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QDifferenceEquations

 Closure
 closure of a q-shift operator with polynomial coefficients

 Calling Sequence Closure(L, Qx, x, q, p, func, options)

Parameters

 L - polynomial in $\mathrm{Qx}$ with coefficients which are polynomials in $x$ over the field of rational functions in $q$ Qx - name, variable denoting the $q$-shift operator $x↦qx$ x - variable name q - either a variable name, or a nonzero constant, or an equation of the form name=constant p - (optional) set of irreducible polynomials in $x$, or a single such polynomial func - (optional) procedure options - (optional) equation(s) of the form 'keyword'=value, where the keyword is either order or maximal

Returns

 • list of polynomials in $\mathrm{Qx}$ with coefficients which are polynomials in $x$ over the field of rational functions in $q$

Description

 • Let $k$ be a field of characteristic 0. Denote by $F$ the $q$-shift polynomial ring consisting of elements, each of which is a polynomial in $\mathrm{Qx}$, with coefficients which are polynomials in $x$ over $k\left(q\right)$. For a given operator $L\in F$, the Closure(L,Qx,x,q) calling sequence constructs a closure of $L$ in the $q$-shift polynomial ring $F$.
 The output is a list of elements in $F$. Each element $R$ in this list represents a generator of the closure of $L$. For example, there exists $f\in k\left(q\right)\left[x\right]$ and $P\in F$, such that the torsion relation $P·L=f·R$ holds.
 • The Closure(L,Qx,x q,p) calling sequence constructs a local closure of $L$ at the irreducible(s) $p$. The output is a list of generators of the local closure of $L$.  For example, each element $R$ in the list is such that the torsion relation $P·L={g}^{i}·R$ holds for some operator $P$, where $g\in p$ and $i\in ℕ$.
 • The parameter q does not have to be a variable. A constant value, such as $q=2$ is possible as well, including the case of a root of unity.
 • The optional argument func, if specified, is applied to the coefficients of the result with respect to $\mathrm{Qx}$; typical examples are expand or factor.
 • Note that setting infolevel[Closure]:=3 will cause some diagnostics to be printed during the computation.

Options

 • 'order'=o, where o is a monomial order
 If this option is given, the Groebner[Basis] command, with respect to the given monomial order, will be applied to the computed closure.
 • 'maximal'=truefalse (default: false)
 This option only has an effect if a local closure is requested. If maximal=true (or maximal for short) is specified, then each element $R$ in the output list is such that the torsion relation $P·L={g\left({q}^{j}x\right)}^{i}·R$ holds for some operator $P$, where $g\in p$, $i\in ℕ$, and $j\in ℤ$. In other words, all generators for the $q$-shift-equivalence class(es) represented by $p$ are computed and returned.

Examples

 > $\mathrm{with}\left(\mathrm{QDifferenceEquations}\right)$
 $\left[{\mathrm{AccurateQSummation}}{,}{\mathrm{AreSameSolution}}{,}{\mathrm{Closure}}{,}{\mathrm{Desingularize}}{,}{\mathrm{ExtendSeries}}{,}{\mathrm{IsQHypergeometricTerm}}{,}{\mathrm{IsSolution}}{,}{\mathrm{PolynomialSolution}}{,}{\mathrm{QBinomial}}{,}{\mathrm{QBrackets}}{,}{\mathrm{QDispersion}}{,}{\mathrm{QECreate}}{,}{\mathrm{QEfficientRepresentation}}{,}{\mathrm{QFactorial}}{,}{\mathrm{QGAMMA}}{,}{\mathrm{QHypergeometricSolution}}{,}{\mathrm{QMultiplicativeDecomposition}}{,}{\mathrm{QPochhammer}}{,}{\mathrm{QPolynomialNormalForm}}{,}{\mathrm{QRationalCanonicalForm}}{,}{\mathrm{QSimpComb}}{,}{\mathrm{QSimplify}}{,}{\mathrm{RationalSolution}}{,}{\mathrm{RegularQPochhammerForm}}{,}{\mathrm{SeriesSolution}}{,}{\mathrm{UniversalDenominator}}{,}{\mathrm{Zeilberger}}\right]$ (1)

