论文标题
Van Hoeij算法的变体，以计算自动复发方程的超小几阶解
A variant of van Hoeij's algorithm to compute hypergeometric term solutions of holonomic recurrence equations
论文作者
论文摘要
据说具有多项式系数的线性均匀复发方程是尸体的。在上个世纪引入了此类方程式，以证明和发现组合和高几何身份。给定特征为零的场k，如果比率a（n+1）/a（n）比k的术语a（n）称为高几何，则与k相比。自变量复发方程的解决方案空间在1990年代从众所周知的Zeilberger的algorithm中获得了更大的兴趣。特别是，MarkoPetkovšek（1993）和Mark Van Hoeij（1999）研究了计算涵盖这些方程式多项式，理性和某些代数解的高几何术语解决方案子空间的算法。后者提出的算法的特征是效率要比其他算法更好得多。它在伽马表示中计算出任何给定的自动复发方程的超几何术语解决方案的子空间的基础，并被视为该领域的最新水平。马克·范·霍伊（Mark van Hoeij）通过命令$ lretools [hypergeomsols] $在计算机代数系统（CAS）枫木中实现了他的算法。 我们提出了van Hoeij算法的一种变体，该变体执行相同的效率，并在不考虑原始版本的某些建议的情况下，就阶乘和转移的阶乘提供了输出。我们为Cass Maxima和Maple的算法实现了。这种实现是最大化的新事物，因此用于通用示例。目前，我们的Maxima代码可作为Maxima的第三方软件包使用。 Van Hoeij的实施与我们的实施情况进行了比较，在2020年Maple 2020中都具有相同的效率，而且对于某些特定情况，我们的代码在$ lretools [hypergeomsols] $失败的情况下找到了结果。
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 Petkovšek (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.
