k_j-k_i}} {1-\frac {X_j}
{X_i}}\\
\times \prod _{i=1} ^{n} \frac {(ABZ/C)_{k_i}\,(BX_i)_{k_i}}
{(q)_{k_i}\,(BZX_i)_{k_i}}q^{k_i (1-i)}\(\frac {C} {B}\)^{k_i}\bigg),
\endmultline\tag1.2$$
provided the series terminate or $\vert q\vert<1$, $\vert Zq^{1-n}\vert<1$,
$\vert Cq^{1-n}/B\vert<1$. Finally, our $U(n)$ extension
(\cite{Gustafson and Krattenthaler [1994], (1.9)}) of the
third Heine transformation reads
$$\multline \sum _{k_1,\dots,k_n\ge0} ^{}\bigg(\prod _{1\le i<j\le n}
^{}\frac {1-\frac {X_j} {X_i} q^{k_j-k_i}} {1-\frac {X_j} {X_i}} \prod
_{i=1} ^{n} \frac {(A)_{k_i}\,(BX_i)_{k_i}}
{(q)_{k_i}\,(CX_i)_{k_i}}q^{k_i (1-i)}Z^{k_i}\bigg)\\
=\prod _{i=1} ^{n}\frac {(ABZq^{1-i}/C)_\infty}
{(Zq^{1-i})_\infty}\sum _{k_1,\dots,k_n\ge0} ^{}
\bigg(\prod _{1\le i<j\le n}
^{}\frac {1-\frac {X_j} {X_i} q^{k_j-k_i}} {1-\frac {X_j}
{X_i}}\\
\times \prod _{i=1} ^{n} \frac {(C/B)_{k_i}\,(CX_i/A)_{k_i}}
{(q)_{k_i}\,(CX_i)_{k_i}}q^{k_i (1-i)}\(\frac {ABZ} {C}\)^{k_i}\bigg),
\endmultline\tag1.3$$
provided the series terminate or $\vert q\vert<1$, $\vert Zq^{1-n}\vert<1$,
$\vert ABZq^{1-n}/C\vert<1$. In particular, from these identities there also
result $U(n)$ extensions of the $q$-Gauss summation formula and the
$q$-binomial theorem, cf\. \cite{Gustafson and Krattenthaler [1994]}.

We proved (1.1)--(1.3) in \cite{Gustafson and Krattenthaler [1994]} by induction with respect to
$n$. The purpose of this paper is to give new, direct proofs of these
identities, avoiding induction. These new proofs, which are described
in section~2 and are shorter
than the original ones, proceed by a combination of Heine's original
$_2\phi_1$-transformations and certain determinant evaluations. One
determinant evaluation is simply the Vandermonde determinant. The
other determinant evaluation, Lemma~1 in section~2, comes from
a determinant lemma (\cite{Krattenthaler [1990a], Lemma~2.2}) (see Lemma~2 in 
section~2) that has been already used successfully in a number of cases
by the second author (\cite{Gessel and Krattenthaler [1995], Krattenthaler [1990a], Krattenthaler [1990b], Krattenthaler
[1995]}) in the
computation of generating functions for tableaux and plane
partitions. In \cite{Gessel and Krattenthaler [1995]} this also lead to find $SU(n)$ basic
hypergeometric identities. However, the particular technique that we
use in the present paper seems to be new in the subject of
multidimensional basic hypergeometric series. Hopefully, it will prove
to be useful in other cases, too. 

As a by-product of our new proofs, we are able to generalize the
identities (1.2) and (1.3). We summarize the new (and old) results in
the Theorem in section~2 (identities (2.4)--(2.6)), and subsequently prove
them. Interestingly, in view of our new technique, the easiest to
prove is the third identity, (2.6). It just
requires the Vandermonde 
determinant and Heine's third $_2\phi_1$-transformation (which
is displayed in (2.3)). Next
in the hierarchy comes the second identity, 
(2.5). It requires a 
fancier determinant evaluation, Lemma~1, and Heine's second
$_2\phi_1$-transformation (which is displayed in (2.2)). The most
difficult one, again in 
view of our technique, is the first identity, (2.4). In fact, we do
not know how to prove (2.4) in a way similar to the proofs of the
other identities. However, once (2.5) and (2.6) are established, (2.4)
follows by simply applying (2.5) and (2.6) in succession.