Compute a closure of the following linear $q$-shift operator when $q=\frac{1}{2}$:

 > $L≔\left(x-3\right)\left(qx-3\right)\mathrm{Qx}+{\left({q}^{2}x-3\right)}^{2}\left({q}^{3}x-3\right)$
 ${L}{≔}\left({x}{-}{3}\right){}\left({q}{}{x}{-}{3}\right){}{\mathrm{Qx}}{+}{\left({{q}}^{{2}}{}{x}{-}{3}\right)}^{{2}}{}\left({{q}}^{{3}}{}{x}{-}{3}\right)$ (2)
 > $C≔\mathrm{Closure}\left(L,\mathrm{Qx},x,q=\frac{1}{2}\right)$
 ${C}{≔}\left[\left(\frac{{1}}{{2}}{}{{x}}^{{2}}{-}\frac{{9}}{{2}}{}{x}{+}{9}\right){}{\mathrm{Qx}}{+}\frac{{{x}}^{{3}}}{{128}}{-}\frac{{3}{}{{x}}^{{2}}}{{8}}{+}\frac{{45}{}{x}}{{8}}{-}{27}{,}\left({-}\frac{{3}{}{x}}{{4}}{+}{9}\right){}{{\mathrm{Qx}}}^{{3}}{+}\left({-}\frac{{3}}{{1024}}{}{{x}}^{{2}}{-}\frac{{9}}{{128}}{}{x}{-}\frac{{405}}{{16}}\right){}{{\mathrm{Qx}}}^{{2}}{+}\left({-}\frac{{9}}{{2048}}{}{{x}}^{{2}}{+}\frac{{81}}{{256}}{}{x}{-}\frac{{81}}{{16}}\right){}{\mathrm{Qx}}{,}\left(\frac{{8}{}{x}}{{63}}{-}\frac{{128}}{{21}}\right){}{{\mathrm{Qx}}}^{{4}}{+}\left(\frac{{1}}{{4032}}{}{{x}}^{{2}}{-}\frac{{5}}{{56}}{}{x}{+}\frac{{191}}{{7}}\right){}{{\mathrm{Qx}}}^{{3}}{+}\left({-}\frac{{399}{}{x}}{{128}}{-}\frac{{63}}{{4}}\right){}{{\mathrm{Qx}}}^{{2}}{+}\left({-}\frac{{51}}{{2048}}{}{{x}}^{{2}}{+}\frac{{243}}{{128}}{}{x}{-}\frac{{135}}{{4}}\right){}{\mathrm{Qx}}{,}\left({-}\frac{{3}{}{x}}{{2}}{+}{9}\right){}{{\mathrm{Qx}}}^{{2}}{+}\left({-}\frac{{3}}{{256}}{}{{x}}^{{2}}{-}\frac{{9}}{{64}}{}{x}{-}\frac{{405}}{{16}}\right){}{\mathrm{Qx}}{-}\frac{{9}{}{{x}}^{{2}}}{{512}}{+}\frac{{81}{}{x}}{{128}}{-}\frac{{81}}{{16}}{,}\left(\frac{{16}{}{x}}{{63}}{-}\frac{{128}}{{21}}\right){}{{\mathrm{Qx}}}^{{3}}{+}\left(\frac{{1}}{{1008}}{}{{x}}^{{2}}{-}\frac{{5}}{{28}}{}{x}{+}\frac{{191}}{{7}}\right){}{{\mathrm{Qx}}}^{{2}}{+}\left({-}\frac{{399}{}{x}}{{64}}{-}\frac{{63}}{{4}}\right){}{\mathrm{Qx}}{-}\frac{{51}{}{{x}}^{{2}}}{{512}}{+}\frac{{243}{}{x}}{{64}}{-}\frac{{135}}{{4}}\right]$ (3)

