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The QDifferenceEquations Package

The  QDifferenceEquations package provides algorithms for solving linear q-difference (q-recurrence) equations or systems in terms of polynomials or rational functions.

Let K be a field and  q an indeterminate over K. A linear q-difference equation with polynomial coefficients has the form ${a}_{n}{Q}^{n}y+{a}_{n-1}{Q}^{n-1}y+\mathrm{...}+{a}_{1}Qy+{a}_{0}y=b$ , where ${a}_{n},{a}_{n-1},\mathrm{...},{a}_{1},{a}_{0},b$  are polynomials in $x$  with coefficients from $K\left(q\right)$  and $Q$  is the q-shift operator $\left({Q}^{i}y\right)\left(x\right)=y\left({q}^{i}x\right)$  for all integers $i$ . (This is a multiplicative analog of the ordinary shift operator $\left({E}^{i}y\right)\left(x\right)=y\left(x+i\right)$ .)

The goal is to find all solutions $y$ that are polynomials or rational functions with coefficients from $K\left(q\right)$. More generally, a system of such equations has the same form as above, but now $y$  is a vector of $m$ unknown functions, a[n], ..., a, which are $m$ by $k$ matrices with polynomial entries, and $b$ is a vector with $k$ polynomial entries. As in the case of (systems of) ordinary difference equations, the polynomial (or rational) solutions form a finite-dimensional vector space over $K\left(q\right)$ .

Note: The Maple LREtools package provides methods for solving ordinary difference equations or systems. For information on solving systems of ordinary or q-difference equations, see the LinearFunctionalSystems package.