Confluent Heun equations: convergence of solutions in series of coulomb wavefunctions

The Leaver solutions in series of Coulomb wavefunctions for the confluent Heun equation are given by two-sided infinite series, that is, by series where the summation index n runs from minus to plus infinity (Leaver 1986 J. Math. Phys. 27 1238). First we show that, in contrast to the D'Alembert test, under certain conditions the Raabe test ensures that the domains of convergence of these solutions include an additional singular point. We also consider solutions for a limit of the confluent Heun equation. For both equations, new solutions are generated by transformations of variables. Finally, we discuss the time dependence of the Klein-Gordon equation in two cosmological models and the spatial dependence of the Schrodinger 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.