sum the solutions of a q-shift operator
AccurateQSummation(L, Q, x)
polynomial in Q over C(q)(x)
name; denote the q-shift operator
name (that Q acts on)
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⁢Δ where Δ=Q−1.
Q is the q-shift operator with respect to x, defined by Q⁡x=q⁢x.
L ≔ −q⁢−1+q2⁢Q2+q2⁢q4−1⁢Q−q5⁢−1+q2
L ≔ −q⁢q2−1⁢Q2+q2⁢q4−1⁢Q−q5⁢q2−1
Ac ≔ AccurateQSummation⁡L,Q,x
Ac ≔ q4q4−q3−q+1−q2+1⁢q⁢Qq4−q3−q+1+Q2q4−q3−q+1,q3+q−1q4−q3−q+1−Qq4−q3−q+1
Lt ≔ op⁡1,Ac;rt ≔ op⁡2,Ac
Lt ≔ q4q4−q3−q+1−q2+1⁢q⁢Qq4−q3−q+1+Q2q4−q3−q+1
rt ≔ q3+q−1q4−q3−q+1−Qq4−q3−q+1
Regarding the meaning of the second element rt in the output of AccurateQSummation, since L is the minimal annihilator of f=q⁢x3+x, g=rt⁡f is an anti-qdifference of f:
A ≔ OreTools:-SetOreRing⁡x,q,'qshift':
f ≔ q⁢x3+x
r ≔ OreTools:-Converters:-FromPolyToOrePoly⁡rt,Q:
g ≔ normal⁡OreTools:-Apply⁡r,f,A
g ≔ q⁢x2+q2+q+1⁢xq3−1
check that Q−1⁢g=f:
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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