Compute a local closure of $L$, now with a symbolic $q$, at $p=-\frac{{q}^{3}x}{3}+1$, with factored coefficients:

 > $p≔-\frac{1}{3}{q}^{3}x+1$
 ${p}{≔}{-}\frac{{{q}}^{{3}}{}{x}}{{3}}{+}{1}$ (4)
 > $C≔\mathrm{Closure}\left(L,\mathrm{Qx},x,q,p,\mathrm{factor}\right)$
 ${C}{≔}\left[\left({x}{-}{3}\right){}\left({x}{}{q}{-}{3}\right){}{\mathrm{Qx}}{+}{\left({{q}}^{{2}}{}{x}{-}{3}\right)}^{{2}}{}\left({{q}}^{{3}}{}{x}{-}{3}\right){,}\left({-}{3}{}{{q}}^{{2}}{}{x}{+}{9}\right){}{{\mathrm{Qx}}}^{{3}}{+}\left({-}{3}{}{{q}}^{{10}}{}{{x}}^{{2}}{-}{9}{}{{q}}^{{7}}{}{x}{+}{18}{}{{q}}^{{6}}{}{x}{+}{9}{}{{q}}^{{5}}{}{x}{+}{9}{}{{q}}^{{4}}{}{x}{-}{27}{}{{q}}^{{4}}{-}{9}{}{{q}}^{{3}}{}{x}{+}{27}{}{{q}}^{{3}}{-}{27}\right){}{{\mathrm{Qx}}}^{{2}}{+}{9}{}\left({q}{-}{1}\right){}\left({{q}}^{{3}}{}{x}{-}{3}\right){}\left({{q}}^{{4}}{}{x}{-}{3}\right){}{{q}}^{{3}}{}{\mathrm{Qx}}{,}\frac{\left({{q}}^{{4}}{}{x}{-}{3}\right){}{{\mathrm{Qx}}}^{{4}}}{{\left({q}{+}{1}\right)}^{{2}}{}\left({{q}}^{{2}}{+}{q}{+}{1}\right){}{\left({q}{-}{1}\right)}^{{2}}{}{q}}{+}\frac{\left({{q}}^{{13}}{}{{x}}^{{2}}{+}{3}{}{{q}}^{{10}}{}{x}{-}{6}{}{{q}}^{{8}}{}{x}{-}{3}{}{{q}}^{{7}}{}{x}{+}{9}{}{{q}}^{{7}}{+}{9}{}{{q}}^{{6}}{-}{9}{}{{q}}^{{5}}{-}{18}{}{{q}}^{{4}}{-}{9}{}{{q}}^{{3}}{+}{9}{}{{q}}^{{2}}{+}{9}{}{q}{+}{9}\right){}{{\mathrm{Qx}}}^{{3}}}{{\left({q}{+}{1}\right)}^{{2}}{}\left({{q}}^{{2}}{+}{q}{+}{1}\right){}{\left({q}{-}{1}\right)}^{{2}}{}{q}}{+}\frac{{9}{}\left({{q}}^{{2}}{+}{q}{+}{1}\right){}\left({{q}}^{{6}}{}{x}{-}{{q}}^{{4}}{}{x}{-}{2}{}{{q}}^{{3}}{}{x}{+}{6}{}{{q}}^{{3}}{+}{3}{}{{q}}^{{2}}{-}{3}\right){}{{\mathrm{Qx}}}^{{2}}}{{q}{+}{1}}{-}\frac{{9}{}\left({{q}}^{{4}}{}{x}{-}{3}\right){}\left({3}{}{{q}}^{{5}}{}{x}{+}{3}{}{{q}}^{{4}}{}{x}{+}{2}{}{{q}}^{{3}}{}{x}{-}{6}{}{{q}}^{{2}}{-}{9}{}{q}{-}{9}\right){}{{q}}^{{3}}{}{\mathrm{Qx}}}{{q}{+}{1}}\right]$ (5)

