On a special nonlinear functional equation

We study analytic solutions of the functional equation h(z) 2 – h(z 2 ) + c = 0, (H) where c > 0 is a parameter, and constant solutions are excluded. It suffices to consider solutions for which h(z) – z -1 is regular in a neighbourhood of z = 0. If 0 < c ≼ ¼, h(z) can be continued as a single-valued analytic function into the unit disc | z | < 1 where its only singularity is the pole at z = 0; the circle | z | = 1 is a natural boundary. On the other hand, if c > ¼, then by analytic continuation h(z) becomes a multiple-valued function with an infinite sequence of quadratic branch points tending to every point of | z | = 1, and no branch of h(z) can be continued beyond this circle. A change of variable transforms (H) into the difference equation g ( Z + 1) – g(Z) = g(Z) 2 + C , where C is a real parameter. The solutions of this equation have properties similar to thóse of (H).