Kavli Affiliate: Yi Zhou
| First 5 Authors: Moulay Barkatou, Mark van Hoeij, Johannes Middeke, Yi Zhou,
| Summary:
We extend Petkovv{s}ek’s algorithm for computing hypergeometric solutions of
scalar difference equations to the case of difference systems $tau(Y) = M Y$,
with $M in {rm GL}_n(C(x))$, where $tau$ is the shift operator.
Hypergeometric solutions are solutions of the form $gamma P$ where $P in
C(x)^n$ and $gamma$ is a hypergeometric term over $C(x)$, i.e.
${tau(gamma)}/{gamma} in C(x)$. Our contributions concern efficient
computation of a set of candidates for ${tau(gamma)}/{gamma}$ which we write
as $lambda = cfrac{A}{B}$ with monic $A, B in C[x]$, $c in C^*$. Factors of
the denominators of $M^{-1}$ and $M$ give candidates for $A$ and $B$, while
another algorithm is needed for $c$. We use the super-reduction algorithm to
compute candidates for $c$, as well as other ingredients to reduce the list of
candidates for $A/B$. To further reduce the number of candidates $A/B$, we
bound the so-called type of $A/B$ by bounding local types. Our algorithm has
been implemented in Maple and experiments show that our implementation can
handle systems of high dimension, which is useful for factoring operators.
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