Q-Difference Equations - Maple Help
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Q-Difference 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 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.

(1)

 

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.

 

(2)

 

Let us verify that  is actually a solution of .

(3)

(4)

(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.

(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:

(7)

(8)

 

This, in fact, is exactly the desingularizing operator from above.


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