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Q-Difference Equations


The QDifferenceEquations package provides tools for studying equations of the form:


Lyx=anxyqnx+an1xyqn1x++a1xyq x+a0xyx=0,


and their solutions yx, where a0, ... ,an are polynomials in the indeterminates x and q. The indeterminate q is considered to be a constant. L=anQn+an1Qn1+  +a1Q+a0 is the associated q-difference operator of order n, where Q represents the q-shift operator Qyx=yq x.


For example, the solutions of the first order q-difference equation Ly=0, where L=x21Qq2 x21, are given by:




where C is an arbitrary constant that is allowed to depend on q, but not on x.


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 L from above.






This operator has singularities at x=±1, where its leading coefficient vanishes. However, the solutions yx=Cx21 satisfying Ly=0 are non-singular at both points, so x=±1 are two apparent singularities. It is possible to remove such apparent singularities by finding a higher order operator M that has the same solutions as L, plus some additional ones. This is what the command Desingularize does.






Let us verify that yx is actually a solution of M.













The closure of an operator L consists of all left "pseudo"-multiples of L, i.e., all operators R for which there exists an operator, P (in Q,x,q) and a polynomial f (in x,q only), such that the following torsion relation holds true:

PL=f R

Basically, this means that PL is a genuine left multiple of L of which one can factor out the content f. Both PL and R have exactly the same solutions, which include all solutions of L. In particular, the desingularizing operator M from above is an element of the closure of L.


The command Closure computes a basis of the closure.





We see that, trivially, L itself belongs to its closure. In addition, the basis contains two second order operators, both of which have fewer and different singularities than L itself, namely, x=q1 and x=q1, respectively. Since these two singularities are different, the two leading coefficients are coprime as polynomials in x, and we can find a linear combination that is monic:

gcdexlcoeffC2,Q,lcoeffC3,Q,x, 's', 't',s,t;







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