Regularization of the total equation with holomorphic coefficients generated by a triangle

Let $D$ be a semicircle. This is the fundamental area of a finite properly discontinuous group of linear fractional transformations. We consider a linear four-element functional equation with coefficients holomorphic in $ D $ generated by this group. The solution is sought in the class of functions that are holomorphic outside the "half" $\partial D $ For vanishing at infinity. To regularize the equation, we introduce an involutive Carleman shift induced by generating transformations of the group. This shift has two fixed points. The condition of equivalence of regularization is satisfied. Applications to the problem of moments for entire functions of exponential type are indicated.