Linear homogeneous recurrence equations with polynomial coefficients are said to be holonomic. Such equations have been introduced in the last century for proving and discovering combinatorial and hypergeometric identities. Given a field K of characteristic zero, a term a(n) is called hypergeometric with respect to K, if the ratio a(n+1)/a(n) is a rational function over K. The solutions space of holonomic recurrence equations gained more interest in the 1990s from the well known Zeilberger's algorithm. In particular, algorithms computing the subspace of hypergeometric term solutions which covers polynomial, rational, and some algebraic solutions of these equations were investigated by Marko Petkov\v{s}ek (1993) and Mark van Hoeij (1999). The algorithm proposed by the latter is characterized by a much better efficiency than that of the other; it computes, in Gamma representations, a basis of the subspace of hypergeometric term solutions of any given holonomic recurrence equation, and is considered as the current state of the art in this area. Mark van Hoeij implemented his algorithm in the Computer Algebra System (CAS) Maple through the command $LREtools[hypergeomsols]$. We propose a variant of van Hoeij's algorithm that performs the same efficiency and gives outputs in terms of factorials and shifted factorials, without considering certain recommendations of the original version. We have implementations of our algorithm for the CASs Maxima and Maple. Such an implementation is new for Maxima which is therefore used for general-purpose examples. Our Maxima code is currently available as a third-party package for Maxima. A comparison between van Hoeij's implementation and ours is presented for Maple 2020. It appears that both have the same efficiency, and moreover, for some particular cases, our code finds results where $LREtools[hypergeomsols]$ fails.

中文翻译：

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van Hoeij算法的一种变体，用于计算完整递归方程的超几何项解

具有多项式系数的线性齐次递归方程被认为是完整的。在上个世纪引入了这样的方程式，以证明和发现组合和超几何恒等式。给定特征为零的场K，如果比率a（n + 1）/ a（n）是关于K的有理函数，则相对于K的项a（n）称为超几何。完整递归方程的解空间众所周知的Zeilberger算法在1990年代引起了更多兴趣。特别是，Marko Petkov \ v {s} ek（1993）和Mark van Hoeij（1999）研究了计算这些方程的多项式，有理数和一些代数解的超几何项解子空间的算法。后者提出的算法的特点是效率比其他算法好得多。它以Gamma表示形式计算任何给定完整递推方程的超几何项解子空间的基础，并被视为该领域的最新技术。Mark van Hoeij通过命令$ LREtools [hypergeomsols] $在计算机代数系统（CAS）Maple中实现了他的算法。我们提出了van Hoeij算法的一种变体，该算法的执行效率相同，并且在阶乘和移位阶乘方面提供输出，而无需考虑原始版本的某些建议。我们已经为CAS的Maxima和Maple实现了算法。这种实现对于Maxima是新的，因此可用于通用示例。我们的Maxima代码目前可以作为Maxima的第三方软件包使用。范·霍伊（van Hoeij）之间的比较