### On a Special Nonlinear Functional Equation

K. Mahler
1981 Proceedings of the Royal Society A
We study analytic solutions of the functional equation where c > 0 is a parameter, and constant solutions are excluded. I t suffices to consider solutions for which h(z) -z~l is regular in a neighbourhood of 2 = 0. If 0 < c ^ t, h(z) can be continued as a single-valued analytic function into the unit disc |s| < 1 where its only singularity is the pole a t 2 = 0; the circle \z\ = 1 is a natural boundary. On the other hand, if c > \, then by analytic continuation h(z) becomes a multiple-valued
more » ... 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 where Ci s a real parameter. The solutions of this equation have properties similar to those of (H). I n t r o d u c t i o n While there are general methods for dealing with linear differential, difference, and functional equations, nonlinear problems of these kinds require special methods, and the solutions often depend rather discontinuously on the occurring parameters. This strange behaviour makes them of particular interest to the pure m athem atician. In the present paper I discuss a special nonlinear functional equation with just one parameter, and I establish the manner in which its main solution depends on this parameter. 1. The functional equation to be considered is a special case of a more general equation studied by myself a few years ago. Denote by pa prime and by H (X ,Y ) = -( X * -Y ) ( Y » -X ) + £ i , E mnX « Y » (E"m = E nm) m = 0 7i = 0 [ 155 1