Multi-Variable Zeilberger and Almkvist-Zeilberger Algorithms and the Sharpening of Wilf-Zeilberger Theory

Mohamud Mohammed^{1} and Doron ZEILBERGER ^{1}

q-Notation. For k integer, [a]_{k} := (1−q^{a})(1−q^{a+1}) ⋯ (1−q^{a+k−1}), if k ≥ 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).

A Multi-Variate q-Zeilberger Algorithm

q-Theorem. Let

F (n; k_{1}, …, k_{r}) = P OL(n; k_{1}, …, k_{r}) · H(n; k_{1}, …, k_{r}) , (qMultiP roperHypergeometric)

where P OL(n; k1, …, k_{r}) is a Laurent polynomial in (q^{n}, q^{k_{1}}, …, q^{k_{r}}), and

H(n; k_{1}, …, k_{r}) = \frac{∏_{j=1}^{A} qRF (a′′_{j} , a′_{j}n + ∑_{i=1}^{r} a_{ji}k_{i})}{∏_{j=1}^{C} qRF (c′′_{j} , c′_{j}n + ∑_{i=1}^{r} c_{ji}k_{i})} · q^{Q(n;k_{1},...,k_{r})} · ∏_{i=1}^{r} z_{i}^{k_{i}} ,

(qMultiP ureHypergeometric)

where the a′_{j}, c′_{j} are non-negative integers and the a_{ji}, c_{ji} are integers, while a′′_{j}, c′′_{j} and z_{1}, …, z_{r} are commuting indeterminates, and Q(n; k_{1}, …, k_{r}) is a quadratic form in (n, k_{1}, …, k_{r}). Then there exists an integer L, to be explicitly constructed in the course of the proof, polynomials in q^{n}, e_{0}(q^{n}), e_{1}(q^{n}), …, e_{L}(q^{n}), not all zero, and r rational functions of (q^{n}, q^{k_{1}}, …, q^{k_{r}}), R_{I}(n; k_{1}, …, k_{r}) (I = 1, …, r) such that

G_{I}(n; k_{1}, …, k_{r}) := R_{I}(n; k_{1}, …, k_{r})F (n; k_{1}, …, k_{r})

satisfy

∑_{i=0}^{L} e_{i}(q^{n})F (n + i, k) = ∑_{I=0}^{r}[G_{I}(n; k_{1}, …, k_{I−1}, k_{I} + 1, k_{I+1}, …, k_{r}) − G_{I}(n; k_{1}, …, k_{r})] .

(qZtuple)

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

Rd., Piscataway, NJ 08854-8019, USA. [mohamudm, zeilberg] at math dot rutgers dot edu , http://www.math.rutgers.edu/~[mohamudm, zeilberg]/. First version: Nov. 15, 2004. Supported in part by the NSF.