Synthesizing Sums

Abstract

Polynomial functions $f:{\mathbb{N}}_{+}⟶{\mathbb{N}}_{+}$ are studied for which sums of arbitrary length $f(1)+f(2)+f(3)+>...+f(n)$, with $n\in {\mathbb{N}}_{+}$, can be expressed by polynomial functions $g:{\mathbb{N}}_{+}⟶{\mathbb{N}}_{+}$ which involve a bounded number of operations, thus not depending on $n$. Open problems and extensions are presented.