We consider linear homogeneous difference equations with rational-function
coefficients.
The search  for solutions in the form of the $m$-interlacing ($1\leq m\leq \ord L$,
where $L$ is a given operator) of finite sums of hypergeometric sequences plays an important role in
the Hendriks-Singer algorithm for constructing all Liouvillian solutions of $L(y)=0$.
We show that the proposed by Hendriks and Singer procedure for finding solutions
in the form of such $m$-interlacing can be simplified.
In addition we show that if an equation has a non-zero solution in the form of such $m$-interlacing  then this equation also has
a non-zero solution in the form of sequence $w_n=f (n/m)$, where $f (x)$ is a finite sum of
analytic functions and each of these functions is a solution of a first-order  linear homogeneous difference equation
with rational-function coefficients.
All such solutions can be found algorithmically.
We also describe adjustments of our implementation of the Hendriks-Singer algorithm
to utilize the presented results.

The results  presented in this talk were obtained by the author
jointly with M.~Barkatou and D.~Khmelnov.