The central two-point connection problem of Heun's class of differential equations

Boundary-value problems of ordinary, linear, homogeneous second-order differential equations belong to the most important and thus well-investigated problems in mathematical physics. This statement is true only as long asirregular singularities of the differential equation at hand are not involved. If singular points of irregular type enter the problem one will hardly find a systematic investigation of such a topic from a practical point of view. This paper is devoted to an outline of an approach to boundary-value problems of the class of Heun's differential equation when irregular singularities may be located at the endpoints of the relevant interval. We present an approach to the central two-point connection problem for all of these equations in a quite uniform manner. The essential point is an investigation of the Birkhoff sets of irregular difference equations, which, on the one hand, gives a detailed insight into the structure of the singularities of the underlying differential equation and, on the other hand, yields the basis of quite convenient algorithms for numerical investigations of the boundary values.