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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 anQny+an1Qn1y+...+a1Qy+a0y=b , where an,an1,...,a1,a0,b  are polynomials in x  with coefficients from Kq  and Q  is the q-shift operator Qiyx=yqix  for all integers i . (This is a multiplicative analog of the ordinary shift operator Eiyx=yx+i .)


The goal is to find all solutions y that are polynomials or rational functions with coefficients from Kq. 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[0], 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 Kq .


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.


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