The prompter method: A treatment for hard-to-solve iterative functional equations

The author developed a new method for obtaining formal series solutions to polynomial-like iterative functional equations of the form ∑n=1Nanfn(x)=g(x), where an∈R,n=1,2,…,N,fn is the n-th iterate of an unknown function f   and where g(x)g(x) is a promptered exponential series, namely, the sum of a Dirichlet series and a linear term called prompter. In this method, a formal composition f1∘f2f1∘f2 of two promptered exponential series f1f1 and f2f2, where the coefficient of the prompter of f2f2 is positive, plays a crucial rôle. We also solve the equation above where g(x)g(x) is a promptered trigonometric series.