
QDifference Equations


 
The QDifferenceEquations package provides tools for studying equations of the form:
and their solutions , where are polynomials in the indeterminates and . The indeterminate is considered to be a constant. is the associated qdifference operator of order , where represents the qshift operator .
For example, the solutions of the first order qdifference 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:
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Closure computes the closure in the ring of linear qdifference operators with polynomial coefficients.

•

Desingularize computes a multiple of a given qdifference operator with fewer singularities.

As an example, let's look at the operator from above.
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 (1) 
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This operator has singularities at , where its leading coefficient vanishes. However, the solutions satisfying are nonsingular 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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 (2) 
Let us verify that is actually a solution of .
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 (3) 
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 (4) 
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 (5) 
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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 (6) 
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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 (7) 
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 (8) 
This, in fact, is exactly the desingularizing operator from above.
