  Integral relations and transformation rules are used to obtain, out of an
asymptotic solution, a new group of four pairs of solutions to the
double-confluent Heun equation. Each pair presents the same series coefficients
but has solutions convergent in different regions of the complex plane.
Integral relations are also established between solutions given by series of
Coulomb wave functions. The Whittaker-Hill equation and another equation are
studied as particular cases of both the double-confluent and the
single-confluent Heun equations. Finally, applications for the Schr\"{o}dinger
equation with certain potentials are discussed, mainly for quasi-exactly
solvable potentials which lead to the above special equations.
