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QDifferenceEquations

 Desingularize
 construct a desingularizing q-shift operator with polynomial coefficients

 Calling Sequence Desingularize(L, Qx, x, q, 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 that is not a root of unity, or an equation of the form name=constant func - (optional) procedure options - (optional) equation of the form 'coeff'=t, where t is one of leading, trailing, or both

Returns

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

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 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.

Options

 • '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.

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)

For the following $q$-shift operator $L$, compute desingularizing operators with respect to the leading coefficient and the trailing coefficient when $q=\frac{1}{3}$:

 > $L≔\left(x-3\right)\left(qx-3\right)\mathrm{Qx}+{\left({q}^{2}x-3\right)}^{2}\left({q}^{3}x-3\right)$
 ${L}{≔}\left({x}{-}{3}\right){}\left({q}{}{x}{-}{3}\right){}{\mathrm{Qx}}{+}{\left({{q}}^{{2}}{}{x}{-}{3}\right)}^{{2}}{}\left({{q}}^{{3}}{}{x}{-}{3}\right)$ (2)
 > $\mathrm{Desingularize}\left(L,\mathrm{Qx},x,q=\frac{1}{3},'\mathrm{coeff}'='\mathrm{leading}',\mathrm{factor}\right)$
 ${{\mathrm{Qx}}}^{{3}}{-}\frac{{3380}{}{{\mathrm{Qx}}}^{{2}}}{{729}}{+}\left(\frac{{2224976}{}{x}}{{4782969}}{+}\frac{{2293408}}{{531441}}\right){}{\mathrm{Qx}}{+}\frac{\left({x}{-}{81}\right){}\left({{x}}^{{2}}{+}{6703830}{}{x}{-}{230132907}\right)}{{10460353203}}$ (3)
 > $\mathrm{Desingularize}\left(L,\mathrm{Qx},x,q=\frac{1}{3},'\mathrm{coeff}'='\mathrm{trailing}',\mathrm{factor}\right)$
 $\frac{{832}{}{x}}{{177147}}{-}\frac{{832}}{{2187}}{+}\left(\frac{{4598}{}{x}}{{6561}}{-}\frac{{406}}{{243}}\right){}{{\mathrm{Qx}}}^{{2}}{-}{{\mathrm{Qx}}}^{{3}}{+}\left(\frac{{511}}{{1594323}}{}{{x}}^{{2}}{-}\frac{{21100}}{{177147}}{}{x}{+}\frac{{92767}}{{6561}}\right){}{\mathrm{Qx}}$ (4)

Note that in the latter case, not all singularities of the trailing coefficient could be removed; the factor ${q}^{3}x-3=\frac{\left(x-81\right)}{27}$ remains.

The following call returns an error since $q=-1$ is a second root of unity:

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

Compatibility

 • The QDifferenceEquations[Desingularize] command was introduced in Maple 18.
 • For more information on Maple 18 changes, see Updates in Maple 18.