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

Linear and homogeneous recurrence equations having polynomial coefficients are said to be holonomic. These equations are useful for proving and discovering combinatorial and hypergeometric identities. Given a field $\mathbb{K}$ of characteristic zero, $a_n$ is a hypergeometric term with respect to $\mathbb{K}$, if the ratio $a_{n+1}/a_n$ is a rational function over $\mathbb{K}$. Two algorithms by Marko Petkov\v{s}ek (1993) and Mark van Hoeij (1999) were proposed to compute hypergeometric term solutions of holonomic recurrence equations. The latter algorithm is more efficient and was implemented by its author in the Computer Algebra System (CAS) Maple through the command \texttt{LREtools[hypergeomsols]}.
We describe a variant of van Hoeij's algorithm that performs with the same efficiency without considering certain recommendations of the original version. We implemented our algorithm in the CASes Maxima and Maple. It also appears for some particular cases that our code finds results where \texttt{LREtools[hypergeomsols]} fails.