
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 $x\mapsto qx$

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 $q=\frac{1}{3}$:
>

$L\u2254\left(x3\right)\left(qx3\right)\mathrm{Qx}+{\left({q}^{2}x3\right)}^{2}\left({q}^{3}x3\right)$

${L}{\u2254}\left({x}{}{3}\right){}\left({q}{}{x}{}{3}\right){}{\mathrm{Qx}}{+}{\left({{q}}^{{2}}{}{x}{}{3}\right)}^{{2}}{}\left({{q}}^{{3}}{}{x}{}{3}\right)$
 (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) 
>

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

$\frac{{832}{}{x}}{{177147}}{}\frac{{832}}{{2187}}{}{{\mathrm{Qx}}}^{{3}}{+}\left(\frac{{4598}{}{x}}{{6561}}{}\frac{{406}}{{243}}\right){}{{\mathrm{Qx}}}^{{2}}{+}\left(\frac{{511}}{{1594323}}{}{{x}}^{{2}}{}\frac{{21100}}{{177147}}{}{x}{+}\frac{{92767}}{{6561}}\right){}{\mathrm{Qx}}$
 (4) 
Note that in the latter case, not all singularities of the trailing coefficient could be removed; the factor ${q}^{3}x3=\frac{\left(x81\right)}{27}$ remains.
The following call returns an error since $q=\mathrm{1}$ 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.