\subhead 2. The $U(n)$ extensions of Heine's
$_2\phi_1$-transformations and their proofs\endsubhead\newline
Before we state the $U(n)$ extensions of Heine's
$_2\phi_1$-transformation formulas that we are going to prove, it
might be convenient to recall Heine's original
$_2\phi_1$-trans\-form\-at\-ions. These are (cf\. \cite{Gasper and Rahman [1990], (1.4.1),
(1.4.5), (1.4.6)})
$$\align
\sum _{k\ge0} ^{}\frac {(a)_k\,(b)_k} {(q)_k\,(c)_k}z^k&=  
  \frac {{( \let\over/ a)_\infty}\,(bz)_\infty}
{{( \let\over/  c)_\infty}\,(z)_\infty} 
\sum _{k\ge0} ^{}\frac {(c/a)_k\,(z)_k} {(q)_k\,(bz)_k}a^k,\tag2.1\\
\sum _{k\ge0} ^{}\frac {(a)_k\,(b)_k} {(q)_k\,(c)_k}z^k&=  
  \frac {{( \let\over/ {c\over b})_\infty}\,(bz)_\infty}
{{( \let\over/  c)_\infty}\,(z)_\infty} 
\sum _{k\ge0} ^{}\frac {(abz/c)_k\,(b)_k} {(q)_k\,(bz)_k}\(\frac {c}
{b}\)^k,\tag2.2\\  
\sum _{k\ge0} ^{}\frac {(a)_k\,(b)_k} {(q)_k\,(c)_k}z^k&=  
  \frac {{( \let\over/ {{a b z}\over c})_\infty}}
{{( \let\over/  z)_\infty}} 
\sum _{k\ge0} ^{}\frac {(c/a)_k\,(c/b)_k} {(q)_k\,(c)_k}\(\frac {abz}
{c}\)^k.\tag2.3\\  
\endalign$$

Our $U(n)$ extensions of these identities are the following. 
\proclaim{Theorem}There hold
$$\multline \sum _{k_1,\dots,k_n\ge0} ^{}\bigg(\prod _{1\le i<j\le n}
^{}\frac {1-\frac {X_j} {X_i} q^{k_j-k_i}} {1-\frac {X_j} {X_i}} \prod
_{i=1} ^{n} \frac {(A)_{k_i}\,(BX_i)_{k_i}}
{(q)_{k_i}\,(CX_i)_{k_i}}q^{k_i (1-i)}Z^{k_i}\bigg)\\
=\prod _{i=1} ^{n}\frac {(Aq^{1-i})_\infty\,(BZX_i)_\infty}
{(Zq^{1-i})_\infty \,(CX_i)_\infty}\sum _{k_1,\dots,k_n\ge0} ^{}
\bigg(\prod _{1\le i<j\le n}
^{}\frac {1-\frac {X_j} {X_i} q^{k_j-k_i}} {1-\frac {X_j}
{X_i}}\\
\times \prod _{i=1} ^{n} \frac {(Z)_{k_i}\,(CX_i/A)_{k_i}}
{(q)_{k_i}\,(BZX_i)_{k_i}}q^{k_i (1-i)}A^{k_i}\bigg),
\endmultline\tag2.4$$
provided the series terminate or $\vert q\vert<1$, $\vert Zq^{1-n}\vert<1$,
$\vert Aq^{1-n}\vert<1$,
$$\multline \sum _{k_1,\dots,k_n\ge0} ^{}\bigg(\prod _{1\le i<j\le n}
^{}\frac {1-\frac {X_j} {X_i} q^{k_j-k_i}} {1-\frac {X_j} {X_i}} \prod
_{i=1} ^{n} \frac {(A_i)_{k_i}\,(BX_i)_{k_i}}
{(q)_{k_i}\,(CX_i)_{k_i}}q^{k_i (1-i)}Z^{k_i}\bigg)\\
=\prod _{i=1} ^{n}\frac {(Cq^{1-i}/B)_\infty\,(BZX_i)_\infty}
{(Zq^{1-i})_\infty \,(CX_i)_\infty}\sum _{k_1,\dots,k_n\ge0} ^{}
\bigg(\prod _{1\le i<j\le n}
^{}\frac {1-\frac {X_j} {X_i} q^{k_j-k_i}} {1-\frac {X_j}
{X_i}}\\
\times \prod _{i=1} ^{n} \frac {(A_iBZ/C)_{k_i}\,(BX_i)_{k_i}}
{(q)_{k_i}\,(BZX_i)_{k_i}}q^{k_i (1-i)}\(\frac {C} {B}\)^{k_i}\bigg),
\endmultline\tag2.5$$
provided the series terminate or $\vert q\vert<1$, $\vert Zq^{1-n}\vert<1$,
$\vert Cq^{1-n}/B\vert<1$,
$$\multline \sum _{k_1,\dots,k_n\ge0} ^{}\bigg(\prod _{1\le i<j\le n}
^{}\frac {1-\frac {X_j} {X_i} q^{k_j-k_i}} {1-\frac {X_j} {X_i}} \prod
_{i=1} ^{n} \frac {(A_i)_{k_i}\,(BX_i)_{k_i}}
{(q)_{k_i}\,(CA_iX_i)_{k_i}}q^{k_i (1-i)}Z^{k_i}\bigg)\\
=\prod _{i=1} ^{n}\frac {(BZq^{1-i}/C)_\infty}
{(Zq^{1-i})_\infty}\sum _{k_1,\dots,k_n\ge0} ^{}
\bigg(\prod _{1\le i<j\le n}
^{}\frac {1-\frac {X_j} {X_i} q^{k_j-k_i}} {1-\frac {X_j}
{X_i}}\\
\times \prod _{i=1} ^{n} \frac {(CA_i/B)_{k_i}\,(CX_i)_{k_i}}
{(q)_{k_i}\,(CA_iX_i)_{k_i}}q^{k_i (1-i)}\(\frac {BZ} {C}\)^{k_i}\bigg),
\endmultline\tag2.6$$
provided the series terminate or $\vert q\vert<1$, $\vert Zq^{1-n}\vert<1$,
$\vert BZq^{1-n}/C\vert<1$.
\endproclaim
\remark{Remark}It should be noted that
(2.4) is identical with (1.1), while (2.5) is more general than (1.2), and
(2.6) is more general than (1.3). In
fact, (1.2) comes out of (2.5) by setting $A_i\equiv A$, and (1.3)
comes out of (2.6) by setting $A_i\equiv A$ and replacing $C$ by $C/A$.
\endremark

