The QDifferenceEquations package provides tools for studying equations of the form:
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+an−1Qn−1+ ⋯ +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=x2−1Q−q2 x2−1, 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=C⋅x2−1 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:
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=−q−1 and x=q−1, 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.
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