polynomial in $\mathrm{Qx}$ with coefficients which are polynomials in $x$ over the field of rational functions in $q$, which maximally desingularizes $L$

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 Desingularize(L,Qx,x,q) calling sequence constructs an operator $R\in F$ that maximally desingularizes the leading coefficient, the trailing coefficient, or both coefficients of $L$, depending on the option coeff. Equivalently, all apparent singularities of the leading coefficient, the trailing coefficient, or both coefficients of $L$ are removed in $R$.

Note that $R$ is right divisible by $L$ over the field $k\left(q\,x\right)$.

The parameter q does not have to be a variable.  A nonzero constant value, such as, $q=2$ is possible as well; provided that it is not a root of unity, and thus satisfies $q^n\ne 1$ for all positive integers $n$.

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[Desingularize]:=3 will cause some diagnostics to be printed during the computation.

'coeff'=t, where t is one of leading, trailing, or both

Indicates whether the desingularization is done with respect to the leading coefficient, the trailing coefficient, or both coefficients of the input operator $L$. The default is leading.

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

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

$L≔\left(x-3\right)\left(qx-3\right)\mathrm{Qx}+{\left(q^{2}x-3\right)}^2\left(q^{3}x-3\right)$

$L≔\left(x-3\right)\left(qx-3\right)\mathrm{Qx}+{\left(q^{2}x-3\right)}^2\left(q^{3}x-3\right)$

$\mathrm{Desingularize}\left(L\,\mathrm{Qx}\,x\,q=\frac{1}{3}\,'\mathrm{coeff}'='\mathrm{leading}'\,\mathrm{factor}\right)$

${\mathrm{Qx}}^3-\frac{3380{\mathrm{Qx}}^2}{729}+\left(\frac{2224976x}{4782969}+\frac{2293408}{531441}\right)\mathrm{Qx}+\frac{\left(x-81\right)\left(x^2+6703830x-230132907\right)}{10460353203}$

$\mathrm{Desingularize}\left(L\,\mathrm{Qx}\,x\,q=\frac{1}{3}\,'\mathrm{coeff}'='\mathrm{trailing}'\,\mathrm{factor}\right)$

$\frac{832x}{177147}-\frac{832}{2187}-{\mathrm{Qx}}^3+\left(\frac{4598x}{6561}-\frac{406}{243}\right){\mathrm{Qx}}^2+\left(\frac{511}{1594323}{x}^2-\frac{21100}{177147}x+\frac{92767}{6561}\right)\mathrm{Qx}$

$\mathrm{Desingularize}\left(L\,\mathrm{Qx}\,x\,q=-1\right)$

Error, (in QDifferenceEquations:-Desingularize) unable to compute a desingularizing operator for the case where q is a root of unity

The QDifferenceEquations[Desingularize] command was introduced in Maple 18.

For more information on Maple 18 changes, see Updates in Maple 18.