  This article deals with two classes of quasi-exactly solvable (QES)
trigonometric potentials for which the one-dimensional Schroedinger equation
reduces to a confluent Heun equation (CHE) where the independent variable takes
only finite values. Power series for the CHE are used to get finite- and
infinite-series eigenfunctions. Finite series occur only for special sets of
parameters and characterize the quasi-exact solvability. Infinite series occur
for all admissible values of the parameters (even values involving finite
series), and are bounded and convergent in the entire range of the independent
variable. Moreover, throughout the article we examine other QES trigonometric
and hyperbolic potentials. In all cases, for a finite series there is a
convergent infinite series.
