This AccurateQSummation(L,Q,x) calling sequence computes an operator M of minimal order such that any solution $f$ of L has an anti-qdifference which is a solution of M.

If the order of L equals the order of M then the output is a list [M, r] such that r(f) is an anti-qdifference of $f$ and also a solution of M for every solution $f$ of L. If the order of L is not equal to M then only M is given in the output. In this case M equals $L\mathrm{\Delta }$ where $\mathrm{\Delta }=Q-1$.

Q is the q-shift operator with respect to x, defined by $Q\left(x\right)=qx$.

$\mathrm{with}\left(\mathrm{QDifferenceEquations}\right)\:$

$L≔\left(-q\left(-1+q^2\right)\right)Q^2+q^2\left(q^4-1\right)Q+\left(-q^5\left(-1+q^2\right)\right)$

$L≔-q\left(q^2-1\right)Q^2+q^2\left(q^4-1\right)Q-q^5\left(q^2-1\right)$

$\mathrm{Ac}≔\mathrm{AccurateQSummation}\left(L\,Q\,x\right)$

$\mathrm{Ac}≔\left[\frac{q^4}{q^4-q^3-q+1}-\frac{\left(q^2+1\right)qQ}{q^4-q^3-q+1}+\frac{Q^2}{q^4-q^3-q+1}\,\frac{q^3+q-1}{q^4-q^3-q+1}-\frac{Q}{q^4-q^3-q+1}\right]$

$\mathrm{Lt}≔\mathrm{op}\left(1\,\mathrm{Ac}\right)\;$$\mathrm{rt}≔\mathrm{op}\left(2\,\mathrm{Ac}\right)$

$\mathrm{Lt}≔\frac{q^4}{q^4-q^3-q+1}-\frac{\left(q^2+1\right)qQ}{q^4-q^3-q+1}+\frac{Q^2}{q^4-q^3-q+1}$

$\mathrm{rt}≔\frac{q^3+q-1}{q^4-q^3-q+1}-\frac{Q}{q^4-q^3-q+1}$

$A≔\mathrm{OreTools}:-\mathrm{SetOreRing}\left(\left[x\,q\right]\,'\mathrm{qshift}'\right)\:$

$f≔q{x}^3+x$

$f≔q{x}^3+x$

$r≔\mathrm{OreTools}:-\mathrm{Converters}:-\mathrm{FromPolyToOrePoly}\left(\mathrm{rt}\,Q\right)\:$

$g≔\mathrm{normal}\left(\mathrm{OreTools}:-\mathrm{Apply}\left(r\,f\,A\right)\right)$

$g≔\frac{\left(q{x}^2+q^2+q+1\right)x}{q^3-1}$

$\mathrm{normal}\left(\mathrm{eval}\left(g\,x=qx\right)-g-f\right)$

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Abramov, S.A., and van Hoeij, M. "Integration of Solutions of Linear Functional Equations." Integral Transformations and Special Functions. Vol. 8 No. 1-2. (1999): 3-12.