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%%a Plain TeX file by Mohamud Mohammed and Doron Zeilberger (xx pages)
 
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\bf
\centerline
{
Multi-Variable Zeilberger and Almkvist-Zeilberger Algorithms 
}
\centerline
{and the}
\centerline
{Sharpening of Wilf-Zeilberger Theory}
\rm
\bigskip
\centerline{ {\it 
Mohamud Mohammed{$^1$} and Doron ZEILBERGER
}\footnote{$^1$}
{\eightrm  \raggedright
Department of Mathematics, Rutgers University (New Brunswick),
Hill Center-Busch Campus, 110 Frelinghuysen Rd., Piscataway,
NJ 08854-8019, USA.
%\break
{\eighttt [mohamudm, zeilberg] at math dot rutgers dot edu} , 
\hfill \break
{\eighttt http://www.math.rutgers.edu/\Tilde [mohamudm, zeilberg]/.}
First version: Nov. 15, 2004.
Supported in part by the NSF.
} 
}

{\bf q-Notation.} For $k$ integer,
$[a]_k:=(1-q^a)(1-q^{a+1})\cdots (1-q^{a+k-1})$, if $k \geq 0$ 
and $[a]_k:=1/[a+k]_{-k}$ if $k<0$.
In order to avoid too many subscripts, we will
denote $[a]_k$ by $qRF(a\,,\,k)$. 

{\bf A Multi-Variate q-Zeilberger Algorithm}

{\bf q-Theorem.} Let
$$
F(n;k_1, \dots, k_r)=POL(n;k_1, \dots, k_r) 
\cdot H(n; k_1, \dots, k_r) \quad ,
\eqno(qMultiProperHypergeometric)
$$
where $POL(n;k1, \dots, k_r)$  is a Laurent polynomial in $(q^n,q^{k_1}, \dots, q^{k_r})$,
and
$$
H(n; k_1, \dots, k_r)=
{{\prod_{j=1}^A qRF(a''_j \, , \, a'_jn+\sum_{i=1}^r a_{ji} k_i)}
\over
{\prod_{j=1}^C qRF(c''_j \, , \, c'_jn+\sum_{i=1}^r c_{ji} k_i)}
}\cdot q^{Q(n;k_1, \dots, k_r)} \cdot\prod_{i=1}^r z_i^{k_i} \quad,
\eqno(qMultiPureHypergeometric)
$$
where the $a'_j,c'_j$ are {\it non-negative} integers and the
$a_{ji}, c_{ji}$ are {\it integers}, while $a''_j,c''_j$ and
$z_1, \dots, z_r$ are {\it commuting indeterminates}, and
$Q(n;k_1, \dots, k_r)$ is a quadratic form in $(n,k_1, \dots, k_r)$.
Then there exists an integer $L$, to be {\it explicitly} constructed in
the course of the proof, polynomials in $q^n$, $e_0(q^n), e_1(q^n), \dots, e_L(q^n)$,
{\it not all zero}, and $r$ rational functions of $(q^n, q^{k_1}, \dots, q^{k_r})$,
$R_I(n;k_1, \dots, k_r)$ ($I=1,\dots, r$) such that 
$$
G_I(n;k_1, \dots, k_r):=R_I(n;k_1, \dots, k_r)F(n;k_1, \dots, k_r)
$$
satisfy
$$
\sum_{i=0}^L e_i(q^n) F(n+i,k)=
\sum_{I=0}^r [G_I(n; k_1, \dots, k_{I-1},k_I+1, k_{I+1}, \dots, k_r)-
G_I(n; k_1, \dots, k_r)] \quad.
\eqno(qZtuple) 
$$


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