  Firstly, we construct kernels of integral relations among solutions of the
confluent Heun equation (CHE) and its limit, the reduced CHE (RCHE). In both
cases we generate additional kernels by systematically applying substitutions
of variables. Secondly, we establish integral relations between known solutions
of the CHE that are power series and solutions that are series of special
functions; and similarly for solutions of the RCHE. Thirdly, by using one of
the integral relations as an integral transformation we obtain a new series
solution of the spheroidal wave equation. From this solution we construct new
solutions of the general CHE, and show that these are suitable for solving the
radial part of the two-center problem in quantum mechanics. Finally, by
applying a limiting process to kernels for the CHEs we obtain kernels for {two}
double-confluent Heun equations. As a result, we deal with kernels of four
equations of the Heun family, each equation presenting a distinct structure of
singularities. In addition, we find that the known kernels for the Mathieu
equation are special instances of kernels of the RCHE.
