The prompter method: A treatment for hard-to-solve iterative functional equations. (July 2016)

• Record Type: Journal Article
• Authors: Izumi, Hideaki
• Is Part Of: Journal of symbolic computation. Volume 75(2016)
• Page Start: 193
• Page End: 208
• Publication Date: 2016-07
• Subjects: Iterative functional equation -- Formal solution -- Iterative root; Mathematics -- Data processing -- Periodicals; Numerical analysis -- Data processing -- Periodicals; Automatic programming (Computer science) -- Periodicals; Mathématiques -- Informatique -- Périodiques; Analyse numérique -- Informatique -- Périodiques; Programmation automatique -- Périodiques; Automatic programming (Computer science); Mathematics -- Data processing; Numerical analysis -- Data processing; Periodicals; Electronic journals; 510.285
• DOI: 10.1016/j.jsc.2015.11.014
• Languages: English
• ISSNs: 0747-7171

Abstract

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.