On accessory coefficients of second order Fuchsian equation with real singular points

In the paper the author considers equations of Fuchs and Schwarz connected with a conformal mapping of the half-plane onto curvilinear polygon. The equations are studied from point of view of spectral theory of automorphic furc tions. A monodromy group is supposed to be a special Fuchsian group. The main question investigated in the paper (following Poincare) is the problem of finding the accessory coefficients as functions of singular points of Fuchsian equation. The principal results of the paper are following ones. In addition to another paper of the author “On explicit formulas for accessory coefficients in Schwarz equation”, Functional, analysis and its appl., 1983, N 2, in this paper the Fourier expansions of Siegel–Selberg series with relation to parabolic subgroups of monodromy group are studied (theorem 1). From here the author receives the formulas for all Fourier coefficients of Klein invariant (theorems 3, 4) and the formulas for singular points of Fuchs equation in geemetric terms of generating polygon (theorem 5). In conclution, the example of quadrangle is given for which the problem of funding the accessory parameters is simplified.