On some addition formulas for homographic type functions

We deal with the functional equation (so called addition formula) of the form f(x + y) = F(f(x),f(y)), where F is an associative rational function. The class of associative rational functions was described by A. Chéritat [1] and his work was followed by a paper of the author. For function F defined by F(x,y) = ϕ−1(ϕ(x) + ϕ(y)), where ϕ is a homographic function, the addition formula is fulfilled by homographic type functions. We consider the class of the associative rational functions defined by formula F(u,v) =uv αuv + u + v, where α is a fixed real numer.