Method for solving an iterative functional equation A²ⁿ(x)=F(x)

2013/02/08 by Dmitry Kruchinin, Kruchinin, Dmitry, Vladimir Kruchinin +1
Mathematics · #39B12 #Classical Analysis and ODEs (math.CA) #Combinatorics (math.CO) #FOS: Mathematics #Functional Analysis (math.FA) #Number Theory (math.NT) #math.CA #math.CO #math.FA #math.NT #msc:39B12

paper · pdf · doi:10.48550/arxiv.1302.1986

11 pages

arxiv created 2013/02/11 · arxiv updated 2013/02/12

Abstract

Using the notion of the composita, we obtain a method of solving iterative functional equations of the form A²ⁿ(x)=F(x), where F(x)=∑_n>0 f(n)xⁿ, f(1)≠ 0. We prove that if F(x)=∑_n>0 f(n)xⁿ has integer coefficients, then the generating function A(x)=∑_n>0a(n)xⁿ, which is obtained from the iterative functional equation 4A(A(x))=F(4x), has integer coefficients. Key words: iterative functional equation, composition of generating functions, composita.

Related

• Composita and its properties
• The method for obtaining expressions for coefficients of reverse generating functions
• Analytical and Numerical Approaches for Finding Functional Iterates and Roots
• Composition of ordinary generating functions
• Application of a composition of generating functions for obtaining explicit formulas of polynomials