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

$\left(\mathrm{Ly}\right)\left(x\right)\=a_n\left(x\right)y\left(q^{n}x\right)\+a_{n-1}\left(x\right)y\left(q^{n-1}x\right)\+\cdots \+a_1\left(x\right)y\left(q x\right)\+a_0\left(x\right)y\left(x\right)\=0\,$

and their solutions $y\left(x\right)$, where $a_0\, ..\. \,a_n$ are polynomials in the indeterminates $x$ and $q$. The indeterminate $q$ is considered to be a constant. $L\=a_n{Q}^n\+a_{n-1}{Q}^{n-1}\+ \cdots \+a_{1}Q\+a_0$ is the associated q-difference operator of order $n$, where $Q$ represents the q-shift operator $\left(\mathrm{Qy}\right)\left(x\right)\=y\left(q x\right)$.

For example, the solutions of the first order q-difference equation $\mathrm{Ly}\=0$, where $L\=\left(x^2-1\right)Q-\left(q^2 x^2-1\right)$, are given by:

$y\left(x\right)\=C\cdot \left(x^2-1\right)$,$$

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:

As an example, let's look at the operator $L$ from above.

This operator has singularities at $x\=\pm 1$, where its leading coefficient vanishes. However, the solutions $y\left(x\right)\=C\cdot \left(x^2-1\right)$ satisfying $\mathrm{Ly}\=0$ are non-singular at both points, so $x\=\pm 1$ 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 $y\left(x\right)$ 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:

$\mathrm{PL}\=f R$

Basically, this means that $\mathrm{PL}$ is a genuine left multiple of $L$ of which one can factor out the content $f$. Both $\mathrm{PL}$ 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:

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