Square root singularities of infinite systems of functional equations

Infinite systems of equations appear naturally in combinatorial counting problems. Formally, we consider functional equations of the form y(x) = F (x,y(x)), where F (x,y) : C × l → l is a positive and nonlinear function, and analyze the behavior of the solution y(x) at the boundary of the domain of convergence. In contrast to the finite dimensional case different types of singularities are possible. We show that if the Jacobian operator of the function F is compact, then the occurring singularities are of square root type, as it is in the finite dimensional setting. This leads to asymptotic expansions of the Taylor coefficients of y(x).