Section method and Frechet polynomials

TL;DR

This paper characterizes solutions to specific functional equations involving roots, sums, and inverse trigonometric functions using the section method, introducing connections with Frechet polynomials.

Contribution

It introduces a novel application of the section method to solve complex functional equations involving roots and trigonometric functions, linking solutions to Frechet polynomials.

Findings

Solutions characterized explicitly for given equations

Connections established between solutions and Frechet polynomials

Method applicable to a class of functional equations involving roots and trigonometry

Abstract

Using the section method we characterize the solutions $ f:U\rightarrow Y$ of the following four equations \begin{equation*} \sum\limits_{i=0}^{n}\left( -1\right) ^{n-i}\tbinom{n}{i}f\left( \sqrt[m]{ u^{m}+iv^{m}}\right) =\left( n!\right) f\left( v\right) \text{, } \end{equation*} \begin{equation*} f\left( u\right) +\sum\limits_{i=1}^{n+1}\left( -1\right) ^{i} \tbinom{n+1}{i}f\left( \sqrt[m]{u^{m}+iv^{m}}\right) =0, \end{equation*} \begin{equation*} \sum\limits_{i=0}^{n}\left( -1\right) ^{n-i}\tbinom{n}{i}f\left( \arcsin \left\vert \sin u\sin ^{i}v\right\vert \right) =\left( n!\right) f\left( v\right) \text{ and } \end{equation*} \begin{equation*} f\left( u\right) +\sum\limits_{i=1}^{n+1}\left( -1\right) ^{i}\tbinom{n+1}{i% }f\left( \arcsin \left\vert \sin u\sin ^{i}v\right\vert \right) =0, \end{equation*} where $m\geq 2$ and $n$ are positive integers,$ \ U\subseteq \mathbb{R} $ is a maximally relevant real domain and $\left( Y,+\right) $ is an $\left( n!\right) $ -divisible Abelian group.