We start with the proof of the third identity.
\demo{Proof of (2.6)}We have
$$\align \prod _{i=1} ^{n}\(X_iq^{k_i}\)^{1-i}\cdot\prod _{1\le i<j\le n}
^{}\left(1-\frac {X_j} {X_i} q^{k_j-k_i}\right) 
&=\prod _{1\le i<j\le n} ^{}\(\frac {1} {X_jq^{k_j}}-
\frac {1} {X_iq^{k_i}}\)\tag2.7\\
&=\det_{1\le s,t\le n}\(\(\frac {1} {X_sq^{k_s}}\)^{t-1}\),\tag2.8
\endalign$$
the second line being due to Vandermonde's determinant. Hence we may
write the left-hand side of (2.6), when multiplied by $\prod _{1\le
i<j\le n} ^{}(1/X_j-1/X_i)$, as
$$\det_{1\le s,t\le n}\bigg(X_s^{1-t}\sum _{k_s\ge0}
^{}\frac {(A_s)_{k_s}\,(BX_s)_{k_s}}
{(q)_{k_s}\,(CA_sX_s)_{k_s}}\(Zq^{1-t}\)^{k_s}\bigg).$$
Now, to the sum inside the determinant we apply the third Heine
transformation, (2.3), with $a=A_s$, $b=BX_s$, $c=CA_sX_s$,
$z=Zq^{1-t}$. Thus we obtain
$$\det_{1\le s,t\le n}\bigg(X_s^{1-t}\frac {(BZq^{1-t}/C)_\infty}
{(Zq^{1-t})_\infty}\sum _{k_s\ge0}
^{}\frac {(CX_s)_{k_s}\,(CA_s/B)_{k_s}}
{(q)_{k_s}\,(CA_sX_s)_{k_s}}\(\frac {BZq^{1-t}} {C}\)^{k_s}\bigg),$$
which, by taking factors out of the determinant and by using linearity
of the determinant with respect to rows, equals
$$\multline
\prod _{i=1} ^{n}\frac {(BZq^{1-i}/C)_\infty}
{(Zq^{1-i})_\infty}\\
\times\sum _{k_1,\dots,k_n\ge0}
^{}\bigg(\prod _{i=1} ^{n}\frac {(CX_i)_{k_i}\,(CA_i/B)_{k_i}}
{(q)_{k_i}\,(CA_iX_i)_{k_i}}\(\frac {BZ} {C}\)^{k_i}\det_{1\le
s,t\le n}\((X_sq^{k_s})^{1-t} \)\bigg).
\endmultline$$
By using (2.8), this last expression is turned into the right-hand side
of (2.6), when multiplied by $\prod _{1\le
i<j\le n} ^{}(1/X_j-1/X_i)$.\quad \quad \qed
\enddemo

For the proof of (2.5) we need the following determinant evaluation.

