We extend Petkov\v{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 $\lambda = c\frac{A}{B}$ 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.

翻译：我们将Petkovšek算法从计算标量差分方程的超几何解推广至差分系统$\tau(Y) = M Y$的情形，其中$M \in {\rm GL}_n(C(x))$，$\tau$为移位算子。超几何解是指形如$\gamma P$的解，其中$P \in C(x)^n$，$\gamma$是$C(x)$上的超几何项，即满足${\tau(\gamma)}/{\gamma} \in C(x)$。我们的贡献在于高效计算${\tau(\gamma)}/{\gamma}$的候选集，将其表示为$\lambda = c\frac{A}{B}$的形式，其中$A, B \in C[x]$为首一多项式，$c \in C^*$。$M^{-1}$与$M$分母的因子给出$A$和$B$的候选，而$c$的计算需要另一算法。我们采用超约化算法计算$c$的候选值，并结合其他技术缩减$A/B$的候选列表。为进一步减少$A/B$的候选数量，我们通过界定局部类型来限定$A/B$的所谓类型界限。该算法已在Maple中实现，实验表明我们的实现能够处理高维系统，这对算子分解具有实用价值。