On solutions of $f(x)+f(a_1x)+...+f(a_Nx)=0$ and related equations

TL;DR

This paper investigates the properties of solutions to a specific functional equation involving sums of a function evaluated at scaled inputs, extending previous research by analyzing particular parameter cases.

Contribution

It provides new insights into the solutions of the functional equation for the case where parameters follow a specific sequence, building on prior work from Mora and others.

Findings

Characterization of solutions for the case a_k=k+1

Extension of known solution properties to new parameter regimes

Connections to earlier studies by Mora and colleagues

Abstract

We study some properties of the solutions of the functional equation $$f(x)+f(a_1x)+...+f(a_Nx)=0,$$ which was introduced in the literature by Mora, Cherruault and Ziadi in 1999, for the case $a_k=k+1$, $k=1,2,...,N$ and studied by Mora in 2008 and Mora and Sepulcre in 2009 and 2011.