The QDifferenceEquations package provides tools for studying equations of the form:
and their solutions 
, where ![a[0], () .. (), a[n]](/products/maple/new_features/images18/QDifferenceEquations_4.gif)
are polynomials in the indeterminates 
and 
. The indeterminate 
is considered to be a constant. 
is the associated q-difference operator of order 
, where 
represents the q-shift operator 
.
For example, the solutions of the first order q-difference equation 
, where 
, are given by:
,
where 
is an arbitrary constant that is allowed to depend on 
but not on 
.
In Maple 18, two new commands were added to this package:
- Closure computes the closure in the ring of linear q-difference operators with polynomial coefficients.
- Desingularize computes a multiple of a given q-difference operator with fewer singularities.
As an example, let's look at the operator 
from above.
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This operator has singularities at 
, where its leading coefficient vanishes. However, the solutions 
satisfying 
are non-singular at both points, so 
are two apparent singularities. It is possible to remove such apparent singularities by finding a higher order operator 
that has the same solutions as 
, plus some additional ones. This is what the command Desingularize does.
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Let us verify that 
is actually a solution of 
.
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![A := SetOreRing([x, q], qshift); -1](/products/maple/new_features/images18/QDifferenceEquations_36.gif) |
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The closure of an operator 
consists of all left "pseudo"-multiples of 
, i.e., all operators 
for which there exists an operator, 
(in 
) and a polynomial 
(in 
only), such that the following torsion relation holds true:
Basically, this means that 
is a genuine left multiple of 
of which one can factor out the content 
. Both 
and 
have exactly the same solutions, which include all solutions of 
. In particular, the desingularizing operator 
from above is an element of the closure of 
.
The command Closure computes a basis of the closure.
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We see that, trivially, 
itself belongs to its closure. In addition, the basis contains two second order operators, both of which have fewer and different singularities than 
itself, namely, 
and 
, respectively. Since these two singularities are different, the two leading coefficients are coprime as polynomials in 
, and we can find a linear combination that is monic:
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![gcdex(lcoeff(C[2], Q), lcoeff(C[3], Q), x, 's', 't'), s, t; 1](/products/maple/new_features/images18/QDifferenceEquations_66.gif) |
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![collect(`+`(`*`(s, `*`(C[2])), `*`(t, `*`(C[3]))), Q, normal)](/products/maple/new_features/images18/QDifferenceEquations_68.gif) |
This, in fact, is exactly the desingularizing operator from above.
