  We find sets of solutions to the generalized spheroidal wave equation (GSWE)
or, equivalently, to the confluent Heun equation. Each set is constituted by
three solutions, one given by a series of ascending powers of the independent
variable, and the others by series of regular and irregular confluent
hypergeometric functions. For a fixed set, the solutions converge over
different regions of the complex plane but present series coefficients
proportional to each other. These solutions for the GSWE afford solutions to a
double-confluent Heun equation by a taking-limit process due to Leaver. Another
procedure, called Whittaker-Ince limit, provides solutions in series of powers
and Bessel functions for two other equations with a different type of
singularity at infinity. In addition, new solutions are obtained for the
Whittaker-Hill and Mathieu equations by considering these as special cases of
both the confluent and double-confluent Heun equations. In particular, we find
that each of the Lindmann-Stieltjes solutions for the Mathieu equation is
associated with two expansions in series of Bessel functions. We also discuss a
set of solutions in series of hypergeometric and confluent hypergeometric
functions for the GSWE and use their Leaver limits to obtain infinite-series
solutions for the Schr\"odinger equation with an asymmetric double-Morse
potential. Finally, the possibility of extending the solutions of the GSWE to
the general Heun equation is briefly discussed.
