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：我们扩展了Petkovšek算法，该算法原用于计算标量差分方程的超几何解，现将其推广至差分系统情况：$\tau(Y) = M Y$，其中$M \in {\rm GL}_n(C(x))$，$\tau$是移位算子。超几何解的形式为$\gamma P$，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候选集合${\tau(\gamma)}/{\gamma}$，将其表示为$\lambda = c\frac{A}{B}$，其中首一多项式$A, B \in C[x]$，$c \in C^*$。$M^{-1}$和$M$分母的因子提供$A$和$B$的候选，而$c$的确定需要另一算法。我们采用超降算法计算$c$的候选，并结合其他方法缩减$A/B$的候选列表。为进一步减少$A/B$的候选，我们通过界定局部类型来限制所谓的$A/B$类型。该算法已在Maple中实现，实验表明我们的实现能处理高维系统，这对算子因式分解具有重要价值。