  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.
