construct a desingularizing q-shift operator with polynomial coefficients
Desingularize(L, Qx, x, q, func, options)
polynomial in Qx with coefficients which are polynomials in x over the field of rational functions in q
name, variable denoting the q-shift operator x↦q⁢x
either a variable name, or a nonzero constant that is not a root of unity, or an equation of the form name=constant
(optional) equation of the form 'coeff'=t, where t is one of leading, trailing, or both
polynomial in 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 Qx, with coefficients which are polynomials in x over k⁡q. For a given operator L∈F, the Desingularize(L,Qx,x,q) calling sequence constructs an operator R∈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⁡q,x.
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 qn≠1 for all positive integers n.
The optional argument func, if specified, is applied to the coefficients of the result with respect to 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.
For the following q-shift operator L, compute desingularizing operators with respect to the leading coefficient and the trailing coefficient when q=13:
L ≔ x−3⁢q⁢x−3⁢Qx+q2⁢x−32⁢q3⁢x−3
Note that in the latter case, not all singularities of the trailing coefficient could be removed; the factor q3⁢x−3=x−8127 remains.
The following call returns an error since q=−1 is a second root of unity:
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.
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