Compute a local closure of $L$ at the q-shift equivalence class represented by $p$:

 > $C≔\mathrm{Closure}\left(L,\mathrm{Qx},x,q,p,'\mathrm{maximal}',\mathrm{factor}\right)$
 ${C}{≔}\left[\left({x}{-}{3}\right){}\left({x}{}{q}{-}{3}\right){}{\mathrm{Qx}}{+}{\left({{q}}^{{2}}{}{x}{-}{3}\right)}^{{2}}{}\left({{q}}^{{3}}{}{x}{-}{3}\right){,}\left({-}{3}{}{{q}}^{{2}}{}{x}{+}{9}\right){}{{\mathrm{Qx}}}^{{3}}{+}\left({-}{3}{}{{q}}^{{10}}{}{{x}}^{{2}}{-}{9}{}{{q}}^{{7}}{}{x}{+}{18}{}{{q}}^{{6}}{}{x}{+}{9}{}{{q}}^{{5}}{}{x}{+}{9}{}{{q}}^{{4}}{}{x}{-}{27}{}{{q}}^{{4}}{-}{9}{}{{q}}^{{3}}{}{x}{+}{27}{}{{q}}^{{3}}{-}{27}\right){}{{\mathrm{Qx}}}^{{2}}{+}{9}{}\left({q}{-}{1}\right){}\left({{q}}^{{3}}{}{x}{-}{3}\right){}\left({{q}}^{{4}}{}{x}{-}{3}\right){}{{q}}^{{3}}{}{\mathrm{Qx}}{,}\frac{\left({{q}}^{{4}}{}{x}{-}{3}\right){}{{\mathrm{Qx}}}^{{4}}}{{\left({q}{+}{1}\right)}^{{2}}{}\left({{q}}^{{2}}{+}{q}{+}{1}\right){}{\left({q}{-}{1}\right)}^{{2}}{}{q}}{+}\frac{\left({{q}}^{{13}}{}{{x}}^{{2}}{+}{3}{}{{q}}^{{10}}{}{x}{-}{6}{}{{q}}^{{8}}{}{x}{-}{3}{}{{q}}^{{7}}{}{x}{+}{9}{}{{q}}^{{7}}{+}{9}{}{{q}}^{{6}}{-}{9}{}{{q}}^{{5}}{-}{18}{}{{q}}^{{4}}{-}{9}{}{{q}}^{{3}}{+}{9}{}{{q}}^{{2}}{+}{9}{}{q}{+}{9}\right){}{{\mathrm{Qx}}}^{{3}}}{{\left({q}{+}{1}\right)}^{{2}}{}\left({{q}}^{{2}}{+}{q}{+}{1}\right){}{\left({q}{-}{1}\right)}^{{2}}{}{q}}{+}\frac{{9}{}\left({{q}}^{{2}}{+}{q}{+}{1}\right){}\left({{q}}^{{6}}{}{x}{-}{{q}}^{{4}}{}{x}{-}{2}{}{{q}}^{{3}}{}{x}{+}{6}{}{{q}}^{{3}}{+}{3}{}{{q}}^{{2}}{-}{3}\right){}{{\mathrm{Qx}}}^{{2}}}{{q}{+}{1}}{-}\frac{{9}{}\left