A new approach to the connection problem for local solutions to the general Heun equation

Abstract

We present new solution of the the connection problem for local solutions to the general Heun equation. Our approach is based on the symmetric form of the Heun's differential equation \cite{Fiziev14,Fiziev16} with four different regular singular points $z_{1,2,3,4}$. The four special regular points in the complex plane: $Z_{123},Z_{234},Z_{341},Z_{412}$ are the centers of the circles, defined by the different triplets $\{z_k,z_l,z_m\}$ with corresponding different indexes and play fundamental role, since the coefficients of the connection matrix can be expressed using the values of local solutions of the general Heun's equation at these points. A special case when all coefficients can be calculated using only one of the points $Z_{klm}$ is also considered.