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QDifferenceEquations

 Zeilberger
 perform Zeilberger's algorithm (q-difference case)

 Calling Sequence Zeilberger(T, n, k, q, Qn)

Parameters

 T - q-hypergeometric term in ${q}^{n}$ and ${q}^{k}$ n - name k - name q - name Qn - name; denote the q-shift operator with respect to ${q}^{n}$

Description

 • 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 $N={q}^{n}$

$L=a[v]\left({q}^{n}\right){\mathrm{Qn}}^{v}+\mathrm{...}+a[1]\left({q}^{n}\right)\mathrm{Qn}+a[0]\left({q}^{n}\right)$

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

$LoT\left({q}^{n},{q}^{k}\right)=G\left({q}^{n},{q}^{k+1}\right)-G\left({q}^{n},{q}^{k}\right)$

 • 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 $L,G$.

Examples

 > $\mathrm{with}\left(\mathrm{QDifferenceEquations}\right):$
 > $T≔{q}^{n+k}\mathrm{QBinomial}\left(n,k,q\right)$
 ${T}{≔}{{q}}^{{n}{+}{k}}{}{\mathrm{QDifferenceEquations:-QBinomial}}{}\left({n}{,}{k}{,}{q}\right)$ (1)
 > $\mathrm{Zpair}≔\mathrm{Zeilberger}\left(T,n,k,q,\mathrm{Qn}\right):$
 > ${\mathrm{Zpair}}_{1}$
 ${{\mathrm{Qn}}}^{{2}}{+}\left({-}{{q}}^{{2}}{-}{q}\right){}{\mathrm{Qn}}{-}{{q}}^{{n}}{}{{q}}^{{4}}{+}{{q}}^{{3}}$ (2)
 > ${\mathrm{Zpair}}_{2}$
 ${-}\frac{{{q}}^{{n}}{}{{q}}^{{4}}{}\left({q}{}{{q}}^{{n}}{-}{1}\right){}\left({-}{1}{+}{{q}}^{{k}}\right){}{{q}}^{{k}}{}{{q}}^{{n}{+}{k}}{}{\mathrm{QDifferenceEquations:-QBinomial}}{}\left({n}{,}{k}{,}{q}\right)}{\left({-}{{q}}^{{n}}{}{{q}}^{{2}}{+}{{q}}^{{k}}\right){}\left({-}{q}{}{{q}}^{{n}}{+}{{q}}^{{k}}\right)}$ (3)
 > $T≔\frac{2{q}^{{k}^{2}}}{\mathrm{QPochhammer}\left(q,q,k\right)\mathrm{QPochhammer}\left(q,q,n-k\right)}$
 ${T}{≔}\frac{{2}{}{{q}}^{{{k}}^{{2}}}}{{\mathrm{QDifferenceEquations:-QPochhammer}}{}\left({q}{,}{q}{,}{k}\right){}{\mathrm{QDifferenceEquations:-QPochhammer}}{}\left({q}{,}{q}{,}{n}{-}{k}\right)}$ (4)
 > $\mathrm{Zpair}≔\mathrm{Zeilberger}\left(T,n,k,q,\mathrm{Qn}\right):$
 > ${\mathrm{Zpair}}_{1}$
 $\left({-}{{q}}^{{n}}{}{{q}}^{{2}}{+}{1}\right){}{{\mathrm{Qn}}}^{{2}}{+}\left({-}{\left({{q}}^{{n}}\right)}^{{2}}{}{{q}}^{{3}}{+}{{q}}^{{n}}{}{{q}}^{{2}}{-}{q}{-}{1}\right){}{\mathrm{Qn}}{+}{q}$ (5)
 > ${\mathrm{Zpair}}_{2}$
 $\frac{{2}{}{{q}}^{{{k}}^{{2}}}{}{\left({{q}}^{{n}}\right)}^{{2}}{}{{q}}^{{4}}{}\left({-}{1}{+}{{q}}^{{k}}\right)}{{\mathrm{QDifferenceEquations:-QPochhammer}}{}\left({q}{,}{q}{,}{k}\right){}{\mathrm{QDifferenceEquations:-QPochhammer}}{}\left({q}{,}{q}{,}{n}{-}{k}\right){}\left({-}{q}{}{{q}}^{{n}}{+}{{q}}^{{k}}\right){}\left({-}{{q}}^{{n}}{}{{q}}^{{2}}{+}{{q}}^{{k}}\right)}$ (6)

References

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