({{q}}^{{4}}{}{x}{-}{3}\right){}\left({3}{}{{q}}^{{5}}{}{x}{+}{3}{}{{q}}^{{4}}{}{x}{+}{2}{}{{q}}^{{3}}{}{x}{-}{6}{}{{q}}^{{2}}{-}{9}{}{q}{-}{9}\right){}{{q}}^{{3}}{}{\mathrm{Qx}}}{{q}{+}{1}}{,}\left({-}{3}{}{x}{}{q}{+}{9}\right){}{{\mathrm{Qx}}}^{{2}}{+}\left({-}{3}{}{{q}}^{{8}}{}{{x}}^{{2}}{-}{9}{}{{q}}^{{6}}{}{x}{+}{18}{}{{q}}^{{5}}{}{x}{+}{9}{}{{q}}^{{4}}{}{x}{-}{27}{}{{q}}^{{4}}{+}{9}{}{{q}}^{{3}}{}{x}{+}{27}{}{{q}}^{{3}}{-}{9}{}{{q}}^{{2}}{}{x}{-}{27}\right){}{\mathrm{Qx}}{+}{9}{}\left({q}{-}{1}\right){}\left({{q}}^{{2}}{}{x}{-}{3}\right){}\left({{q}}^{{3}}{}{x}{-}{3}\right){}{{q}}^{{3}}{,}\frac{\left({{q}}^{{3}}{}{x}{-}{3}\right){}{{\mathrm{Qx}}}^{{3}}}{{\left({q}{+}{1}\right)}^{{2}}{}\left({{q}}^{{2}}{+}{q}{+}{1}\right){}{\left({q}{-}{1}\right)}^{{2}}{}{q}}{+}\frac{\left({{q}}^{{11}}{}{{x}}^{{2}}{+}{3}{}{{q}}^{{9}}{}{x}{-}{6}{}{{q}}^{{7}}{}{x}{+}{9}{}{{q}}^{{7}}{-}{3}{}{{q}}^{{6}}{}{x}{+}{9}{}{{q}}^{{6}}{-}{9}{}{{q}}^{{5}}{-}{18}{}{{q}}^{{4}}{-}{9}{}{{q}}^{{3}}{+}{9}{}{{q}}^{{2}}{+}{9}{}{q}{+}{9}\right){}{{\mathrm{Qx}}}^{{2}}}{{\left({q}{+}{1}\right)}^{{2}}{}\left({{q}}^{{2}}{+}{q}{+}{1}\right){}{\left({q}{-}{1}\right)}^{{2}}{}{q}}{+}\frac{{9}{}\left({{q}}^{{2}}{+}{q}{+}{1}\right){}\left({{q}}^{{5}}{}{x}{-}{{q}}^{{3}}{}{x}{+}{6}{}{{q}}^{{3}}{-}{2}{}{{q}}^{{2}}{}{x}{+}{3}{}{{q}}^{{2}}{-}{3}\right){}{\mathrm{Qx}}}{{q}{+}{1}}{-}\frac{{9}{}\left({{q}}^{{3}}{}{x}{-}{3}\right){}\left({3}{}{{q}}^{{4}}{}{x}{+}{3}{}{{q}}^{{3}}{}{x}{+}{2}{}{{q}}^{{2}}{}{x}{-}{6}{}{{q}}^{{2}}{-}{9}{}{q}{-}{9}\right){}{{q}}^{{3}}}{{q}{+}{1}}\right]$ (6)

