Convergence of (generalized) power series solutions of functional equations

TL;DR

This paper investigates the convergence of formal power series solutions to nonlinear functional equations, including differential, q-difference, and Mahler equations, focusing on their summability and existence of actual solutions.

Contribution

It provides new insights into the convergence criteria of formal power series solutions for various classes of nonlinear functional equations.

Findings

Established conditions for convergence of formal solutions

Analyzed summability of series with complex exponents

Extended results to differential, q-difference, and Mahler equations

Abstract

Solutions of nonlinear functional equations are generally not expressed as a finite number of combinations and compositions of elementary and known special functions. One of the approaches to study them is, firstly, to find formal solutions (that is, series whose terms are described and ordered in some way but which do not converge apriori) and, secondly, to study the convergence or summability of these formal solutions (the existence and uniqueness of actual solutions with the given asymptotic expansion in a certain domain). In this paper we deal only with the convergence of formal functional series having the form of an infinite sum of power functions with (complex, in general) power exponents and satisfying analytical functional equations of the following three types: a differential, $q$-difference or Mahler equation.