Section method and Frechet polynomials

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.