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QDifferenceEquations

 AccurateQSummation
 sum the solutions of a q-shift operator

 Calling Sequence AccurateQSummation(L, Q, x)

Parameters

 L - polynomial in Q over C(q)(x) Q - name; denote the q-shift operator x - name (that Q acts on)

Description

 • 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 $Q\left(x\right)=qx$.

Examples

 > $\mathrm{with}\left(\mathrm{QDifferenceEquations}\right):$
 > $L≔-q\left(-1+{q}^{2}\right){Q}^{2}+{q}^{2}\left({q}^{4}-1\right)Q-{q}^{5}\left(-1+{q}^{2}\right)$
 ${L}{≔}{-}{q}{}\left({{q}}^{{2}}{-}{1}\right){}{{Q}}^{{2}}{+}{{q}}^{{2}}{}\left({{q}}^{{4}}{-}{1}\right){}{Q}{-}{{q}}^{{5}}{}\left({{q}}^{{2}}{-}{1}\right)$ (1)
 > $\mathrm{Ac}≔\mathrm{AccurateQSummation}\left(L,Q,x\right)$
 ${\mathrm{Ac}}{≔}\left[\frac{{{q}}^{{4}}}{{{q}}^{{4}}{-}{{q}}^{{3}}{-}{q}{+}{1}}{-}\frac{\left({{q}}^{{2}}{+}{1}\right){}{q}{}{Q}}{{{q}}^{{4}}{-}{{q}}^{{3}}{-}{q}{+}{1}}{+}\frac{{{Q}}^{{2}}}{{{q}}^{{4}}{-}{{q}}^{{3}}{-}{q}{+}{1}}{,}\frac{{{q}}^{{3}}{+}{q}{-}{1}}{{{q}}^{{4}}{-}{{q}}^{{3}}{-}{q}{+}{1}}{-}\frac{{Q}}{{{q}}^{{4}}{-}{{q}}^{{3}}{-}{q}{+}{1}}\right]$ (2)
 > $\mathrm{Lt}≔\mathrm{op}\left(1,\mathrm{Ac}\right);$$\mathrm{rt}≔\mathrm{op}\left(2,\mathrm{Ac}\right)$
 ${\mathrm{Lt}}{≔}\frac{{{q}}^{{4}}}{{{q}}^{{4}}{-}{{q}}^{{3}}{-}{q}{+}{1}}{-}\frac{\left({{q}}^{{2}}{+}{1}\right){}{q}{}{Q}}{{{q}}^{{4}}{-}{{q}}^{{3}}{-}{q}{+}{1}}{+}\frac{{{Q}}^{{2}}}{{{q}}^{{4}}{-}{{q}}^{{3}}{-}{q}{+}{1}}$
 ${\mathrm{rt}}{≔}\frac{{{q}}^{{3}}{+}{q}{-}{1}}{{{q}}^{{4}}{-}{{q}}^{{3}}{-}{q}{+}{1}}{-}\frac{{Q}}{{{q}}^{{4}}{-}{{q}}^{{3}}{-}{q}{+}{1}}$ (3)

Regarding the meaning of the second element rt in the output of AccurateQSummation, since $L$ is the minimal annihilator of $f=q{x}^{3}+x$, $g=\mathrm{rt}\left(f\right)$ is an anti-qdifference of $f$:

 > $A≔\mathrm{OreTools}:-\mathrm{SetOreRing}\left(\left[x,q\right],'\mathrm{qshift}'\right):$
 > $f≔q{x}^{3}+x$
 ${f}{≔}{q}{}{{x}}^{{3}}{+}{x}$ (4)
 > $r≔\mathrm{OreTools}:-\mathrm{Converters}:-\mathrm{FromPolyToOrePoly}\left(\mathrm{rt},Q\right):$
 > $g≔\mathrm{normal}\left(\mathrm{OreTools}:-\mathrm{Apply}\left(r,f,A\right)\right)$
 ${g}{≔}\frac{\left({q}{}{{x}}^{{2}}{+}{{q}}^{{2}}{+}{q}{+}{1}\right){}{x}}{{{q}}^{{3}}{-}{1}}$ (5)

check that $\left(Q-1\right)g=f$:

 > $\mathrm{normal}\left(\genfrac{}{}{0}{}{g}{\phantom{x=qx}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}|\phantom{\rule[-0.0ex]{0.1em}{0.0ex}}\genfrac{}{}{0}{}{\phantom{g}}{x=qx}-g-f\right)$
 ${0}$ (6)

References

 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.

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