For a specified q-hypergeometric term $T\left(q^n\,q^k\right)$ of $q^n$ and $q^k$, the Zeilberger(T, n, k, q, Qn) calling sequence constructs for $T\left(q^n\,q^k\right)$ a Z-pair $L,G$ that consists of a linear q-difference operator with coefficients that are polynomials of $N=q^n$

and a q-hypergeometric term $G\left(q^n\,q^k\right)$ of $q^n$ and $q^k$ such that

Qn is the q-shift operator with respect to $q^n$, defined by $\mathrm{Qn}\left(F\left(q^n\,q^k\right)\right)=F\left(q^{n+1}\,q^k\right)$.

By assigning values to the global variables _MINORDER and _MAXORDER, the algorithm is restricted to finding a Z-pair $L,G$ for $T\left(q^n\,q^k\right)$ such that the order of L is between _MINORDER and _MAXORDER (the default value of _MAXORDER is 6).

The output from the Zeilberger command is a list of two elements $\left[L\,G\right]$ representing the computed Z-pair $L,G$.

$\mathrm{with}\left(\mathrm{QDifferenceEquations}\right)\:$

$T≔q^{n+k}\mathrm{QBinomial}\left(n\,k\,q\right)$

$T≔q^{n+k}\mathrm{QBinomial}\left(n\,k\,q\right)$

$\mathrm{Zpair}≔\mathrm{Zeilberger}\left(T\,n\,k\,q\,\mathrm{Qn}\right)\:$

$\mathrm{Zpair}\left[1\right]$

${\mathrm{Qn}}^2+\left(-q^2-q\right)\mathrm{Qn}-q^n{q}^4+q^3$

$\mathrm{Zpair}\left[2\right]$

$-\frac{q^n{q}^4\left(q{q}^n-1\right)\left(-1+q^k\right)q^k{q}^{n+k}\mathrm{QBinomial}\left(n\,k\,q\right)}{\left(-q{q}^n+q^k\right)\left(-q^n{q}^2+q^k\right)}$

$T≔\frac{2q^{k^2}}{\mathrm{QPochhammer}\left(q\,q\,k\right)\mathrm{QPochhammer}\left(q\,q\,n-k\right)}$

$T≔\frac{2q^{k^2}}{\mathrm{QPochhammer}\left(q\,q\,k\right)\mathrm{QPochhammer}\left(q\,q\,n-k\right)}$

$\mathrm{Zpair}≔\mathrm{Zeilberger}\left(T\,n\,k\,q\,\mathrm{Qn}\right)\:$

$\mathrm{Zpair}\left[1\right]$

$\left(-q^n{q}^2+1\right){\mathrm{Qn}}^2+\left(-{\left(q^n\right)}^2{q}^3+q^n{q}^2-q-1\right)\mathrm{Qn}+q$

$\mathrm{Zpair}\left[2\right]$

$\frac{2q^{k^2}{\left(q^n\right)}^2{q}^4\left(-1+q^k\right)}{\mathrm{QPochhammer}\left(q\,q\,k\right)\mathrm{QPochhammer}\left(q\,q\,n-k\right)\left(-q{q}^n+q^k\right)\left(-q^n{q}^2+q^k\right)}$

Petkovsek, M.; Wilf, H.; and Zeilberger, D. A=B. Wellesley, Massachusetts: A K Peters, Ltd., 1996.