Verify the torsion property:

 > $A≔\mathrm{OreTools}:-\mathrm{SetOreRing}\left(\left[x,q\right],'\mathrm{qshift}'\right):$
 > $L≔\mathrm{OreTools}:-\mathrm{Converters}:-\mathrm{FromPolyToOrePoly}\left(L,\mathrm{Qx}\right)$
 ${L}{≔}{\mathrm{OrePoly}}{}\left({\left({{q}}^{{2}}{}{x}{-}{3}\right)}^{{2}}{}\left({{q}}^{{3}}{}{x}{-}{3}\right){,}\left({x}{-}{3}\right){}\left({x}{}{q}{-}{3}\right)\right)$ (7)
 > $\mathrm{R2}≔\mathrm{OreTools}:-\mathrm{Converters}:-\mathrm{FromPolyToOrePoly}\left(C\left[2\right],\mathrm{Qx}\right)$
 ${\mathrm{R2}}{≔}{\mathrm{OrePoly}}{}\left({0}{,}{9}{}\left({q}{-}{1}\right){}\left({{q}}^{{3}}{}{x}{-}{3}\right){}\left({{q}}^{{4}}{}{x}{-}{3}\right){}{{q}}^{{3}}{,}{-}{3}{}{{q}}^{{10}}{}{{x}}^{{2}}{-}{9}{}{{q}}^{{7}}{}{x}{+}{18}{}{{q}}^{{6}}{}{x}{+}{9}{}{{q}}^{{5}}{}{x}{+}{9}{}{{q}}^{{4}}{}{x}{-}{27}{}{{q}}^{{4}}{-}{9}{}{{q}}^{{3}}{}{x}{+}{27}{}{{q}}^{{3}}{-}{27}{,}{-}{3}{}{{q}}^{{2}}{}{x}{+}{9}\right)$ (8)
 > $\mathrm{OreTools}:-\mathrm{Remainder}\left['\mathrm{right}'\right]\left(\mathrm{R2},L,A,'\mathrm{P2}'\right)$
 ${\mathrm{OrePoly}}{}\left({0}\right)$ (9)
 > $\mathrm{P2}≔\mathrm{OreTools}:-\mathrm{Converters}:-\mathrm{FromOrePolyToPoly}\left(\mathrm{P2},\mathrm{Qx}\right)$
 ${\mathrm{P2}}{≔}\frac{{9}{}{{q}}^{{3}}{}\left({q}{-}{1}\right){}{\mathrm{Qx}}}{{{q}}^{{3}}{}{x}{-}{3}}{-}\frac{{3}{}{{\mathrm{Qx}}}^{{2}}}{{{q}}^{{3}}{}{x}{-}{3}}$ (10)
 > $\mathrm{denom}\left(\mathrm{P2}\right)$
 ${{q}}^{{3}}{}{x}{-}{3}$ (11)
 > $\mathrm{R4}≔\mathrm{OreTools}:-\mathrm{Converters}:-\mathrm{FromPolyToOrePoly}\left(C\left[4\right],\mathrm{Qx}\right)$
 ${\mathrm{R4}}{≔}{\mathrm{OrePoly}}{}\left({9}{}\left({q}{-}{1}\right){}\left({{q}}^{{2}}{}{x}{-}{3}\right){}\left({{q}}^{{3}}{}{x}{-}{3}\right){}{{q}}^{{3}}{,}{-}{3}{}{{q}}^{{8}}{}{{x}}^{{2}}{-}{9}{}{{q}}^{{6}}{}{x}{+}{18}{}{{q}}^{{5}}{}{x}{+}{9}{}{{q}}^{{4}}{}{x}{-}{27}{}{{q}}^{{4}}{+}{9}{}{{q}}^{{3}}{}{x}{+}{27}{}{{q}}^{{3}}{-}{9}{}{{q}}^{{2}}{}{x}{-}{27}{,}{-}{3}{}{x}{}{q}{+}{9}\right)$ (12)
 > $\mathrm{OreTools}:-\mathrm{Remainder}\left['\mathrm{right}'\right]\left(\mathrm{R4},L,A,'\mathrm{P4}'\right)$
 ${\mathrm{OrePoly}}{}\left({0}\right)$ (13)
 > $\mathrm{P4}≔\mathrm{OreTools}:-\mathrm{Converters}:-\mathrm{FromOrePolyToPoly}\left(\mathrm{P4},\mathrm{Qx}\right)$
 ${\mathrm{P4}}{≔}\frac{{9}{}{{q}}^{{3}}{}\left({q}{-}{1}\right)}{{{q}}^{{2}}{}{x}{-}{3}}{-}\frac{{3}{}{\mathrm{Qx}}}{{{q}}^{{2}}{}{x}{-}{3}}$ (14)
 > $\mathrm{denom}\left(\mathrm{P4}\right)$
 ${{q}}^{{2}}{}{x}{-}{3}$ (15)