\proclaim{Lemma~1}Let $B,C,X_1,\dots,X_n$ be indeterminates. Then
there holds
$$\multline
\det_{1\le s,t\le n}\(\frac {(BX_s)_{k_s+t-1}} {(CX_s)_{k_s+t}}\)
=B^{\binom n2}q^{\sum _{i=1} ^{n}\binom{i-1}2}\\
\times\prod _{i=1}
^{n}\bigg(\(X_iq^{k_i}\)^{n-1}\frac {(BX_i)_{k_i}\,(Cq/B)_{i-1}}
{(CX_i)_{k_i+n}}\bigg)
\prod _{1\le i<j\le n} ^{}\(\frac {1}
{X_jq^{k_j}}-\frac {1} {X_iq^{k_i}}\).
\endmultline
\tag2.9$$
\endproclaim
This determinant evaluation follows easily from a determinant lemma
(\cite{Krattenthaler [1990a], Lem\-ma~2.2}), which we state here without proof.

\proclaim{Lemma~2}Let $X_1,\dots,X_n$, $A_2,\dots,A_n$, and $B_2,
\dots B_n$ be indeterminates. Then there holds
$$\multline \det_{1\le s,t\le n}\Big((X_s+A_n)\cdots(X_s+A_{t+1})
(X_s+B_t)\cdots (X_s+B_2)\Big)\\
\hskip2cm =\prod _{1\le i<j\le n} ^{}(X_i-X_j)\prod _{2\le i\le j\le n}
^{}(B_i-A_j). \quad \quad \qed
\endmultline\tag2.10$$
\endproclaim

