On the binomial equation on topological groups

We consider a class of functional equations in one variable that, in some settings, describe polynomials on groups. Namely, a binomial equation of order n, for functions from a group G to an Abelian group K, is the equation of the form $$ \sum _{k=0}^n(-1)^{n-k}\left( {\begin{array}{c}n\\ k\end{array}}\right) f(x^k) = 0 \qquad \text {for any}\ x \in G. $$ We call solutions of this equation binomial functions of order n. Our aim is to consider diverse connections between binomial functions and (semi)polynomials on G. We prove that all sufficiently smooth $$\mathbb {R}$$ -valued binomial functions on $$\mathbb {R}^d$$ are polynomials. Furthermore, we show that continuous binomial functions on groups with dense union of compact subgroups (e.g. on the groups of all triangular matrices whose diagonal elements are of module 1) are constant. On the other hand, we construct examples showing that, in distinction to the case of semipolynomials, there are non-constant binomial functions on some groups topologically generated by compact subgroups, e.g. on the groups $$SL(n,\mathbb {R})$$ of unimodular matrices.