Integral relations for solutions of the confluent Heun equation

Abstract

Firstly, we construct kernels for integral relations among solutions of the confluent Heun equation (CHE). Additional kernels are systematically generated by 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. Thirdly, by using one of the integral relations as an integral transformation we obtain a new series solution of the ordinary spheroidal wave equation (a particular CHE). From this solution we construct new series solutions of the general CHE, and show that these are suitable for solving the radial part of the two-center problem in quantum mechanics.