  The Leaver solutions in series of Coulomb wave functions for the confluent
Heun equation (CHE) are given by two-sided infinite series, that is, by series
where the summation index $n$ runs from minus to plus infinity [E. W. Leaver,
J. Math. Phys. 27, 1238 (1986)]. First we show that, in contrast to the
D'Alembert test, under certain conditions the Raabe test assures that the
domains of convergence of these solutions include an additional singular point.
Further, by using a limit proposed by Leaver, we obtain solutions for the
double-confluent Heun equation (DCHE). In addition, we get solutions for the
so-called Whittaker-Ince limit of the CHE and DCHE. For these four equations,
new solutions are generated by transformations of variables. In the second
place, for each of the above equations we obtain one-sided series solutions
($n\geq 0$) by truncating on the left the two-sided series. Finally we discuss
the time dependence of the Klein-Gordon equation in two cosmological models and
the spatial dependence of the Schr\"{o}dinger equation to a family of
quasi-exactly solvable potentials. For a subfamily of these potentials we
obtain infinite-series solutions which converge and are bounded for all values
of the independent variable, in opposition to a common belief}.
