The prompter method: A treatment for hard-to-solve iterative functional equations. (July 2016) - Izumi, Hideaki
Abstract: The author developed a new method for obtaining formal series solutions to polynomial-like iterative functional equations of the form ∑ n = 1 N a n f n ( x ) = g ( x ), where a n ∈ R, n = 1, 2, …, N, f n is the n -th iterate of an unknown function f and where 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 f 1 ∘ f 2 of two promptered exponential series f 1 and f 2, where the coefficient of the prompter of f 2 is positive, plays a crucial rôle. We also solve the equation above where g ( x ) is a promptered trigonometric series.