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Linear homogeneous recurrence equations with polynomial coefficients are saidto be holonomic. Such equations have been introduced in the last century forproving and discovering combinatorial and hypergeometric identities. Given afield K of characteristic zero, a term a(n) is called hypergeometric withrespect to K, if the ratio a(n+1)/a(n) is a rational function over K. Thesolutions space of holonomic recurrence equations gained more interest in the1990s from the well known Zeilberger's algorithm. In particular, algorithmscomputing the subspace of hypergeometric term solutions which coverspolynomial, rational, and some algebraic solutions of these equations wereinvestigated by Marko Petkov\v{s}ek (1993) and Mark van Hoeij (1999). Thealgorithm proposed by the latter is characterized by a much better efficiencythan that of the other; it computes, in Gamma representations, a basis of thesubspace of hypergeometric term solutions of any given holonomic recurrenceequation, and is considered as the current state of the art in this area. Markvan Hoeij implemented his algorithm in the Computer Algebra System (CAS) Maplethrough the command $LREtools[hypergeomsols]$.  We propose a variant of van Hoeij's algorithm that performs the sameefficiency and gives outputs in terms of factorials and shifted factorials,without considering certain recommendations of the original version. We haveimplementations of our algorithm for the CASs Maxima and Maple. Such animplementation is new for Maxima which is therefore used for general-purposeexamples. Our Maxima code is currently available as a third-party package forMaxima. A comparison between van Hoeij's implementation and ours is presentedfor Maple 2020. It appears that both have the same efficiency, and moreover,for some particular cases, our code finds results where$LREtools[hypergeomsols]$ fails.

A variant of van Hoeij's algorithm to compute hypergeometric term solutions of holonomic recurrence equations