\demo{Proof of Lemma~1}By taking some factors out of the determinant,
we write the left-hand side of (2.9) as
$$\align
\prod _{i=1} ^{n}&\frac {(BX_i)_{k_i}} {(CX_i)_{k_i+n}}
\det_{1\le s,t\le n}\Big((1-BX_sq^{k_s})\cdots(1-BX_sq^{k_s}q^{t-2})\\
&\hskip4cm\cdot(1-CX_sq^{k_s}q^t)\cdots (1-CX_sq^{k_s}q^{n-1})\Big)\\
&=\prod _{i=1} ^{n}\frac {(BX_i)_{k_i}}
{(CX_i)_{k_i+n}}\(X_iq^{k_i}\)^{n-1}
\det_{1\le s,t\le n}\bigg(\(\frac {1} {X_sq^{k_s}}-B\)\cdots \(\frac {1}
{X_sq^{k_s}}-Bq^{t-2}\)\\
&\hskip4cm\cdot\(\frac {1} {X_sq^{k_s}}-Cq^{t}\)\cdots \(\frac {1}
{X_sq^{k_s}}-Cq^{n-1}\)\bigg).
\endalign$$
Now we apply Lemma~2 with the replacements $X_s\to 1/X_sq^{k_s}$,
$A_t\to -Cq^{t-1}$, $B_t\to -Bq^{t-2}$. A few simplifications then
give the right-hand side of (2.9).\quad \quad \qed
\enddemo
Now we are in the position to prove our second identity.
\demo{Proof of (2.5)}By (2.7) and Lemma~1 with $C$ replaced by
$Cq^{-n}$, the left-hand side of (2.5), when multiplied by $\prod _{1\le
i<j\le n} ^{}(1/X_j-1/X_i)$, equals
$$\multline B^{-\binom n2}q^{-\sum _{i=1} ^{n}\binom {i-1}2}\prod
_{i=1} ^{n} \frac {(CX_iq^{-n})_n} {(Cq^{1-n}/B)_{i-1}}\\
\times \det_{1\le s,t\le n}\bigg(\sum _{k_s\ge0} ^{}\frac
{(A_s)_{k_s}\,(BX_s)_{k_s+t-1}}
{(q)_{k_s}\,(CX_sq^{-n})_{k_s+t}}\(X_sq^{k_s}\)^{1-n}Z^{k_s}\bigg).
\endmultline$$
Now, to the sum inside the determinant we apply the second Heine
transformation, (2.2), with $a=A_s$, $b=BX_sq^{t-1}$, $c=CX_sq^{t-n}$,
$z=Zq^{1-n}$.  Thus we obtain
$$\align 
&B^{-\binom n2}q^{-\sum _{i=1} ^{n}\binom {i-1}2}\prod
_{i=1} ^{n} \frac {(CX_iq^{-n})_n} {(Cq^{1-n}/B)_{i-1}}
 \det_{1\le s,t\le n}\bigg(\frac {(BX_s)_{t-1}} {(CX_sq^{-n})_{t}}X_s^{1-n}\\
&\quad \cdot\frac {(Cq^{1-n}/B)_\infty\,(BZX_sq^{t-n})_\infty}
{(CX_sq^{t-n})_\infty\, (Zq^{1-n})_\infty}
\sum _{k_s\ge0} ^{}\frac
{(A_sBZ/C)_{k_s}\,(BX_sq^{t-1})_{k_s}}
{(q)_{k_s}\,(BZX_sq^{t-n})_{k_s}}\(\frac {Cq^{1-n}} {B}\)^{k_s}\bigg)\\
&=B^{-\binom n2}q^{-\sum _{i=1} ^{n}\binom {i-1}2}\prod
_{i=1} ^{n} 
 \frac {(Cq^{i-n}/B)_\infty\,(BZX_iq^{-n})_\infty}
{(CX_i)_\infty\, (Zq^{1-n})_\infty}\\
&\ \times\sum _{k_1,\dots,k_n\ge0} ^{}\bigg(\prod _{i=1} ^{n} \frac
{(A_iBZ/C)_{k_i}} {(q)_{k_i}}\(\frac {C}
{B}\)^{k_i}\(X_iq^{k_i}\)^{1-n} \det_{1\le s,t\le n}\bigg( 
\frac {(BX_s)_{k_s+t-1}}
{(BZX_sq^{-n})_{k_s+t}}\bigg)\bigg).
\endalign$$
The determinant in the last line can be evaluated by means of Lemma~1,
with $C$ replaced by $BZq^{-n}$. Subsequently, a few simplifications lead to
the right-hand side of (2.5), when multiplied by $\prod _{1\le
i<j\le n} ^{}(1/X_j-1/X_i)$.\quad \quad \qed
\enddemo

Finally, we are able to prove our first identity.
\demo{Proof of 2.4}To the left-hand side of (2.4) we apply (1.2)
(which is (2.5) with $A_i\equiv A$). To the resulting series we
apply (1.3) (which is (2.6) with $A_i\equiv A$ and $C$ replaced by
$C/A$), with the replacements $A\to ABZ/C$, $C\to BZ$, $Z\to C/B$. 
Thus we arrive at the right-hand side of (2.4).\quad \quad \qed
\enddemo







\Refs

\ref\by G.    Gasper and M. Rahman \yr 1990 \book Basic
hypergeometric series\publ Encyclopedia of Mathematics And Its
Applications~35, Cambridge University Press\publaddr Cambridge\endref 

\ref\by I. M. Gessel and C. Krattenthaler \yr 1995 \paper
Cylindric partitions\paperinfo preprint\jour \vol \pages \endref 

\ref\by R. A. Gustafson and C. Krattenthaler\yr 1994 \paper Heine
transformations for a new kind of basic hypergeometric series in
$U(n)$\paperinfo preprint\endref 

\ref\by C.    Krattenthaler \yr 1990 \paper Generating
functions for plane partitions of a given shape\jour Manuscripta
Math\.\vol 69\pages 173--202\endref 

\ref\by C.    Krattenthaler \yr 1990 \paper A determinant
evaluation and some enumeration results for plane partitions\inbook
Number-Theoretic Analysis\eds E.~Hlawka, R.~F.~Tichy\publ Lect\. Notes
in Math. 1452, Springer-Verlag\publaddr Berlin, New York\pages
121--131\endref 

\ref\by C.    Krattenthaler \yr 1995 \book The major counting of
nonintersecting lattice paths and generating functions for
tableaux\publ Mem\. Amer\. Math\. Soc\. 115, no.~552\publaddr Amer\.
Math\. Soc\., Providence, R.~I \endref


\endRefs




\enddocument