This is, in fact, a negative $q$-shift of $p$:

 > $\mathrm{numer}\left(\mathrm{eval}\left(p,x=\frac{x}{q}\right)\right)$
 ${-}{{q}}^{{2}}{}{x}{+}{3}$ (16)

Compute a closure of the following operator when $q=-1$, a second root of unity:

 > $L≔{\left(x+2\right)}^{2}+\left(x+1\right){\left(x-2\right)}^{2}\mathrm{Qx}$
 ${L}{≔}{\left({x}{+}{2}\right)}^{{2}}{+}\left({x}{+}{1}\right){}{\left({x}{-}{2}\right)}^{{2}}{}{\mathrm{Qx}}$ (17)
 > $\mathrm{infolevel}\left[\mathrm{Closure}\right]≔3:$
 > $C≔\mathrm{Closure}\left(L,\mathrm{Qx},x,-1\right)$
 Closure:   "-1 is a 2 root of unity" Closure:   "compute the matrix representation of the input operator" Closure:   "the matrix representation is Matrix(2, 2, [[(x+2)^2,(x+1)*(x-2)^2],[(-x+1)*(-x-2)^2*eta,(-x+2)^2]])" Closure:   "compute the candidate primes and bounds for their multiplicities" Closure:   "the candidate primes and bounds for their multiplicities are [[x-2, 2], [x+2, 2]]" Closure:   "compute the local closures" Closure:   "compute P such that P.L = (x-2)^j.R, 1<=j<=2" Closure:   "compute P such that P.L = (x+2)^j.R, 1<=j<=2"
 ${C}{≔}\left[{\left({x}{+}{2}\right)}^{{2}}{+}\left({x}{+}{1}\right){}{\left({x}{-}{2}\right)}^{{2}}{}{\mathrm{Qx}}{,}\left({{x}}^{{2}}{-}{x}{-}{2}\right){}{{\mathrm{Qx}}}^{{3}}{+}\left({-}{{x}}^{{2}}{-}{4}{}{x}{-}{4}\right){}{{\mathrm{Qx}}}^{{2}}{+}\left({x}{-}{2}\right){}{\mathrm{Qx}}{,}\left({{x}}^{{2}}{-}{1}\right){}{{\mathrm{Qx}}}^{{3}}{+}{\mathrm{Qx}}{,}\left({{x}}^{{2}}{-}{x}{-}{2}\right){}{{\mathrm{Qx}}}^{{3}}{+}\left({-}{{x}}^{{2}}{-}{4}{}{x}{-}{4}\right){}{{\mathrm{Qx}}}^{{2}}{+}\left({x}{-}{2}\right){}{\mathrm{Qx}}{,}\left({-}{{x}}^{{2}}{-}{x}{+}{2}\right){}{{\mathrm{Qx}}}^{{2}}{+}\left({{x}}^{{2}}{-}{4}{}{x}{+}{4}\right){}{\mathrm{Qx}}{+}{x}{+}{2}{,}\left({-}{x}{+}{1}\right){}{{\mathrm{Qx}}}^{{2}}{+}\left({{x}}^{{2}}{-}{4}{}{x}{+}{4}\right){}{\mathrm{Qx}}{+}{x}{+}{3}{,}\left({-}{{x}}^{{2}}{-}{x}{+}{2}\right){}{{\mathrm{Qx}}}^{{2}}{+}\left({{x}}^{{2}}{-}{4}{}{x}{+}{4}\right){}{\mathrm{Qx}}{+}{x}{+}{2}\right]$ (18)

Compatibility

 • The QDifferenceEquations[Closure] command was introduced in Maple 18.